RESEARCH © Yellow Dog Productions/Iconica/Getty Images
SNAPSHOT
Interested in Retirement? It Often
Depends on Your Age
Chi-square tests are used often in business research. Consider The x2 value can be computed as shown below:
a business that sponsors a program to educate employees on
retirement issues. They need to plan the number and types of Expected Observed O 2 E (O 2 E)2 (O 2 E)2/E
activities that should be the focus of the training and devel-
opment seminars. One question is whether or not an equal Younger 100 78 222 484 48.4
number of younger versus older employees will come to the Older 100 122 22 484 48.4
sessions. They decide to observe the relative frequencies of 96.8
younger versus older employees based upon the number of
sign-ups they receive in the first week since the program was The HR managers want to be sure a difference exists before
announced, with a cutoff set at 200. The results are shown in investing resources into activities designed for younger or older
the bar chart below: employees only. Therefore, the acceptable level of Type I error
is set at 0.01. Rather than referring to a critical value table, the
120 Younger p-value associated with a x2 value and the associated degrees of © Cengage Learning 2013
100 Older freedom can be found on any one of several statistical calcula-
80 tors found on the Internet. In this case, the researcher uses the
60 Observed calculator found at http://faculty.vassar.edu/lowry/tabs
40 .html#csq. By simply plugging in the observed value of 96.8
20 and the number of degrees of freedom as indicated, 1 in this
case, the calculator returns a p-value. In this case, the p-value
0 returned is less than 0.0001. Therefore, since the p-value is less
Expected than the acceptable level of risk, the researcher reaches the con-
clusion that the older workers are much more likely to attend
the retirement seminar. They can design the seminar to meet
the needs associated with the number and type of attendees.
Alternatively, the calculation can be followed in tabular form:
Location Oi Ei (Oi 2 Ei) (Oi 2 Ei)2
Ei
Standalone 60 50 10
210 100/50 5 2.0
Shopping Center 40 50 100/50 5 2.0
0
Total 100 100 x2 5 4.0
■■ Like many other probability distributions, the x2 distribution is not a single probability curve, 523
but a family of curves. These curves vary slightly with the degrees of freedom. In this case, the
degrees of freedom can be computed as
df 5 k 2 1
where
k 5 number of cells associated with column or row data.
Thus, the degrees of freedom equal 1 (df 5 2 2 1 5 1).
■■ Now the computed x2 value needs to be compared with the critical chi-square values associ-
ated with the 0.05 probability level with 1 degree of freedom. In Table A.4 of the appen-
dix the critical x2 value is 3.84. Since the calculated x2 is larger than the tabular chi-square,
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•524 PART SIX Data Analysis and Presentation
the conclusion is that the observed values do not equal the expected values. Therefore, the
hypothesis is supported. More Papa John’s restaurants are located in standalone locations.
We discuss the chi-square test further in Chapter 22 as it is also frequently used to analyze
contingency tables.
Hypothesis Test of a Proportion
hypothesis test of a Researchers often test univariate statistical hypotheses about population proportions.The p opulation
proportion proportion (p) can be estimated on the basis of an observed sample proportion (p). Conducting
a hypothesis test of a proportion is conceptually similar to hypothesis testing of the mean. Math-
A test that is conceptually ematically, however, the formulation of the standard error of the proportion differs somewhat.
similar to the one used when
the mean is the characteristic Consider the following example. A state legislature is considering a proposed right-to-work law.
of interest but that differs in One legislator has hypothesized that more than 50 percent of the state’s labor force is unionized.In other
the mathematical formulation words, the hypothesis to be tested is that the proportion of union workers in the state is greater than 0.5.
of the standard error of the The researcher formulates the hypothesis that the population proportion (p) exceeds
proportion. 50 p ercent (0.5):
H1: p . 0.5
Suppose the researcher conducts a survey with a random sample of 100 workers and calculates
p 5 0.6. While we know that 0.6 is greater than 0.5, we need to test if the observed difference might
be attributed to our sample.While the population proportion is unknown, a large sample allows use
of a Z-test (rather than the t-test). If the researcher decides that the decision rule will be set at the
0.01 level of significance, the critical Z-value of 2.57 is used for the hypothesis test. Using the fol-
lowing formula, we can calculate the observed value of Z given a certain sample proportion:
Zobs 5 p 2 p
Sp
where
p 5 sample proportion
p 5 hypothesized population proportion
Sp 5 estimate of the standard error of the proportion
The formula for Sp is
Sp 5 pq or Sp 5 p(1 2 p)
n A n
where A
Sp 5 estimate of the standard error of the proportion
p 5 proportion of successes
q 5 1 2 p, proportion of failures
In our example,
Sp 5 A (0.6)(0.4)
100
5 0.24
A 100
5 20.0024
5 0.04899
Zobs can now be calculated: p 2 p
Sp
Zobs 5
5 0.6 2 0.5
0.04899
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•CHAPTER 21 Univariate Statistical Analysis 525
0.1
5 0.04899
5 2.04
The Zobs value of 2.04 is less than the critical value of 2.57, so the hypothesis is not supported.
Additional Applications of Hypothesis Testing
The discussion of statistical inference in this chapter has been restricted to examining the difference
between an observed sample mean and a population or prespecified mean, a 2 test examining the
difference between an observed frequency and the expected frequency for a given distribution, and
Z-tests to test hypotheses about sample proportions when sample sizes are large. Other hypothesis
tests for population parameters estimated from sample statistics exist but are not mentioned here.
Many of these tests are no different conceptually in their methods of hypothesis testing. However,
the formulas are mathematically different. The purpose of this chapter has been to discuss basic
statistical concepts. Once you have learned the terminology in this chapter, you should have no
problem generalizing to other statistical problems.
The key to understanding statistics is learning the basics of the language. For this chapter, we
begin to adopt a more practical perspective by focusing on the p-values to determine whether a
hypothesis is supported rather than discussing null and alternative hypotheses. In more cases than
not, low p-values (below the specified ) support researchers’ hypotheses.5 It is hoped that some of
the myths about statistics have been shattered and that they are becoming easier to use.
SUMMARY
1. Implement the hypothesis-testing procedure. Hypothesis testing can involve univariate,
bivariate, or multivariate statistics. In this chapter, the focus is on univariate statistics. These are
tests that involve one variable. Usually, this means that the observed value for one variable will be
compared to some benchmark or standard. Statistical analysis is needed to test hypotheses when
sample observations are used to draw an inference about some corresponding population. The
research establishes an acceptable significance level, representing the chance of a Type I error, and
then computes the statistic that applies to the situation. The exact statistic that must be computed
depends largely on the level of scale measurement.
2. Use p-values to test for statistical significance. A p-value is the probability value associated with
a statistical test. The probability in a p-value is the probability that the expected value for some test dis-
tribution is true. In other words, for a t-test, the expected value of the t-distribution is 0. If a researcher
is testing whether or not a variable is significantly different from 0, then the p-value that results from
the corresponding computed t-value represents the probability that the true population mean is actu-
ally 0. For most research hypotheses, a low p-value supports the hypothesis. If a p-value is lower than
the researcher’s acceptable significance level (a), then the hypothesis is usually supported.
3. Test a hypothesis about an observed mean compared to some standard. Researchers often
have to compare an observed sample mean with some specified value. The appropriate statistical
test to compare an interval or ratio level variable’s mean with some value is either the Z- or t-test.
The Z-test is most appropriate when the sample size is large or the population standard deviation
is known. The t-test is most appropriate when the sample size is small or the population standard
deviation is not known. In most practical applications the t-test and Z-test will result in the same
conclusion. The t-test is used more often in practice.
4. Know the difference between Type I and Type II errors. A Type I error occurs when a researcher
reaches the conclusion that some difference or relationship exists within a population when in fact none
exists. In the context of a univariate t-test, the researcher may conclude that some mean value for a vari-
able is greater than 0 when in fact the true value for that variable in the population being considered is 0.
A Type II error is the opposite situation. When the researcher reaches the conclusion that no difference
exists when one truly does exist in the population, the researcher has committed a Type II error. More
attention is usually given to Type I errors. Type II errors are very sensitive to sample size.
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526 Part 6: Data Analysis and Presentation
5. Know the univariate 2 test and how to conduct one. A 2 test is one of the most basic tests
for statistical significance. The test is particularly appropriate for testing hypotheses about frequen-
cies arranged in a frequency or contingency table. The 2 test value is a function of the observed
value for a given entry in a frequency table minus the statistical expected value for that cell. The
observed statistical value can be compared to critical values to determine the p-value with any test.
The 2 test is often considered a goodness-of-fit test because it can test how well an observed matrix
represents some theoretical standard.
KEY TERMS AND CONCEPTS
bivariate statistical analysis, 507 multivariate statistical analysis, 507 t-test, 517
chi-square (2) test, 521 nonparametric statistics, 516 Type I error, 514
critical values, 512 parametric statistics, 516 Type II error, 514
degrees of freedom (df ), 517 p-value, 509 univariate statistical
goodness-of-fit (GOF), 521 significance level, 509
hypothesis test of a proportion, 524 t-distribution, 517 analysis, 507
QUESTIONS FOR REVIEW AND CRITICAL THINKING
1. What is the purpose of a statistical hypothesis? program. Formulate a hypothesis for a chi-square test and the
2. What is a significance level? How does a researcher choose a way the variable would be created.
12. Give an example in which a Type I error may be more serious
significance level? than a Type II error.
3. What is the difference between a significance level and a p-value? 13. Refer to the pizza store location 2 data on page 523. What
4. How is a p-value used to test a hypothesis? statistical decisions could be made if the 0.01 significance level
5. Distinguish between a Type I and Type II error. were selected rather than the 0.05 level?
6. What are the factors that determine the choice of the appropri- 14. Determine a hypothesis that the following data may address and
perform a 2 test on the survey data.
ate statistical technique? a. The X Factor should be broadcast before 9 p.m.
7. A researcher is asked to determine whether or not a productivity
Agree 40
objective (in dollars) of better than $75,000 per employee is pos- Neutral 35
sible. A productivity test is done involving 20 employees. What Disagree 25
conclusion would you reach? The sales results are as follows: 100
a. 28,000 105,000 58,000 93,000 96,000 b. Political affiliations of a group indicate
b. 67,000 82,500 75,000 81,000 59,000
c. 101,000 60,500 77,000 72,500 48,000 Republicans 102
d. 99,000 78,000 71,000 80,500 78,000 Democrats 98
200
8. Assume you have the following data: H1: m ∞ 200, S = 30,
n = 64, and X = 218. Conduct a two-tailed hypothesis test at 15. A researcher hypothesizes that 15 percent of the people in a test-
the 0.05 significance level. market will recall seeing a particular advertisement. In a sample
of 1,200 people, 20 percent say they recall the ad. Perform a
9. If the data in question 8 had been generated with a sample of hypothesis test.
25 (n = 25), what statistical test would be appropriate?
10. The answers to a researcher’s question will be nominally scaled.
What statistical test is appropriate for comparing the sample data
with hypothesized population data?
11. A researcher plans to ask employees whether they favor, oppose,
or are indifferent about a change in the company retirement
RESEARCH ACTIVITIES weather related websites such as http://www.weather.com
or through each community’s local news website. Record the
1. ’NET What is the ideal climate? Fill in the following blanks: The data in a spreadsheet or statistical package such as SPSS. Using
lowest temperature in January should be no lower than _____ the benchmark (preferred population low temperature) you
degrees. At least _____ days should be sunny in January. filled in above, test whether the sample places that you would
a. List at least 15 places where you would like to live. Using like to live have an ideal January minimum temperature.
the Internet, find the average low temperature in January
for each place. This information is available through various
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Chapter 21: Univariate Statistical Analysis 527
b. Using the same website, record how many days in January a. What is the p-value? Is the hypothesis supported?
are typically sunny. Test whether or not the number of b. Write the 95% confidence interval which corresponds to an
sunny days meets your standard.
a of 0.05.
c. For each location, record whether or not there was measur- c. Technically, since the sample size is greater than 30, a Z-test
able precipitation yesterday. Test the following hypothesis:
might be more appropriate. However, since the t-test result
H1: Among places you would like to live, there is less than a is readily available with SPSS, the research presents this result.
33.3 percent chance of rain/snow on a given day ( five days out Is there an ethical problem in using the one-sample t-test?
of fifteen).
One-Sample Statistics
2. ETHICS Examine the statistical choices under Analyze in SPSS.
Click on Compare Means. To compare an observed mean to Mean Std. Std. Error
some benchmark or hypothesized population mean, the avail- N Deviation Mean
able choice is a one-sample t-test. A researcher is preparing a
report and finds the following result testing a hypothesis that 1997–2000 67 14.5337 16.02663 1.95796
suggests the sample mean did not equal 14.
Test Value = 14
95% Confidence Interval of the
Difference
t df Sig. (two-tailed) Mean Difference Lower Upper
–3.3755 4.4429
1997–2000 0.273 66 0.786 0.53373
Premier Motorcars
CASE Premier Motorcars is the new Fiat dealer in Delavan, CASE EXHIBIT 21.1-1
21.1 Illinois. Premier Motorcars has been regularly advertis-
Miles per Gallon Information
ing in its local market area that the new Fiat 500 aver-
ages 30 miles to a gallon of gas and mentions that this figure may Purchaser Miles per Purchaser Miles per
vary with driving conditions. A local consumer group wishes to ver- 1 Gallon 13 Gallon
ify the advertising claim. To do so, it selects a sample of recent pur- 2 30.9 14 27.0
chasers of the Fiat 500. It asks them to drive their cars until two tanks 3 24.5 15 26.7
of gasoline have been used up and to record the mileage. The group 31.2 16 31.0
then calculates and records the miles per gallon for each purchaser. 17 23.5
The data in Case Exhibit 21.1-1 portray the results of the tests. 18 29.4
19 26.3
4 28.7 20 27.5
21 28.2
5 35.1 22 28.4
23 29.1
© Zoran Karapanceev/Shutterstock 6 29.0 24 21.9
30.9
7 28.8
8 23.1
Questions 9 31.0
1. Formulate a statistical hypothesis appropriate for the consumer 10 30.2
11 28.4
group’s purpose. 12 29.3
2. Calculate the mean average miles per gallon. Compute the
sample variance and sample standard deviation.
3. Construct the appropriate statistical test for your hypothesis,
using a 0.05 significance level.
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22CHAPTER Bivariate Statistical
Analysis
Differences Between
Two Variables
LEARNING OUTCOMES
After studying this chapter, you should be able to
1. Recognize when a particular bivariate statistical test is appropriate
2. Calculate and interpret a x2 test for a contingency table
3. Calculate and interpret an independent samples t-test comparing
two means
4. Understand the concept of analysis of variance (ANOVA)
5. Interpret an ANOVA table
Chapter Vignette:
Gender Differences and Double Standards
in Ethical Perceptions
W hat if you went to trade in your car, standards than men and that the “double standard” would
knowing that it had an oil leak “which is exist with respondents perceiving customer actions as more
not very noticeable and doesn’t require ethical than business behavior.
immediate attention,” but would require
While we would need to talk to every man and woman to
$200 to have fixed in the near future?1 Would you feel a moral actually know if their ethical perceptions were different, as
researchers we understand that a sample of the population has
obligation to tell the car dealer? How about if you went to buy
a car with the same issue? Would it be ethical for
the dealer to sell you the car without mentioning the
oil leak?
Ethical conduct, both of businesses and consum-
ers, is an important issue in the business world.
Recent research conducted in Flanders, a European
region that includes parts of Belgium, France, and
the Netherlands, investigated two aspects of ethical
perceptions with relevance to business.2 First, is there
a difference between women and men in their ethi-
cal perceptions? Second, is there an ethical double © Holbox/Shutterstock
standard—that consumers view an action performed
by a customer as more ethical than the same action
performed by a business? The researchers hypoth-
esized that women would report higher ethical
528
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•CHAPTER 22 Bivariate Statistical Analysis: Differences Between Two Variables 529
to be used in most situations. In this case, business researchers while the other half had a business performing the act. The
asked 127 respondents to evaluate a series of ethical scenarios respondents were presented the scenarios and then asked to
(short stories with ethical implications) including the car with indicate how ethical they thought the act described was on
the oil leak mentioned above (Scenario 2). These scenarios were a scale from 1 indicating “totally unethical” to 7 indicating
split so that half of them had a consumer engaging in the act “totally ethical.” Across the four scenarios, the results show:
Gender Scenario 1 Scenario 2 Scenario 3 Scenario 4
Male 5.59 4.38 5.83 6.24
Female 5.21 3.12 4.88 5.71
While all four scenarios indicate that men rated the activity as three of the four scenarios, with only Scenario 2 showing a
more ethical than women, to generalize these results from the s tatistically significant difference at the 0.05 level.
sample to the population we need to perform statistical tests.
These tests show that there is not a significant difference in When testing for the presence of the proposed double
s tandard, the results show:
Source Scenario 1 Scenario 2 Scenario 3 Scenario 4
Consumer 5.38 3.70 5.32 5.95
Corporate 3.36 1.67 3.38 4.97
In this case, there is a statistically significance difference on all these results are likely to be found across the population? This
four scenarios. In other words, people perceive the same act as c hapter focuses on this question when we are examining differ-
less ethical when performed by a business than a consumer. ences between two variables.
How do we do these statistical tests? How can we deter-
mine if the results we see might be unique to the sample, or if
Introduction
The Chapter Vignette is just one illustration of business researchers’ desire to test hypotheses tests of differences
stating that two groups differ. In business research, differences in behavior, characteristics, beliefs, The investigation of hypotheses
opinions, emotions, or attitudes are commonly examined. For example, in the most basic experi- stating that two (or more)
mental design, the researcher tests differences between subjects assigned to an experimental group groups differ with respect to
and subjects assigned to the control group. The experiment illustration presented in Chapter 12 measures on a variable.
on self-efficacy is an example of this approach. A survey researcher may be interested in whether
male and female consumers purchase a product in the same amount. Business researchers may also ‘‘You got to be
test whether or not business units in Europe are as profitable as business units in the United States.
Such tests are bivariate tests of differences when they involve only two variables: a variable that careful if you don’t
acts like a dependent variable and a variable that acts as a classification variable. These bivariate tests know where you’re
of differences are the focus of this chapter. going, because
What Is the Appropriate Test of Difference? ’’you might not get
Exhibit 22.1 illustrates that the type of measurement, the nature of the comparison, and the num- there.
ber of groups to be compared influence the statistical choice. Often researchers are interested in
testing differences in mean scores between groups or in comparing how two groups’ scores are —Yogi Berra
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
•530 PART SIX Data Analysis and Presentation
distributed across possible response categories.We will focus our attention on these issues.3 The rest
of the chapter focuses on how to choose the right statistic for two-group comparisons and perform
the corresponding test. Exhibit 22.1 provides a frame of reference for the rest of the chapter by
illustrating various possible comparisons involving a few golfers.
Construction of contingency tables for x2 analysis gives a procedure for comparing observed
frequencies of one group with the frequencies of another group.This is a good starting point from
which to discuss testing of differences.
EXHIBIT 22.1
Some Bivariate Hypotheses
Golfer Hypothesis Level of
or Research Measurement
Information Dolly Lori Mel Question Involved Statistic Used Comment Result
Average 135 150 185 Lori hits her drive Golfer 5 Nominal; Independent Data for Lori and Supported
driver distance
(meters) further than Dolly Drive distance 5 samples t-test to Dolly are used (t 5 2.07,
Ratio compare mean df 5 56,
distance p , .05)
s 30 25 30
Average 140 145 150 Mel hits her driver Club 5 Nominal Paired-samples Only the data for Supported
7-wood
distance futher than her (7-wood or driver); t-test to compare Mel are used (std of (t 5 6.39,
(meters)
7-wood 7-wood distance 5 mean distances for diff 5 30) df 5 29,
Ratio Mel p , .05)
s 30 30 30
Sample size 28 30 drives 29 drives A relationship Golfer 5 Nominal; One-way ANOVA All data for 7-wood Not
(number of drives 28 28 exists Distance 5 Ratio to compare means distance are used supported
balls hit) 28 7-woods 7-woods between for the three (MSE 5 30) (F 5 0.83, ns)
7-woods golfers and groups
7-wood distance
Number of 4 22 11 Mel drives the ball Golfer 5 Nominal; Cross-tabulation Resulting cross Supported
drives in more accurately Accuracy 5 Nominal with x2 statistic tabulation table (x2 5 10.3,
fairway than Dolly (right, fairway, left) is 2 rows 3 3 df 5 3,
columns p , .05)
(rows 5 golfer and
columns 5 accuracy
(right, fairway, left))
Drives missing 16 7 9 A relationship Golfer 5 Nominal; Cross-tabulation Cross-tabulation is Supported
right of exists between Accuracy 5 Nominal with x2 statistic now 3 rows 3 (x2 5 23.7, © Cengage Learning 2013
fairway golfers and (right, fairway, left 3 columns df 5 4,
accuracy p , .05)
Drives missing 8 1 9
left of fairway
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
STUHRVISEY! Courtesy of Qualtrics.com
Our survey includes data that can be appropriately
analyzed with the techniques discussed in this chap-
ter. After reading the chapter, access the online
data and answer the following three questions:
1. Is there a relationship between student
gender and their major? Does one gender
select certain majors more than another? Use
cross-tabulations and the x2 test to examine the relationship between gender and major. What did you find?
2. Is there a difference between those respondents that are currently employed and those that are not currently employed
regarding their goal achievement and life satisfaction? Respondents indicated whether or not they were currently employed.
Use a t-test to examine the differences between the employed/not employed respondents on the six questions which ask:
a. I am energetically pursuing my goals.
b. I really can’t see any way around my problems.
c. I am meeting the goals I set for myself.
d. I am simply not being very successful these days.
e. I know there are many ways to achieve my goals.
f. My life could hardly be any better
What did you find?
3. Is there a difference among the various student classifications and their attitude regarding their goal achievement and
life satisfaction (the questions identified in part 2 above)? One question asks the respondents to indicate their level
as a student (lower-level undergrad, upper-level undergrad, etc.). Use ANOVA to examine any differences in attitudes
across the student classification groups. What did you find?
Cross-Tabulation Tables: The x2 Test
for Goodness-of-Fit
Cross-tabulation is one of the most widely used statistical techniques among business researchers. Cross-
tabulations are intuitive, easily understood, and lend themselves well to graphical analysis using tools
like bar charts. Cross-tabs are appropriate when the variables of interest are less-than interval in nature.
As we discussed in Chapter 20, a cross-tabulation, or contingency table, is a joint frequency
distribution of observations on two or more variables. Researchers generally rely on two-variable
cross-tabulations the most since the results can be easily communicated. Cross-tabulations are much
like tallying.When two variables exist, each with two categories, four cells result.The x2 distribution
provides a means for testing the statistical significance of a contingency table. In other words, the bivar-
iate x2 test examines the statistical significance of relationships between two less-than interval variables.
The x2 test for a contingency table involves comparing the observed frequencies (Oi) with
the expected frequencies (Ei) in each cell of the table.The goodness- (or closeness-) of-fit of the
observed distribution with the expected distribution is captured by this statistic. Remember that
the convention is that the row variable is considered the independent variable and the column
variable is considered the dependent variable.
Recall that in Chapter 21 we used a x2 test to examine whether or not Papa John’s restaurants in
California were more likely to be located in a standalone location or in a shopping center.The univariate
(one-dimensional) analysis suggests that the majority of the locations (60 percent) are standalone units:
Location One-Way Frequency Table
Standalone 60 stores
Shopping Center 40 stores
Total 100 stores
531
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•532 PART SIX Data Analysis and Presentation
Recall that the x2 5 4.0 with 1 degree of freedom (p , 0.01).
Is there any effect of location on Papa John’s restaurants? Suppose the researcher wishes to
examine the following hypothesis:
Standalone locations are more likely to be profitable than are shopping center locations.
While the researcher is unable to obtain the dollar figures for profitability of each unit, a press
release indicates which Papa John’s units were profitable and which were not. Cross-tabulation
using a x2 test is appropriate because
■■ The independent variable (location) is less-than interval.
■■ The dependent variable (profitable/not profitable) is less-than interval.
The data can be recorded in the following 2 3 2 contingency table:
Location Profitable Not Profitable Total
Standalone 50 10 60
Shopping Center 15 25 40
Totals 65 35 100
Several conclusions appear evident. One, it seems that more stores are profitable than not
profitable (65 versus 35, respectively). Secondly, more of the profitable restaurants seem to be in
standalone locations (50 of the 65). However, is the difference strong enough to be statistically
significant?
Is the observed difference between standalone and shopping center locations the result of
chance variation due to random sampling? Is the discrepancy more than sampling variation? The
x2 test allows us to conduct tests for significance in the analysis of the R 3 C contingency table
(where R 5 row and C 5 column).The formula for the x2 statistic is the same as that for one-way
frequency tables (see Chapter 21): Ox2 5
where (Oi 2 Ei )2
Ei
x2 5 chi-square statistic
Oi 5 observed frequency in the ith cell
Ei 5 expected frequency in the ith cell
Again, as in the univariate x2 test, a frequency count of data that nominally identify or categorically
rank groups is acceptable.
If the researcher’s hypothesis is true, the frequencies shown in the contingency table should
not resemble a random distribution. In other words, if location has no effect on profitability, the
profitable and unprofitable stores would be spread evenly across the two location categories.This
is really the logic of the test in that it compares the observed frequencies with the theoretical
expected values for each cell.
After obtaining the observations for each cell, the expected values for each cell must be
obtained. The expected values are what we would find if there is no relationship between the
two variables. In this case, that the location of the pizza store has no relationship with whether
or not the store is profitable.The expected values for each cell can be computed easily using this
formula:
Eij 5 RiCj
n
where
Ri 5 total observed frequency count in the ith row
Cj 5 total observed frequency count in the jth column
n 5 sample size
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•CHAPTER 22 Bivariate Statistical Analysis: Differences Between Two Variables 533
Only the total column and total row values are needed for this calculation.Thus, the calcula-
tion could be performed before the data are even tabulated. The following values represent the
expected values for each cell:
Location Profitable Not Profitable Total
Standalone (60 3 65)/100 5 39 (60 3 35)/100 5 21 60
Shopping Center (65 3 40)/100 5 26 (40 3 35)/100 5 14 40
Totals 100
65 35
Notice that the row and column totals are the same for both the observed and expected contin-
gency matrices.These values also become useful in providing the substantive interpretation of the
relationship. Significant variation from the expected value indicates a relationship and tells us the
direction.
The actual bivariate x2 test value can be calculated in the same manner as for the univariate
test. The one difference is that the degrees of freedom are now obtained by multiplying the
n umber of rows minus one (R 2 1) times the number of columns minus one (C 2 1):
Ox2 5 (Oi 2 Ei )2
Ei
with (R 2 1)(C 2 1) degrees of freedom.The observed and expected values can be plugged into
the formula as follows:
(50 2 39)2 (10 2 21)2 (15 2 26)2 (25 2 14)2
x2 5 39 1 21 1 26 1 14
5 3.102 1 5.762 1 4.654 1 8.643
5 22.16
The number of degrees of freedom equals 1:
(R 2 1)(C 2 1) 5 (2 2 1)(2 2 1) 5 1
From Table A.4 in the appendix, we see that the critical value at the 0.05 probability level
with 1 df is 3.84.Thus, we are very confident that the observed values are not equal to the expected
values. Before the hypothesis can be supported, however, the researcher must check and see that the
deviations from the expected values are in the hypothesized direction. Since the difference between
the standalone locations’ observed profitability and the expected values for that cell are positive, the
hypothesis is supported. Location is associated with profitability. The Research S napshot “Accurate
Information? How About a Chi-Square Test?” provides another example of cross-tabs and a x2 test.
Thus, testing the hypothesis involves two key steps:
1. Examine the statistical significance of the observed contingency table.
2. Examine whether the differences between the observed and expected values are consistent
with the hypothesized prediction.
The examples provided both have 2-by-2 contingency tables (that is, two levels of two
variables). However, cross-tabulations and the x2 test can be used regardless of the number of
levels. For instance, if the Papa John’s locations were instead standalone, shopping center, and
delivery only, we would have a 3-by-2 contingency table. Or, perhaps we want to look at the
distribution of our male and female sales reps across our five product lines, which would give
us a 2-by-5 contingency table. The number of cells is not limited. However, proper use of the
x2 test requires that each expected cell frequency (E) have a value of at least 5. If this sample
size requirement is not met, the researcher should take a larger sample or combine (collapse)
response categories.
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RESEARCH
SNAPSHOT
Accurate Information? How About
a Chi-Square Test?
When is a cross-tabulation with a x2 test appropriate? The instance, this might
answer to this question can be determined by answering these involve the adoption of
questions: a new technology or the © Comstock Images/Jupiter Images
■■ Are multiple variables expected to be related to one another? effect of training. For
■■ Is the independent variable nominal or ordinal? instance, c onsider the
■■ Is the dependent variable nominal or ordinal? following contingency
data represented in bar
When the answer to all of these questions is yes, cross-tabulation charts to the right.
with a x2 test will address the research question. One common The data show
application involves the effect of some workplace change. For whether or not the
Technology and Accuracy adoption of a new
60 information system produced accurate or inaccurate informa-
tion. The 2-by-2 contingency table underlying this bar chart
produces a x2 value of 5.97 with 1 degree of freedom. The
50 p-value is less than 0.05; thus, the new technology does seem
to have changed accuracy. However, we must examine
40 the actual cell counts to see exactly what this effect has
New Technology been. In this case, the bar chart indicates that the new
30 Old Technology technology is associated with more incidences of accu-
20 Inaccurate rate rather than inaccurate information.
10
0 Sources: For examples of research involving this type of analysis, see
G ohmann, S. E., R. M. Barker, D. J. Faulds, and J. Guan, “Salesforce
Accurate A utomation, Perceived Information Accuracy and User Satisfaction,”
Journal of Business and Industrial Marketing 20 (2005), 23–32; Makela, C. J.
and S. Peters, “Consumer Education: Creating Consumer Awareness among
Adolescents in Botswana,” International Journal of Consumer Studies 28
(September 2004), 379–387.
The t-Test for Comparing Two Means
Cross-tabulations and the x2 test are appropriate when both variables are less-than interval level.
However, researchers often want to compare one interval or ratio level variable across categories
of respondents.The Chapter Vignette describes such a situation.The researchers are interested in
comparing the ethical perceptions between genders.When a researcher needs to compare means
for a variable grouped into two categories based on some less-than interval variable, a t-test is
appropriate. One way to think about this is testing the way a dichotomous (two-level) independent
variable is associated with changes in a continuous dependent variable. Several variations of the
t-test exist.
Independent Samples t-Test
independent samples Most typically, the researcher will apply the independent samples t-test, which tests the differences
t-test between means taken from two independent samples or groups. So, for example, if we measure the
price for some designer jeans at 30 different retail stores, of which 15 are Internet-only stores (pure
A test for hypotheses which clicks) and 15 are traditional stores, we can test whether or not the prices are different based on
compares the mean scores for store type with an independent samples t-test.The t-test for difference of means assumes the two
two groups comprised of some samples (one Internet and one traditional store) are drawn from normal distributions and that the
interval- or ratio-scaled variable variances of the two populations are approximately equal (homoscedasticity).
using a less-than interval
classificatory variable.
534
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•CHAPTER 22 Bivariate Statistical Analysis: Differences Between Two Variables 535
Independent Samples t-Test Calculation
The t-test actually tests whether or not the differences between two means is zero. Not surprisingly,
this idea can be expressed as the difference between two population means:
m1 5 m2, which is equivalent to m1 2 m2 5 0
However, since this is inferential statistics, we test the idea by comparing two sample means
(X1 2 X2 ).
A verbal expression of the formula for t is:
t 5 Sample mean 1 2 Sample mean 2
Variability of random means
In almost all situations, we will see from the calculation of the two sample means that they are not
exactly equal.The question is actually whether the observed differences have occurred by chance,
or likely exist in the population. The t-value is a ratio with information about the difference
between means (provided by the sample) in the numerator and the standard error in the denomi-
nator.To calculate t, we use the following formula:
t 5 X1 2 X2
SX1 2 X2
where
X1 5 mean for group 1 pooled estimate of the
X2 5 mean for group 2 standard error
SX12X2 5 pooled, or combined, standard error of difference between means
An estimate of the
A pooled estimate of the standard error is a better estimate of the standard error than one standard error for a t-test
based on the variance from either sample. The pooled standard error of the difference between of independent means that
means of independent samples can be calculated using the following formula: assumes the variances of both
groups are equal.
Sx1 2 x2 5 a (n1 2 1)S12 1 (n2 2 1)S 2 b a 1 1 1
D n1 1 n2 2 2 2 n1 n2 b
where
S 2 5 variance of group 1
1
S 2 5 variance of group 2
2
n1 5 sample size of group 1
n2 5 sample size of group 2
Are business majors or sociology majors more positive about a career in business? A t-test can be
used to test the difference between sociology majors and business majors on scores on a scale mea-
suring attitudes toward business careers. We will assume that the attitude scale is an interval scale.
The result of the simple random sample of these two groups of college students is shown below:
Business Students Sociology Students
X1 5 16.5 X2 5 12.2
S1 5 2.1 S2 5 2.6
n1 5 21 n2 5 14
A high score indicates a favorable attitude toward business.We can see in the sample that busi-
ness students report a higher score (16.5) than sociology students (12.2).This particular t-test tests
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•536 PART SIX Data Analysis and Presentation
whether the difference in attitudes between sociology and business students is significant.That is, is
the sample result due to chance or do we expect this difference to exist in the population? A higher
t-value is associated with a lower p-value.As the t-value gets higher and the p-value gets lower, the
researcher has more confidence that the means are truly different.The relevant data computation is
Sx1 2 x2 5 a (n1 2 1)S12 1 (n2 2 1)S22 b 1 1 1
D n1 1 n2 22 a n1 n2 b
(20)(2.1)2 1 (13)(2.6)2 1 1
5 Da 33 b a 21 1 14 b
5 0.797
The calculation of the t-statistic is
t 5 X1 2 X2
Sx1 2 x2
t 5 16.5 2 12.2
0.797
5 4.3
0.797
5 5.395
In a test of two means, degrees of freedom are calculated as follows:
df 5 n 2 k
where
n 5 n1 1 n2
k 5 number of groups
In our example df equals 33((21 1 14) 2 2). If the 0.01 level of significance is selected, reference
to Table A.3 in the appendix yields the critical t-value.The t-value of 2.75 must be surpassed by the
observed t-value if the hypothesis test is to be statistically significant at the 0.01 level.The calculated
value of t, 5.39, far exceeds the critical value of t for statistical significance, so it is significant at
a 5 0.01.The p-value is less than 0.01. In other words, this research shows that business students have
significantly more positive attitudes toward business than do sociology students.The Research Snap-
shot “Expert ‘T-eeze’ ” describes the situation when an independent samples t-test should be used.
Practically Speaking
While it is good to understand the process involved, in practice computer software is used to com-
pute the t-test results. Exhibit 22.2 displays a typical t-test printout.These particular results examine
the following research question:
RQ: Does religion relate to price sensitivity?
This question was addressed in the context of restaurant and wine consumption by allowing 100
consumers to sample a specific wine and then tell the researcher how much they would be will-
ing to pay for a bottle of the wine.The sample included 57 Catholics and 43 Protestants. Because
no direction of the relationship is stated (no hypotheses is offered), a two-tailed test is appropriate.
Although instructors still find some value in having students learn to perform t-test calculations,
this procedure is usually computer generated and interpreted today. Using SPSS, the click-through
sequence would be:
Analyze → Compare Means → Independent-Samples t-test
Then, the variable used to categorize the respondent as either Catholic or Protestant would be
entered as the grouping variable and the variable with the amount the respondent was willing to
pay as the test variable.
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•CHAPTER 22 Bivariate Statistical Analysis: Differences Between Two Variables 537
EXHIBIT 22.2
Independent Samples t-Test Results
rel Group Statistics Std. Std. Error 1. Shows mean, standard deviation, and standard
Deviation Mean error for each group (Catholic and Protestant)
Catholic N Mean
price Protestant 57 61.00 43.381 5.746
43 50.27 64.047 9.767
NOTE: Top row shows Levene’s Test for Independent Samples Test
results assuming equal Equality of Variances t-Test for Equality of Means
variances. Bottom row
95% Confidence
assumes variance is Interval of the
different in each.
Difference
Sig. Mean Std. Error
F Sig. t d.f. (2-tailed) Difference Difference Lower Upper
price Equal
variances
assumed .769 .383 .998 98 .321 10.734 10.752 −10.603 32.070
10.374
Equal .947 69.829 .347 11.332 −11.868 33.336
variances
© Cengage Learning 2013
not
assumed
2. Computed t-test value shown in 3. p-value for t-value and associated 4. Confidence intervals for a 5 0.05
this column (t 5 0.998). degrees of freedom (t 5 0.998, 98 d.f.) (100% − 95%). In this case, it includes 0.
The interpretation of the t-test is made simple by focusing on either the p-value or the confi-
dence interval and the group means. Here are the basic steps:
1. Examine the difference in means to find the “direction” of any difference. In this case,
Catholics are willing to pay nearly $11 more than Protestants.
2. Compute or locate the computed t-test value. In this case, t 5 0.998.
3. Find the p-value associated with this t and the corresponding degrees of freedom. Here, the
p-value (two-tailed significance level) is 0.321. This suggests a 32 percent chance that the
means are actually equal given the observed sample means. In other words, the difference we
see may be due to this sample of 100 respondents rather than being found in the population.
Assuming a 0.05 acceptable Type I error rate (a), the appropriate conclusion is that the means
are not significantly different.
4. The difference can also be examined using the 95 percent confidence interval
( 2 10.603 , X1 2 X2 , 32.070). Since the confidence interval includes 0, we lack suf-
ficient confidence that the true difference between the population means is not 0.
A few points are worth noting about this particular result. First, strictly speaking, the t-test
assumes that the two population variances are equal.A slightly more complicated formula exists that
will compute the t-statistic assuming the variances are not equal.4 SPSS provides both results when
an independent samples t-test is performed.The sample variances appear considerably different in
this case (43.4, 64.0). Nonetheless, the conclusions are the same using either assumption. In business
research, we often deal with values that have variances close enough to assume equal variance.This
isn’t always the case in the physical sciences where variables may take on values of drastically different
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RESEARCH
SNAPSHOT
Expert “T-eeze” © Image Source/Jupiter Images
When is an independent samples t-test appropriate? Once
again, we can find out by answering some simple questions:
■■ Is the dependent variable interval or ratio? Decision Time (seconds)
■■ Can the dependent variable scores be grouped based upon 140
some categorical variable? 120
■■ Does the grouping result in scores drawn from
100
independent samples?
■■ Are two groups involved in the research question? 80 Series 1
When the answer to all questions is yes, an independent 60
samples t-test is appropriate. Often, business researchers may
wish to examine how some process varies between novices and 40
experts (or new employees and current employees). Consider
the following example. 20
Researchers looked at the difference in decision speed for 0 Experts
expert and novice salespeople faced with the same situation. Novices
Decision speed is a ratio dependent variable and the scores are
grouped based on whether or not the salesperson is an expert Source: Shepherd, D. G., S. F. Gardial, M. G. Johnson, and J. O. Rentz,
or a novice. Thus, this categorical variable produces two groups. “C ognitive Insights into the Highly Skilled or Expert Salesperson,”
The results across 40 respondents, 20 experts, and 20 novices, Psychology & Marketing 23 (February 2006), 115–138. Reprinted with
are shown at the top right. permission of John Wiley & Sons, Inc.
The average difference in decision time is 38 seconds.
Is this significantly different from 0? The calculated t-test is
2.76 with 38 df. The one-tailed p-value is 0.0045; thus the
conclusion is reached that experts do take less time to make a
decision than do novices.
magnitude.Thus, the rule of thumb in business research is to use the equal variance assumption. In
the vast majority of cases, the same conclusion will be drawn using either assumption.
Second, notice that even though the means appear to be not so close to each other, the statisti-
cal conclusion is that they are the same.The substantive conclusion is that Catholics and Protestants
would not be expected to pay different prices. Why is it that means do not appear to be similar,
yet that is the conclusion? The answer lies in the variance. Respondents tended to provide very
wide ranges of acceptable prices. Notice how large the standard deviations are compared to the
mean for each group. Since the t-statistic is a function of the standard error, which is a function of
the standard deviation, a lot of variance means a smaller t-value for any given observed difference.
When this occurs, the researcher may wish to double-check for outliers. A small number of wild
price estimates could be inflating the variance for one or both groups.An additional consideration
would be to increase the sample size and test again.
Third, a t-test is used even though the sample size is greater than 30. Strictly speaking, a Z-test
could be used to test this difference. Researchers often employ a t-test even with large samples.As
samples get larger, the t-test and Z-test will tend to yield the same result. Although a t-test can be
used with large samples, a Z-test should not be used with small samples.Also, a Z-test can be used
in instances where the population variance is known ahead of time.
As another example, consider 11 sales representatives categorized as either young (1) or old (2)
on the basis of their ages in years, as shown in Exhibit 22.3.The exhibit presents a SAS c omputer
output that compares the mean sales volume for these two groups.We can see that the mean for the
young group is 61,879 and that of the old group is 86,962, which appears considerably d ifferent.
Again, though, this difference is not statistically significant at the 0.05 level as the p-value is 0.3218.
In this case, the very small sample size (11 in total) drastically limits the statistical power.
538
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•CHAPTER 22 Bivariate Statistical Analysis: Differences Between Two Variables 539
EXHIBIT 22.3
SAS t-Test Output
t-Test Procedure Variable: CR Sales
Standard Standard
Age n Mean Deviation Error Minimum Maximum Variances t d.f. Prob > |T |
1 6 61879.33333 22356.20845 9126.88388 41152.00000 103059.0000 Unequal 20.9758 5.2 0.3729 © Cengage Learning 2013
2 5 86961.80000 53734.45098 24030.77708 42775.00000 172530.0000 Equal 21.0484 9.0 0.3218
For H0: Variances are equal, F 5 5.78 with 4 and 5 d.f., Prob. . F 5 0.0815.
Paired-Samples t-Test
What happens when means need to be compared that are not from independent samples? Such paired-samples t-test
might be the case when the same respondent is measured twice—for instance, when the respon-
dent is asked to rate both how much he or she likes shopping on the Internet and how much he An appropriate test for
or she likes shopping in traditional stores. Since the liking scores are both provided by the same comparing the scores of two
person, the assumption that they are independent is not realistic. Additionally, if one compares interval variables drawn from
the prices the same retailers charge in their stores with the prices they charge on their websites, related populations.
the samples cannot be considered independent because each pair of observations is from the same
sampling unit.
A paired-samples t-test is appropriate in this situation. The idea behind the paired-samples
t-test can be seen in the following computation:
t 5 d
sd / 1n
where d is the difference between means, sd is the standard deviation of the observed differences,
and n is the number of observations. Researchers also can compute the paired-samples t-test using
statistical software. For example, using SPSS, the click-through sequence would be:
Analyze → Compare Means → Paired-Samples t-test
A dialog box then appears in which the “paired variables” should be entered. When a paired-
samples t-test is appropriate, the two numbers being compared are usually scored as separate
variables.
Exhibit 22.4 displays a paired-samples t-test result.A sample of 143 young adult consumers was
asked to rate how likely they would be to consider purchasing an engagement ring (or want their
ring purchased) via (a) an Internet retailer and (b) a well-known jewelry store. Each respondent
provided two responses, much as in a within-subjects experimental design.The bar chart depicts
the means for each variable (Internet purchase likelihood and store purchase likelihood).The t-test
results suggest that the average difference of –42.4 is associated with a t-value of –16.0. As can be
seen using either the p-value (0.000 rounded to 3 decimals) or the confidence interval –47.6 ,
d , −37.1), which does not include 0, the difference is significantly different from 0. Therefore,
the results suggest a higher likelihood to buy a wedding ring in a well-known brick-and-mortar
retail store than via an Internet merchant. For those of you considering marriage, this might be a
good tip!
Management researchers have used paired-samples t-tests to examine the effect of downsiz-
ing on employee morale. For instance, job satisfaction for a sample of employees can be measured
immediately after the downsizing. Some months later, employee satisfaction can be measured again.
The difference between the satisfaction scores can be compared using a paired-samples t-test.
Results suggest that the employee satisfaction scores increase within a few months of the downsiz-
ing as evidenced by statistically significant paired-samples t-values.5
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•540 PART SIX Data Analysis and Presentation
EXHIBIT 22.4 Shopping Intentions
Example Results for a Paired-Samples t-Test
50
40
30
20
10
0 Store
Internet
Paired-Samples Test
Paired Differences
95% Confidence
Interval of the
Difference
Mean Std. Std. Error Lower Upper Sig.
Deviation Mean t d.f. (2-tailed)
Pair 1 int1 2 int2 242.388 31.660 2.648 247.622 237.154 216.011 142 0.000
1. The average difference 3. 95% confidence 2. The computed t-value, © Cengage Learning 2013
between observations interval shown d.f., and p-value are
is shown here. here. shown here.
The Z-Test for Comparing Two Proportions
Z-test for differences of What type of statistical comparison can be made when the observed statistics are proportions?
proportions Suppose a researcher wishes to test the hypothesis that wholesalers in Asian and European markets
differ in the proportion of sales they make to discount retailers. Testing whether the population
A technique used to test the proportion for group 1 (p1) equals the population proportion for group 2 (p2) is conceptually the
hypothesis that proportions same as the t-test of two means. This section illustrates a Z-test for differences of proportions,
are significantly different for which requires a sample size greater than 30.
two independent samples or
The test is appropriate for a hypothesis of this form:
groups.
H0:p1 5 p2
which may be restated as
H0:p1 2 p2 5 0
Comparison of the observed sample proportions p1 and p2 allows the researcher to ask whether
the difference between two large (greater than 30) random samples occurred due to chance alone.
The Z-test statistic can be computed using the following formula:
Z 5 ( p1 2 p2) 2 (p1 2 p2)
Sp1 2 p2
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•CHAPTER 22 Bivariate Statistical Analysis: Differences Between Two Variables 541
where
p1 5 sample proportion of successes in group 1
p2 5 sample proportion of successes in group 2
p1 2 p2 5 hypothesized population proportion 1 minus hypothesized population proportion 2
Sp12p2 5 pooled estimate of the standard error of differences in proportions
The statistic normally works on the assumption that the value of p1 2 p2 is zero, so this formula
is actually much simpler than it looks at first inspection. Readers also may notice the similarity
between this and the paired-samples t-test.
To calculate the standard error of the differences in proportions, use the formula
Sp1 2 p2 5 p q A 1 1 1 B
A n1 n2
where
p 5 pooled estimate of proportion of successes in a sample
q 5 1 − p, or pooled estimate of proportion of failures in a sample
n1 5 sample size for group 1
n2 5 sample size for group 2
To calculate the pooled estimator, p, use the formula
p 5 n1 p1 1 n2 p2
n1 1 n2
Suppose the survey data are as follows:
Asian Wholesalers European Wholesalers
p1 5 0.35 p2 5 0.40
n1 5 100 n2 5 100
First, the standard error of the difference in proportions is
A BS 5p12p2 1 1
4 pq n1 1 n2
5 A (0.375)(0.625) A 1 1 1 B 5 0.068
100 100
where
p 5 (100)(0.35) 1 (100)(0.40) 5 0.375
100 1 100
If we wish to test the two-tailed question of no difference, we must calculate an observed Z-value.
Thus,
Z 5 ( p1 2 p2) 2 (p1 2 p2)
Sp1 2 p2
5 (0.35 2 0.40) 2 (0)
0.068
5 20.73
In this example the idea that the proportion of sales differs between the Asian and European
regions is not supported.The calculated Z-value is less than the critical Z-value of 1.96.Therefore,
the p-value associated with the test is greater than 0.05.
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•542 PART SIX Data Analysis and Presentation
Analysis of Variance (ANOVA)
What Is ANOVA?
analysis of variance So far, we have discussed tests for differences between two groups. However, what happens when
(ANOVA) we have more than two groups? For example, what if we want to test and see if employee turnover
differs across our five production plants? When the means of more than two groups or popula-
Analysis involving the tions are to be compared, one-way analysis of variance (ANOVA) is the appropriate statistical tool.
investigation of the effects ANOVA involving only one grouping variable is often referred to as one-way ANOVA because
of one treatment variable on only one independent variable is involved. Another way to define ANOVA is as the appropriate
an interval-scaled dependent statistical technique to examine the effect of a less-than interval independent variable on an at-least
variable—a hypothesis-testing interval dependent variable.Thus, a categorical independent variable and a continuous dependent
variable are involved.An independent samples t-test can be thought of as a special case of ANOVA
technique to determine in which the independent variable has only two levels. When more levels exist, the t-test alone
whether statistically significant cannot handle the problem.
differences in means occur The statistical null hypothesis for ANOVA is stated as follows:
between two or more groups.
m1 5 m2 5 m3 5 Á 5 mk
The symbol k is the number of groups or categories for an independent variable. In other words,
all group means are equal.The substantive hypothesis tested in ANOVA is:6
At least one group mean is not equal to another group mean.
As the term analysis of variance suggests, the problem requires comparing variances to make
inferences about the means.
The Papa John’s example considered locations that were standalone and shopping center, com-
pared to the categorical variable of profitable or not profitable. However, if we knew the exact
amount of profit or loss for each store, this becomes a good example of a t-test. Specifically, the
independent variable could be thought of as “location,” meaning either standalone or shopping
center. The dependent variable is the amount of profit/loss. Since only two groups exist for the
independent variable, either an independent samples t-test or one-way ANOVA could be used.
The results would be identical.This is shown in the Research Snapshot “More Than One-Way”.
However, assume further that location involved three group levels. Profit would now be com-
pared based on whether the store was standalone, shopping center, or delivery only. The t-test
would not be appropriate; one-way ANOVA would be the choice for this analysis.
Simple Illustration of ANOVA
ANOVA’s logic is fairly simple. Look at the data table below that describes how much coffee respon-
dents report drinking each day based on which shift they work (day shift, second shift, or nights).
Day 1
Day 3
Day 4
Day 0
Day 2
Second 7
Second 2
Second 1
Second 6
Night 6
Night 8
Night 3
Night 7
Night 6
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RESEARCH
SNAPSHOT
More Than One-Way
© Cengage Learning 2013
© fbmadeira/ShutterstockAn independent samples t-test is a special case of one-waynew Italian restaurant in their town. Sex2 is dummy coded so
ANOVA. When the independent variable in ANOVA has only that 0 5 men and 1 5 women. Excitement was measured on a
two groups, the results for an independent samples t-test and scale ranging from 0 to 6.
ANOVA will be the same.
The two sets of statistical results below demonstrate this
fact. Both outputs are taken from the same data. The test
c onsiders whether men or women are more excited about a
Independent Samples t-Test Results:
Group Statistics
Sex2 N Mean Std. Deviation Std. Error Mean
Excitement 0.00 69 2.64 2.262 0.272
1.00 73 2.32 2.140 0.250
Independent Samples Test
t-Test for Equality of Means
Levene’s Test 95% Confidence
for Equality of Interval of the
Variances Difference
Excitement Equal F Sig. t df Sig. Mean Std. Error Lower Upper
variances 1.768 0.186 0.873 140 (two-tailed) Difference Difference 20.408 1.053
assumed
Equal 0.384 0.323 0.369
variances
not assumed 0.872 138.265 0.385 0.323 0.370 20.409 1.054
In this case, we would conclude that men and women are equally excited—or unexcited as the case may be. The t of 0.873 with
140 df is not significant (p 5 0.384).
ANOVA Results:
Descriptives
95% Confidence Interval
for Mean
N Mean Std. Deviation Std. Error Lower Bound Upper Bound Minimum Maximum
Excitement 0.00 69 2.64 2.262 0.272 2.09 3.18 0 7
1.00 73 2.32 2.140 0.250 1.82 2.81 0 7
Total 142 2.47 2.198 0.184 2.11 2.84 0 7
ANOVA F Sig.
Sum of Squares df Mean Square
Excitement Between Groups 3.692 1 3.692 0.763 0.384
Within Groups 677.695 140 4.841
Total 681.387 141
Notice that the F-ratio shown in the ANOVA table is associated with the same p-value as is the t-value above. This is no accident since
the F and t are mathematical functions of one another. So, when two groups are involved, the researcher can skin the cat either way!
543
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•544 PART SIX Data Analysis and Presentation
The following table displays the means for each group and the overall mean:
Shift Mean Std. Deviation N
Day 2.00 1.58 5
Second 4.00 2.94 4
Night 6.00 1.87 5
Total 4.00 2.63 14
Exhibit 22.5 plots each observation with a bar.The long vertical line illustrates the total range of
observations.The lowest is 0 cups and the highest is 8 cups of coffee for a range of 8. The overall
mean is 4 cups. Each group mean is shown with a different colored line that matches the bars cor-
responding to the group.The day shift averages 2 cups of coffee a day, the second shift 4 cups, and
the night shift 6 cups of coffee per day.
Here is the basic idea of ANOVA. Look at the dark double-headed arrow in Exhibit 22.5.This
line represents the range of the differences between group means. In this case, the lowest mean
is 2 cups and the highest mean is 6 cups. Thus, the middle vertical line corresponds to the total
variation (range) in the data and the thick double-headed black line corresponds to the variance
accounted for by the group differences.As the thick black line accounts for more of the total vari-
ance, then the ANOVA model suggests that the group means are not all the same, and in particular,
not all the same as the overall mean. This also means that the independent variable, in this case
work shift, explains the dependent variable. Here, the results suggest that knowing when someone
works explains how much coffee they drink. Night-shift workers drink the most coffee.
Partitioning Variance in ANOVA
Total Variability
An implicit question with the use of ANOVA is, “How can the dependent variable best be pre-
dicted?”Absent any additional information, the error in predicting an observation is minimized by
choosing the central tendency, or mean for an interval variable. For the coffee example, if no infor-
mation was available about the work shift of each respondent, the best guess for coffee drinking
EXHIBIT 22.5
Illustration of ANOVA Logic
8 SST
SSB
Mean for the 7
Cups
night shift. 6
The overall or “grand” 5
mean, as well as
mean for second shift. 4
Mean for the 3
day shift. 2
1 © Cengage Learning 2013
0
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•CHAPTER 22 Bivariate Statistical Analysis: Differences Between Two Variables 545
consumption would be 4 cups.The sum of squares total (SST) or variability that would result from grand mean
using the grand mean, meaning the mean over all observations, can be thought of as
The mean of a variable over all
SST 5 Total of (Observed Value 2 Value Grand Mean)2 observations.
Although the term error is used, this really represents how much total variation exists among the
measures.
Using the first observation, the error of observation would be
(1 cup 2 4 cups)2 5 9
The same squared error could be computed for each observation and these squared errors totaled
to give SST.
Between-Groups Variance
ANOVA tests whether “grouping” observations explains variance in the dependent variable. In between-groups variance
Exhibit 22.5, the three colors reflect three levels of the independent variable, work shift. Given
this additional information about which shift a respondent works, the prediction changes. Now, The sum of differences
instead of guessing the grand mean, the group mean would be used. So, once we know that some- between the group mean and
one works the day shift, the prediction would be that he or she consumes 2 cups of coffee per day. the grand mean summed over
Similarly, the second and night-shift predictions would be 4 and 6 cups, respectively. Thus, the all groups for a given set of
between-groups variance or sum of squares between groups (SSB) can be found by taking the observations.
total sum of the weighted difference between group means and the overall mean as shown:
SSB 5 Total of ngroup(Group Mean 2 Grand Mean)2
The weighting factor (ngroup) is the specific group sample size. Let’s consider the first observa-
tion once again. Since this observation is in the day shift, we predict 2 cups of coffee will be con-
sumed. Looking at the day shift group observations in Exhibit 22.5, the new error in prediction
would be
(2 cups 2 4 cups)2 5 (2)2 5 4
The error in prediction has been reduced from 3 using the grand mean to 2 using the group
mean.This squared difference would be weighted by the group sample size of 5, to yield a contri-
bution to SSB of 20.
Next, the same process could be followed for the other groups yielding two more contribu-
tions to SSB. Because the second shift group mean is the same as the grand mean, that group’s
contribution to SSB is 0. Notice that the night-shift group mean is also 2 different than the grand
mean, like the day shift, so this group’s contribution to SSB is likewise 20. The total SSB then
represents the variation explained by the experimental or independent variable. In this case, total
SSB is 40.The reader may look at the statistical results shown in Exhibit 22.6 to find this value in
the sums of squares column.
Within-Group Error within-group error or
variance
Finally, error within each group would remain. Whereas the group means explain the variation
between the total mean and the group mean, the distance from the group mean and each indi- The sum of the differences
vidual observation remains unexplained.This distance is called within-group error or variance or between observed values and
the sum of squares error (SSE).The values for each observation can be found by the group mean for a given set
of observations; also known as
SSE 5 Total of (Observed Mean 2 Group Mean)2 total error variance.
Again, looking at the first observation, the SSE component would be
SSE 5 (1 cup 2 2 cups)2 5 1 cup
This process could be computed for all observations and then totaled. The result would be the total
error variance—a name used to refer to SSE since it is variability not accounted for by the group
means.These three components are used in determining how well an ANOVA model explains a
dependent variable.
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•546 PART SIX Data Analysis and Presentation
EXHIBIT 22.6
Interpreting ANOVA
Tests of Between-Subjects Effects (Dependent Variable: Coffee)
Source Type III Sum Mean F Sig.
of Squares d.f. Square
Corrected Model 40.000a 2 20.000 4.400 .039 1. This row shows overall F-value
testing whether all group means
Intercept 221.538 1 221.538 48.738 .000 are equal. The sum of squares
column calculates the SST, SSE, and
Shift 40.000 2 20.000 4.400 .039 SSB (shift row).
Error 50.000 11 4.545 2. This column shows the group
means for each level of the
Total 314.000 14 independent variable.
aR Squared 5 .444 (Adjusted R Squared 5 .343)
95% Confidence Interval
Shift Mean Std. Error Lower Bound Upper Bound
Day 2.000 .953
Second 4.000 1.066 2.099 4.099 © Cengage Learning 2013
Night 6.000 .953 1.654 6.346
3.901 8.099
The F-Test
F-test The F-test is the key statistical test for an ANOVA model.The F-test determines whether there is
more variability in the scores of one sample than in the scores of another sample.The key ques-
A procedure used to determine tion is whether the two sample variances are different from each other or whether they are from
whether there is more the same population.Thus, the test breaks down the variance in a total sample and illustrates why
ANOVA is analysis of variance.
variability in the scores of one
sample than in the scores of The F-statistic (or F-ratio) can be obtained by taking the larger sample variance and divid-
another sample. ing by the smaller sample variance. Using Table A.5 or A.6 in the appendix is much like using the
tables of the Z- and t-distributions that we have previously examined. These tables portray the
F-distribution, which is a probability distribution of the ratios of sample variances. These tables
indicate that the distribution of F is actually a family of distributions that change quite drastically
with changes in sample sizes.Thus, degrees of freedom must be specified. Inspection of an F-table
allows the researcher to determine the probability of finding an F as large as a calculated F.
Using Variance Components to Compute F-Ratios
In ANOVA, the basic consideration for the F-test is identifying the relative size of variance com-
ponents.The three forms of variation described briefly above are:
1. SSE—variation of scores due to random error or within-group variance due to individual dif-
ferences from the group mean. This is the error of prediction.
2. SSB—systematic variation of scores between groups due to manipulation of an experimental
variable or group classifications of a measured independent variable or between-groups
variance.
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•CHAPTER 22 Bivariate Statistical Analysis: Differences Between Two Variables 547
3. SST—the total observed variation across all groups and individual observations.
Thus, we can partition total variability into within-group variance (SSE) and between-groups total variability
v ariance (SSB).
The sum of within-group
The F-distribution is a function of the ratio of these two sources of variances: variance and between-groups
variance.
SSB
F 5 f a SSE b
A larger ratio of variance between groups to variance within groups implies a greater value of F. If
the F-value is large, the results are likely to be statistically significant.
A Different but Equivalent Representation
F also can be thought of as a function of the between-groups variance and total variance:
F 5 f SSB
a SST 2 SSB b
In this sense, the ratio of the thick black line to the middle line representing the total range of data
presents the basic idea of the F-value.Appendix 22A explains the calculations in more detail with
an illustration.
Practically Speaking
Exhibit 22.6 displays the ANOVA result for the coffee-drinking example. Again, one advantage of
living in modern times is that even a simple problem like this one need not be hand computed.
Even though this example presents a small problem, one-way ANOVA models with more observa-
tions or levels would be interpreted similarly.
The first thing to check is whether or not the overall model F is significant. In this case, the
computed F 5 4.40 with 2 and 11 degrees of freedom.The p-value associated with this value is
0.039. Thus, we have high confidence in concluding that the group means are not all the same.
Second, the researcher must remember to examine the actual means for each group to properly
interpret the result. Doing so, the conclusion reached is that the night-shift people drink the most
coffee, followed by the second-shift workers, and then lastly, the day-shift workers.
As there are three groups, we may wish to know whether or not group 1 is significantly dif-
ferent than group 3 or group 2, and so on. In a later chapter, we will describe ways of examining
specifically which group means are different from one another.
SUMMARY
1. Recognize when a particular bivariate statistical test is appropriate. Bivariate statistical
techniques analyze scores on two variables at a time. Tests of differences investigate hypotheses
stating that two (or more) groups differ with respect to a certain behavior, characteristic, or
attitude. Both the type of measurement and the number of groups to be compared influence
researchers’ choices of the type of statistical test. When both variables are less-than interval level,
a contingency table and a x2 test are appropriate. When one variable is less-than interval with
two levels and the other variable is interval or ratio level, a t-test is appropriate. When one vari-
able is less-than interval with three or more levels and the other variable is interval or ratio level,
ANOVA is the appropriate statistical technique.
2. Calculate and interpret a x2 test for a contingency table. A x2 test is used in conjunction
with cross-classification or cross-tabs. Thus, when an independent variable is ordinal or nominal
and a dependent variable is likewise ordinal or nominal, a x2 test can examine whether a rela-
tionship exists between the row variable and column variable. A x2 test is computed by examin-
ing the squared differences between observed cell counts and the expected value for each cell
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548 Part 6: Data Analysis and Presentation
in a contingency table. Higher x2 values are generally associated with lower p-values, meaning
there is a greater chance that the relationship between the row and column variable is statistically
significant.
3. Calculate and interpret an independent samples t-test comparing two means. When a
researcher needs to compare means for a variable grouped into two categories based on some
less-than interval variable, a t-test is appropriate. An independent samples t-test examines whether
a dependent variable like job satisfaction differs based on a grouping variable like biological sex.
Statistically, the test examines whether the difference between the mean for men and women is
different from 0. A paired-samples t-test examines whether or not the means from two variables
that are not independent are different. A common situation calling for this test is when the two
observations are from the same respondent. A simple before-and-after test calls for a paired-
samples t-test so long as the dependent variable is continuous.
4. Understand the concept of analysis of variance (ANOVA). ANOVA is the appropriate sta-
tistical technique to examine the effect of a less-than interval independent variable with three
or more categories on an at-least interval dependent variable. Conceptually, ANOVA partitions
the total variability into three types: total variation, between-groups variation, and within-group
variation. As the explained variance represented by SSB becomes larger relative to SSE or SST, the
ANOVA model is more likely to be significant, indicating that at least one group mean is different
from another group mean.
5. Interpret an ANOVA table. An ANOVA table provides essential information. Most impor-
tantly, the ANOVA table contains the model F-ratio. The researcher should examine this value
along with the corresponding p-value. Generally, as F increases, p decreases, meaning that a
statistically significant ANOVA model is more likely.
KEY TERMS AND CONCEPTS
analysis of variance (ANOVA), 542 independent samples t-test, 534 total variability, 547
between-groups variance, 545 paired-samples t-test, 539 within-group error or variance, 545
F-test, 546 pooled estimate of the standard error, 535 Z-test for differences of
grand mean, 545 tests of differences, 529
proportions, 540
QUESTIONS FOR REVIEW AND CRITICAL THINKING
1. What tests of difference are appropriate in the following 2. Perform a x2 test on the following data:
situations? a. Regulation is the best way to ensure safe products.
a. Average contributions (in $) to a marketing campaign for
healthy living for men and women are to be compared. Agree Disagree No Opinion
b. Average contributions (in $) to a marketing campaign for
healthy living for people who are 20–30 years old, 30–40 Managers 58 66 8
years old, and 40–60 years old are to be compared. Line Employees 34 24 10
c. Human resource managers and chief executive officers Totals 92 90 18
have responded “yes,” “no,” or “not sure” to an attitude
q uestion. The HR and CEO responses are to be compared. b. Ownership of residence
d. One-half of a sample received an incentive in a mail survey
while the other half did not. A comparison of response rates Yes No
is desired. 20
e. A researcher believes that married men will push the Male 25 14
grocery cart when grocery shopping with their wives. How Female 16
would the hypothesis be tested?
f. A manager wishes to compare the job performance of a
salesperson before ethics training with the performance of
that same salesperson after ethics training.
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Chapter 22: Bivariate Statistical Analysis: Differences Between Two Variables 549
3. Interpret the following computer cross-tab output including a x2 test. Variable COMMUTE is “How did you get to work last week?”
Variable GENDER is “Are you male or female?” Comment on any particular problems with the analysis.
COMMUTE * GENDER Cross-Tabulation
GENDER
Female Male Total
COMMUTE At Home Count 6 10 16
Total Bus % within COMMUTE 37.5% 62.5% 100.0%
Drive % within GENDER 17.9%
Passenger % of Total 7.0% 11.3%
Walk Count 4.2% 7.0% 11.3%
% within COMMUTE 16
% within GENDER 16 32
% of Total 50.0% 50.0% 100.0%
Count 18.6% 28.6%
% within COMMUTE 11.3% 11.3% 22.5%
% within GENDER 22.5%
% of Total 32 17
Count 65.3% 34.7% 49
% within COMMUTE 37.2% 30.4% 100.0%
% within GENDER 22.5% 12.0%
% of Total 34.5%
Count 24 9 34.5%
% within COMMUTE 72.7% 27.3%
% within GENDER 27.9% 16.1% 33
% of Total 16.9% 100.0%
Count 6.3%
% within COMMUTE 8 4 23.2%
% within GENDER 66.7% 23.3%
% of Total 33.3%
9.3% 7.1% 12
5.6% 2.8% 100.0%
56
86 8.5%
60.6% 39.4% 8.5%
100.0% 100.0%
60.6% 142
39.4% 100.0%
100.0%
100.0%
x2 Tests H1: Internet retailers offer lower prices for Blu-ray players than do
traditional in-store retailers.
Value df Asymp. Sig. Blu-ray Player
(two-sided) Average
Price
Pearson Chi-Square 7.751a 4 0.101 Retail Type Standard n
4 0.102 Deviation 25
25
Likelihood Ratio 7.725
N of Valid Cases 142 Internet Retailers $109.95 $20.00
Multichannel Retailers $159.30 $45.00
a1 cells (10.0%) have expected count less than 5. The minimum expected count is 4.73.
4. A store manager’s computer-generated list of all retail sales 6. How does an independent sample t-test differ from the
employees indicates that 70 percent are full-time employees, following?
20 percent are part-time employees, and 10 percent are fur- a. one-way ANOVA
loughed or laid-off employees. A sample of 50 employees from b. paired-samples t-test
the list indicates that there are 40 full-time employees, 6 part- c. a x2 test
time employees, and 4 furloughed/laid-off employees. Conduct d. a Z-test for differences
a statistical test to determine whether the sample is representa-
tive of the population. 7. Are t-tests or Z-tests used more often in business research? Why?
8. A sales force received some management-by-objectives training.
5. Test the following hypothesis using the data summarized in the
table below. Interpret your result: Are the before/after mean scores for salespeople’s job perfor-
mance statistically significant at the 0.05 level? The results from
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550 Part 6: Data Analysis and Presentation
a sample of employees are as follows (use your computer and 10. In an experiment with wholesalers, a researcher
statistical software to solve this problem): manipulated perception of task difficulty and measured
level of aspiration for performing the task a second time.
Skill Skill Skill Skill Group 1 was told the task was very difficult, group 2 was told
Name Before After Name Before After the task was somewhat difficult but attainable, and group 3
was told the task was easy. Perform an ANOVA on the
Ed 4.84 5.43 Kathy 4.00 5.00 resulting data:
Mark 5.24 5.51 Susie 4.67 4.50
Jason 5.37 5.42 Ron 4.95 4.40 Level of Aspiration (10-Point Scale)
Raj 3.69 4.50 Jen 4.00 5.95
Heidi 5.95 5.90 Matt 3.75 3.50 Subjects Group 1 Group 2 Group 3
Donna 4.75 5.25 Doug 3.85 4.00
Rob 3.90 4.50 Bob 5.00 4.10 1655
2746
3575
9. Conduct a Z-test to determine whether the following two 4864
samples indicate that the population proportions are significantly
different at the 0.05 level: 5872
6673
Sample 1 Sample 2 Cases 6 6 6
Sample Proportion 0.77 0.68
Sample Size 55 46
11. Interpret the following output examining group differences for purchase intentions. The three groups refer to consumers from three
states: Illinois, Louisiana, and Texas.
Tests of Between-Subjects Effects
Dependent Variable: int2
Source Type III Sum of Squares df Mean Square F Sig.
Corrected Model 6681.746a 2 3340.873 3.227 0.043
Intercept 308897.012 1 308897.012 298.323 0.000
State 2 3340.873 0.043
Error 6681.746 143 1035.444 3.227
148068.543 146
145
Total 459697.250
Corrected Total 154750.289
aR Squared 5 0.043 (Adjusted R Squared 5 0.030)
Law 95% Confidence Interval
Dependent Variable: int2
State Mean Lower Bound Upper Bound
Std. Error
IL 37.018
LA 50.357 4.339 28.441 45.595
TX 51.459 4.965 40.542 60.172
4.597 42.373 60.546
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Chapter 22: Bivariate Statistical Analysis: Differences Between Two Variables 551
RESEARCH ACTIVITIES
1. ETHICS/’NET How ethical is it to do business in different coun- b. Are there differences among the corruption indices in the
tries around the world? An international organization, Trans-
parency International, keeps track of the perception of ethical past five years (between 2003 and 2008)?
practices in different countries and computes the corruption 2. ‘NET The Bureau of Labor Statistics of the United States
perceptions index (CPI). Visit the website and search for the
latest CPI (http://www.transparency.org/policy_research/ Government maintains a website that contains current and his-
surveys_indices/cpi). Using the data found here, test the fol- toric data related to the United States economy. Navigate to the
lowing research questions. BLS website at http://www.bls.gov. In the “Latest Numbers”
a. Are nations from Europe and North America perceived to be section, select the small dinosaur next to the Unemployment
more ethical than nations from Asia, Africa, and South America? Rate link. Select a year and its 12-month unemployment data,
and then select the most current year for comparison. What
statistical tests are appropriate?
Old School versus New School Sports Fans
CASE Three academic researchers investigated the idea that, notions of the team before the player, sportsmanship, and loyalty
22.1 in American sports, there are two segments with above all else, and competition simply for “love of the game.”
opposing views about the goal of competition (i.e., New school/old school was measured by asking agreement with
winning versus self-actualization) and the acceptable/desirable way ten attitude statements.The scores on these statements were com-
of achieving this goal.7 Persons who believe in “winning at any cost” bined. Higher scores represent an orientation toward old school
are proponents of sports success as a product and can be labeled new values. For purposes of this case study, individuals who did not
school (NS) individuals. The new school is founded on notions of answer every question were eliminated from the analysis. Based on
the player before the team, loyalty to the highest bidder, and high- their summated scores across the 10 items, respondents were grouped
tech production and consumption of professional sports. On the into low score, middle score, and high score groups. Exhibit 22.1-1
other hand, persons who value the process of sports and believe that shows the SPSS computer output of a cross-tabulation to relate the
“how you play the game matters” can be labeled old school (OS) gender of the respondent (GENDER) with the new school/old
individuals. The old school emerges from old-fashioned American school grouping (OLDSKOOL).
CASE EXHIBIT 22.1-1
SPSS Output
OLDSKOOL * GENDER Cross-Tabulation
GENDER Men Total
Women
OLDSKOOL high Count 9 17 26
% within OLDSKOOL 34.6% 65.4% 100.0%
low % within GENDER 10.6%
% of Total 9.2% 9.6%
middle 3.3% 6.3% 9.6%
Count
Total % within OLDSKOOL 45 70 115 © Cengage Learning 2013
% within GENDER 39.1% 60.9% 100.0%
% of Total 52.9% 37.8%
16.7% 25.9% 42.6%
Count 42.6%
% within OLDSKOOL 31 98
% within GENDER 24.0% 76.0% 129
% of Total 36.5% 53.0% 100.0%
11.5% 36.3%
Count 47.8%
% within OLDSKOOL 85 185 47.8%
% within GENDER 31.5% 68.5%
% of Total 100.0% 100.0% 270
31.5% 68.5% 100.0%
100.0%
100.0%
(Continued )
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552 Part 6: Data Analysis and Presentation
CASE EXHIBIT 22.1-1 (Continued)
SPSS Output
Chi-Square Tests
Value df Asymp. Sig. (two-sided)
Pearson Chi-Square 6.557a 2 .038
Likelihood Ratio 6.608 2 .037
N of Valid Cases 270
a0 cells (.0%) have expected count less than 5. The minimum expected count is 8.19.
Questions 2. Is the analytical approach used here appropriate?
3. Describe an alternative approach to the analysis of the original
1. Interpret the computer output. What do the results presented
above indicate? data. Which of these two analyses would you suggest using?
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APPENDIX 22A
Manual Calculation of an
F-Statistic
Manual calculations are almost unheard of these days. However, understanding the calculations
can be very useful in gaining a thorough understanding of ANOVA. The data in Exhibit 22A-1
are from a hypothetical packaged-goods company’s test-market experiment on pricing. Three
pricing treatments were administered in four separate areas (12 test areas, A–L, were required).
These data will be used to illustrate ANOVA.
Terminology for the variance estimates is derived from the calculation procedures, so an expla-
nation of the terms used to calculate the F-ratio should clarify the meaning of the analysis of variance
technique.The calculation of the F-ratio requires that we partition the total variation into two parts:
Within-group Between-groups
Total sum of squares 5 sum of squares 1 sum of squares
(SST) (SSE) (SSB)
or
SST 5 SSE 1 SSB
SST is computed by squaring the deviation of each score from the grand mean and summing these
squares: OOn c
SST 5 (Xij 2 X )2
i51 j51
Sales in Units (thousands) EXHIBIT 22A-1
Regular Price, Reduced Price, Cents-Off Coupon, A Test-Market Experiment on
$.99 $.89 Regular Price Pricing
Test-Market A, B, or C 130 145 153 © Cengage Learning 2013
Test-Market D, E, or F 118 143 129
Test-Market G, H, or I 120
Test-Market J, K, or L 87 131 96
Mean 84 X2 5 134.75 99
Grand Mean X1 5 104.75 X3 5 119.25
X 5 119.58
553
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554 Part 6: Data Analysis and Presentation
where
X 5 individual score—that is, the ith observation or test unit in the jth group
X 5 grand mean
n 5 number of all observations or test units in a group
c 5 number of jth groups (or columns)
In our example,
SST 5 (130 2 119.58)2 1 (118 2 119.58)2 1 (87 2 119.58)2
1 (84 2 119.58)2 1 (145 2 119.58)2 1 (143 2 119.58)2
1 (120 2 119.58)2 1 (131 2 119.58)2 1 (153 2 119.58)2
1 (129 2 119.58)2 1 (96 2 119.58)2 1 (99 2 119.58)2
5 5,948.93
SSE, the variability that we observe within each group, or the error remaining after using the
groups to predict observations, is calculated by squaring the deviation of each score from its group
mean and summing these scores: OOn c
SSE 5 (Xij 2 Xj)2
i51 j51
where
X 5 individual score
Xj 5 group mean for the jth group
n 5 number of observations in a group
c 5 number of jth groups
In our example,
SSE 5 (130 2 104.75)2 1 (118 2 104.75)2 1 (87 2 104.75)2
1 (84 2 104.75)2 1 (145 2 134.75)2 1 (143 2 134.75)2
1 (120 2 134.75)2 1 (131 2 134.75)2 1 (153 2 119.25)2
1 (129 2 119.25)2 1 (96 2 119.25)2 1 (99 2 119.25)2
5 4,148.25
SSB, the variability of the group means about a grand mean, is calculated by squaring the
deviation of each group mean from the grand mean, multiplying by the number of items in the
group, and summing these scores: Oc
SSB 5 nj(Xj 2 X )2
where j51
Xj 5 group mean for the jth group
X 5 grand mean
nj 5 number of items in the jth group
In our example,
SSB 5 4(104.75 2 119.58)2 1 4(134.75 2 119.58)2
1 4(119.25 2 119.58)2
5 1,800.68
The next calculation requires dividing the various sums of squares by their appropriate degrees
of freedom. These divisions produce the variances, or mean squares. To obtain the mean square
between groups, we divide SSB by c – 1 degrees of freedom:
MSB 5 SSB
c21
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Chapter 22: Bivariate Statistical Analysis: Differences Between Two Variables 555
In our example,
MSB 5 1,800.68 5 1,800.68 5 900.34
321 2
To obtain the mean square within groups, we divide SSE by cn – c degrees of freedom:
MSE 5 SSE
cn 2 c
In our example,
MSE 5 4,148.25 5 4,148.25 5 460.91
12 2 3 9
Finally, the F-ratio is calculated by taking the ratio of the mean square between groups to the
mean square within groups. The between-groups mean square is the numerator and the within-
groups mean square is the denominator:
F 5 MSB
MSE
In our example,
F 5 900.34 5 1.95
460.91
There will be c – 1 degrees of freedom in the numerator and cn – c degrees of freedom in the
denominator:
c21 5 321 5 2
cn 2 c 3(4) 2 3 9
In Table A.5 in the text appendix, the critical value of F at the 0.05 level for 2 and 9 degrees of
freedom indicates that an F of 4.26 would be required to reject the null hypothesis.
In our example, we conclude that we cannot reject the null hypothesis. It appears that all the
price treatments produce approximately the same sales volume.
The information produced from an analysis of variance is traditionally summarized in table
form. Exhibits 22A-2 and 22A-3 summarize the formulas and data from our example.
Source of Sum of Squares Degrees of Mean Square F-Ratio EXHIBIT 22A-2
Variation Freedom
Between groups ANOVA Summary Table
Within groups
Total Oc c21 MSB 5 SSB —
cn 2 c c21
SSB 5 nj(Xj 2 X)2 cn 2 1
OOj51
nc SSE MSB
cn 2 MSE
SSE 5 (Xij 2 Xj)2 MSE 5 c F 5
OOi51 j51nc © Cengage Learning 2013
SST 5 (Xij 2 Xj)2 — —
i51 j51
where c 5 number of groups
n 5 number of observations in a group
Source of Sum of Squares Degrees of Mean Square F-Ratio EXHIBIT 22A-3
Variation 1,800.68 Freedom 900.34 —
Between groups 4,148.25 460.91 Pricing Experiment ANOVA
Within groups 5,948.93 2 — 1.953 Table
Total 9 —
11 © Cengage Learning 2013
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APPENDIX 22B
ANOVA for Complex
Experimental Designs
To test for statistical significance in a randomized block design, or RBD (see Chapter 12), another
version of analysis of variance is utilized.The linear model for the RBD for an individual observa-
tion is*
Yij 5 m + aj + bi + ´ij
where
Yij 5 individual observation on the dependent variable
m 5 grand mean
aj 5 jth treatment effect
bi 5 ith block effect
´ij 5 random error or residual
The statistical objective is to determine whether significant differences exist among treatment
means and block means.This is done by calculating an F-ratio for each source of effects.
The same logic that applies in single-factor ANOVA—using variance estimates to test for dif-
ferences among means—applies in ANOVA for randomized block designs. Thus, to conduct the
ANOVA, we partition the total sum of squares (SStotal) into non-overlapping components.
SStotal 5 SStreatments 1 SSblocks 1 SSerror
The sources of variance are defined as follows.
Total sum of squares: OOr c
where
SStotal 5 (Yij 2 Y )2
i51 j51
Yij 5 individual observation
Y 5 grand mean
r 5 number of blocks (rows)
c 5 number of treatments (columns)
*We assume no interaction effect between treatments and blocks.
556
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Chapter 22: Bivariate Statistical Analysis: Differences Between Two Variables 557
Treatment sum of squares: OOr
SStreatments 5 c
(Yj 2 Y )2
i51 j51
where
Yj 5 jth treatment mean
Y 5 grand mean
Block sum of squares: OOr c
where
SSblocks 5 (Yi 2 Y )2
i51 j51
Yi 5 ith block mean
Y 5 grand mean
Sum of squares error: OOr c
SSerror 5 (Yij 2 Yi 2 Yj 2 Y )2
i51 j51
The SSerror may also be calculated in the following manner:
SSerror 5 SStotal 2 SStreatments 2 SSblocks
The degrees of freedom for SStreatments are equal to c – 1 because SStreatments reflects the dispersion
of treatment means from the grand mean, which is fixed. Degrees of freedom for blocks are
r – 1 for similar reasons. SSerror reflects variations from both treatment and block means.Thus, df 5
(r – 1)(c – 1).
Mean squares are calculated by dividing the appropriate sum of squares by the corresponding
degrees of freedom.
Exhibit 22B-1 is an ANOVA table for the randomized block design. It summarizes what has
been discussed and illustrates the calculation of mean squares.
Source of Variation Sum of Squares Degrees of Mean Squares EXHIBIT 22b-1
Between blocks SSblocks Freedom
Between treatments SSblocks ANOVA Table for Randomized
Error SStreatments r21 r21 Block Designs
Total SSerror c21
SStotal (r 2 1)(c 2 1) SStreatments © Cengage Learning 2013
rc 2 1 c21
SSerror
(r 2 1)(c 2 1)
—
F-ratios for treatment and block effects are calculated as follows:
Ftreatment 5 Mean square treatment
Mean square error
Fblocks 5 Mean square blocks
Mean square error
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558 Part 6: Data Analysis and Presentation
Factorial Designs
There is considerable similarity between the factorial design (see Chapter 12) and the one-way
analysis of variance. The sum of squares for each of the treatment factors (rows and columns) is
similar to the between-groups sum of squares in the single-factor ANOVA model. Each treatment
sum of squares is calculated by taking the deviation of the treatment means from the grand mean.
Determining the sum of squares for the interaction is a new calculation because this source of vari-
ance is not attributable to the treatment sum of squares or the error sum of squares.
ANOVA for a Factorial Experiment
In a two-factor experimental design the linear model for an individual observation is
Yijk5m+ bi +aj+Iij+´ijk
where
Yijk 5 individual observation on the dependent variable
m 5 grand mean
bi 5 ith effect of factor B—row treatment
aj 5 jth effect of factor A—column treatment
Iij 5 interaction effect of factors A and B
´ijk 5 random error or residual
Partitioning the Sum of Squares for a Two-Way ANOVA
Again, the total sum of squares can be allocated into distinct and overlapping portions:
Sum of Sum of Sum of squares Sum of Sum of
squares = squares rows + columns + squares + squares
total (treatment B) (treatment A) interaction error
or
SStotal 5 SSRtreatment B 1 SSCtreatment A 1 SSinteraction 1 SSerror
Sum of squares total: SStotal 5 OOOr c n (Yijk 2 Y )2
where
i51 j51 k51
Yijk 5 individual observation on the dependent variable
Y 5 grand mean
j 5 level of factor A
i 5 level of factor B
k 5 number of an observation in a particular cell
r 5 total number of levels of factor B (rows)
c 5 total number of levels of factor A (columns)
n 5 total number of observations in the sample
Sum of squares rows (treatment B): Or
SSRtreatment B 5 (Yi 2 Y )2
i51
where
Yj 5 mean of ith treatment—factor B
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Chapter 22: Bivariate Statistical Analysis: Differences Between Two Variables 559
OSum of squares columns (treatment A): c
SSCtreatment A 5 (Yj 2 Y )2
j51
where
Yj 5 mean of jth treatment—factor A
OOOSum of squares interaction:
r cn
SSinteraction 5 (Yij 2 Yi 2 Yj 2 Y )2
i51 j51 k51
The above is one form of calculation. However, SSinteraction generally is indirectly computed in
the following manner:
SSinteraction 5 SStotal 2 SSRtreatment B 2 SSCtreatment A 2 SSerror
Sum of squares error: SSerror 5 OOOr c n (Yijk 2 Yij)2
where
i51 j51 k51
Yij 5 mean of the interaction effect
These sums of squares, along with their respective degrees of freedom and mean squares, are sum-
marized in Exhibit 22B-2.
Source of Sum of Squares Degrees of Mean Square F-Ratio EXHIBIT 22B-2
Variation Freedom
Treatment B SSRtreatment B MSRtreatment B ANOVA Table for Two-Factor
Treatment A SSCtreatment A r21 MSerror Design
Interaction SSinteraction c21
Error (r 2 1)(c 2 1) MSRtreatment B 5 SSRtreatment B MSCtreatment A
Total SSerror rc(n 2 1) r21 MSerror
SStotal rcn 2 1
MSCtreatment A 5 SSCtreatment A MSinteraction
c21 MSerror
MSinteraction 5 (r SSinteraction 1)
2 1)(c 2
MSerror 5 SSerror © Cengage Learning 2013
rc(n 2 1)
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23CHAPTER Bivariate Statistical
Analysis
Measures of Association
LEARNING OUTCOMES
After studying this chapter, you should be able to
1. Apply and interpret simple bivariate correlations
2. Interpret a correlation matrix
3. Understand simple (bivariate) regression
4. Understand the least-squares estimation technique
5. Interpret regression output including the tests of hypotheses tied to
specific parameter coefficients
Chapter Vignette: © Vicki Beaver
Bringing Your Work to Your Home
(and Bringing Your Home to Work)
D o you “bring work home?” Do your family
experiences and demands affect your work
responsibilities? When you think about the
stress you may have faced in a particular work
situation, it is easy to see how this may, in fact, be the case.
For many years, there was a belief that employees universally
separated their work roles from their home/family roles. This
belief generally centered on the idea that “what happens at
home” doesn’t matter in the workplace. As an employee, you
were simply there to perform your job, and your family was
not part of those responsibilities. Likewise, the responsibilities
you have with your family were not affected by work—you left
those challenges and stresses “at work.” Our understanding
of the work and family interface has changed substantially in
recent years.
The idea that work roles and family roles could be at odds
with one another is nowadays referred to as work-family con-
flict (WFC).1 It is typically defined as conflict that results when
the demands and responsibilities of one role “spill over” into
the other role. For example, it is easy to see how a manager,
simultaneously facing a project deadline and a family reunion in
the same week, may allow some of his or her frustrations and
stress to affect one (or both) of these work and family roles.
560
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•CHAPTER 23 Bivariate Statistical Analysis: Measures of Association 561
There are numerous ways that WFC can be created for an variables) that can predict WFC (a dependent variable), with the
employee. Work demands can include inflexible schedules, goal of providing insights into the causes and consequences
project timelines, and even an abusive supervisor. With regard of this phenomenon.3 Think about your own work or school
to family demands, any number of family demands can “spill and how it interferes with your family obligations. What would
over” into the work role, to include child or elder care and dual be some of the demands on you that lead to WFC? How have
career relationships.2 these demands affected your job satisfaction and/or family
harmony? It is not hard to see that “bringing home work” is a
Researchers have begun to examine and explore the many common problem we face in today’s society.
different work and family characteristics (i.e. independent
Introduction
When business researchers develop and implement a research survey, it is often conducted with
several goals in mind. Most important, however, is the goal of answering a particular research
question, using the data to justify the result. Finding the “answer” is critically important in this
regard. This chapter is designed to familiarize you with how bivariate business statistics are
used to help accomplish this task. With bivariate analysis we often times distinguish between
the independent variable (the cause) and the dependent variable (the effect).The mathematical
symbol X is commonly used for an independent variable, and Y typically denotes a dependent
variable.
The Basics
Business research involves many different professional disciplines. For management, the ques- measure of association
tions regarding conflict, employee satisfaction, and employee turnover are often key dependent A general term that refers to a
variables of interest. As the chapter vignette outlines, this question can be a complex one, with number of bivariate statistical
many different work and family independent variables affecting work-family conflict. In market- techniques used to measure
ing, sales volume is often the dependent variable managers want to predict. Independent variables the strength of a relationship
are typically marketing mix elements such as price, number of salespeople, and the amount of between two variables.
advertising, that are related to sales volume. In finance, earnings reports are heavily linked with
stock prices. correlation coefficient
A statistical measure of the
Exhibit 23.1 shows that measurement characteristics influence which measure of associa- covariation, or association,
tion is most appropriate. The chi-square (x2) test provides information about whether two or between two at-least interval
more less-than interval variables are interrelated. For example, a x2 test between the color of variables.
the package and the product chosen for purchase provides information about the independence covariance
or interrelationship of the two variables. This chapter also describes simple correlation (Pear- The extent to which two
son’s product-moment correlation coefficient, r) and bivariate or simple regression analysis. variables are associated
Correlation analysis is most appropriate for interval or ratio variables. While regression can systematically with each other.
accommodate less-than interval independent variables, the dependent variable must be con-
tinuous. Over the years, psychological statisticians have developed several other techniques that
demonstrate empirical association. These techniques are for advanced students who have
specific needs.4
Simple Correlation Coefficient
The most popular technique for indicating the relationship of one variable to another is correla-
tion. A correlation coefficient is a statistical measure of covariation, or association between two
variables. Covariance is the extent to which a change in one variable corresponds systematically to
a change in another. Correlation can be thought of as a standardized covariance.
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
STUHRVISEY! Courtesy of Qualtrics.com
The Survey This! feature contains many
variables that you might think would
be related to a particular outcome for
a person, such as satisfaction. Based
upon the variables list, do the following:
(1) Choose three variables (independent
variables) that you think would predict
satisfaction (dependent variable);
(2) conduct a bivariate correlation analy-
sis for all of your selected variables—do
they show the correct sign (positive or negative)? Are they significantly related? (3) Using those same independent and
dependent v ariables, c onduct a simple regression analysis. What do you find?
When correlations estimate relationships between continuous variables, the Pearson
p roduct-moment correlation is appropriate. The correlation coefficient, r, ranges from 21.0
to 11.0. If the value of r equals 11.0, a perfect positive relationship exists. Perhaps the two
variables are one and the same! If the value of r equals 21.0, a perfect negative relationship
exists. The implication is that one variable is a mirror image of the other. As one goes up,
the other goes down in proportion and vice versa. No correlation is indicated if r equals 0.
EXHIBIT 23.1 Association for 2
Variables
Bivariate Analysis—Common
Procedures for Testing
Association
What is the
measurement
level?
Nominal Ordinal Interval/Ratio
Example: Does Example: Does Example: Does number
nationality relate to color importance ranking of radio spots relate
relate to brand choice? to unit sales?
preference? Statistical Choice:
Pearson’s r or simple
Statistical Choice: Statistical Choice: regression © Cengage Learning 2013
Cross-tabulation with Cross-tabulation with 2 test
or Spearman rank correlation
2 test
562
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•CHAPTER 23 Bivariate Statistical Analysis: Measures of Association 563
A correlation coefficient indicates both the magnitude of the linear relationship and the direc-
tion of that relationship. For example, if we find that r 5 20.92, we know we have a very strong
inverse relationship—that is, the greater the value measured by variable X, the lower the value
measured by variable Y.
The formula for calculating the correlation coefficient for two variables X and Y is as follows:
O(Xi 2 X )(Yi 2 Y )
n
rxy 5 ryx 5 O OA i51 (Xi 2 X )2 i51 (Yi 2 Y )2i51n
n
where the symbols X and Y represent the sample averages of X and Y, respectively. An alternative
way to express the correlation formula is
rxy 5 ryx 5 sxy
2sx2s2y
where
s 2 5 variance of X
x
s 2 5 variance of Y
y
sxy 5 covariance of X and Y
with On
(Xi 2 X )(Yi 2 Y )
sxy 5 i51 n
If associated values of Xi and Yi differ from their means in the same direction, their covariance will ‘‘Statistics are like
be positive. If the values of Xi and Yi tend to deviate in opposite directions, their covariance will
be negative. a bikini. What they
reveal is suggestive,
The Pearson correlation coefficient is a standardized measure of covariance. Covariance coef-
ficients retain information about the absolute scale ranges so that the strength of association for ’’but what they
scales of different possible values cannot be compared directly. Researchers find the correlation
coefficient useful because they can compare two correlations without regard for the amount of conceal is vital.
variance exhibited by each variable separately.
—AARON Levenstein
Exhibit 23.2 illustrates the correlation coefficients and scatter diagrams for several sets of data.
Notice that in the no-correlation condition, the observations are scattered rather evenly about inverse (negative)
the space. In contrast, when correlations are strong and positive, the observations lie mostly in relationship
quadrants II and IV formed by inserting new axes though X and Y . If correlation was strong and Covariation in which the
negative, the observations would lie mostly in quadrants I and III. a ssociation between variables is
in opposing directions. As one
An Example goes up, the other goes down.
The correlation coefficient can be illustrated with a simple example. As you might expect, research-
ers today do not need to calculate correlation manually. However, the calculation process helps
illustrate exactly what is meant by correlation and covariance. Consider an investigation made to
determine whether the average number of hours worked in manufacturing industries is related to
unemployment.A correlation analysis of the data is carried out in Exhibit 23.3.
The correlation between the two variables is 20.635, indicating an inverse (negative) rela-
tionship.When number of hours goes up, unemployment comes down.This makes intuitive sense.
If factories are increasing output, regular workers will typically work more overtime and new
employees will be hired (reducing the unemployment rate). Both variables are probably related to
overall economic conditions.
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•564 PART SIX Data Analysis and Presentation
EXHIBIT 23.2 r ϭ .30 r ϭ .80 r ϭϩ1.0
Y Y Y
Scatter Diagram to Illustrate
Correlation Patterns
Low Positive X High Positive X Perfect Positive X
Correlation Correlation Correlation
r ϭ0 r ϭ Ϫ.60 r ϭ Ϫ1.0
Y Y Y
X X Perfect Negative X © Cengage Learning 2013
No Correlation Moderate Negative Correlation
Correlation
Correlation, Covariance, and Causation
Recall that concomitant variation is one condition needed to establish a causal relationship
between two variables. When two variables covary, they display concomitant variation. This sys-
tematic covariation does not in and of itself establish causality. Remember that the relationship
would also need to be nonspurious and that any hypothesized “cause” would have to occur before
any subsequent “effect.” For example, work experience displays a significant correlation with job
performance.5 However, in a retail context, workers with more experience often get assigned to
newer stores.Thus, the researcher would need to sort out to what extent age of the store may also
be responsible for causing store performance.
Coefficient of Determination
coefficient of If we wish to know the proportion of variance in Y that is explained by X, we can calculate the
coefficient of determination (R2) by squaring the correlation coefficient:
determination (R2)
R2 5 Explained variance
A measure obtained by Total variance
squaring the correlation
coefficient; the proportion of The coefficient of determination, R2, measures that part of the total variance of Y that is accounted
the total variance of a variable for by knowing the value of X. In the example about hours worked and unemployment,
accounted for by another value r 5 20.635; therefore, R2 5 0.403. About 40 percent of the variance in hours worked can be
explained by the variance in unemployment.As can be seen, R-squared really is just r squared!
of another variable.
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•CHAPTER 23 Bivariate Statistical Analysis: Measures of Association 565
EXHIBIT 23.3
Correlation Analysis of Number of Hours Worked in Manufacturing Industries with Unemployment Rate
Unemployment Number of Hours Xi 2 X (Xi 2 X )2 Yi 2 Y (Yi 2 Y )2 (Xi 2 X)(Yi 2 Y )
Rate (Xi) Worked (Yi) 2.3621
5.5 2.2301
4.4 39.6 .51 .2601 2.71 .5041 2.0801
4.1 .3519
4.3 40.7 2.59 .3481 .39 .1521
6.8 22.0091
5.5 40.4 2.89 .7921 .09 .0081 2.0051
5.5 2.3111
6.7 39.8 2.69 .4761 2.51 .2601 2.8721
5.5 .0459
5.7 39.2 1.81 3.2761 21.11 1.2321 .1349
5.2 .0819
4.5 40.3 .51 .2601 2.01 .0001 2.4361
3.8
3.8 39.7 .51 .2601 2.61 .3721 21.1781
3.6 2.3451
3.5 39.8 1.71 2.9241 2.51 .2601 2.5421
4.9 2.4321
5.9 40.4 .51 .2601 .09 .0081 .0459
5.6 2.3731
40.5 .71 .5041 .19 .0361 .1769
40.7 .21 .0441 .39 .1521
41.2 2.49 .2401 .89 .7921
41.3 21.19 1.4161 .99 .9801
40.6 21.19 1.4161 .29 .0841
40.7 21.39 1.9321 .39 .1521
40.6 21.49 2.2201 .29 .0841
39.8 2.09 .0081 2.51 .2601
39.9 .91 .8281 2.41 .1681
40.6 .61 .3721 .29 .0841
X 5 4.99
Y 5 40.31
a (Xi 2 X)2 5 17.8379
a (Yi 2 Y)2 5 5.5899 © Cengage Learning 2013
a (Xi 2 X)(Yi 2 Y) 5 26.3389
r 5 a (Xi 2 X)(Yi 2 Y) 5 2 6.3389 5 26.3389 5 2.635
2 a (Xi 2 X)2 a (Yi 2 Y)2 2(17.8379)(5.5899) 299.712
Correlation Matrix
A correlation matrix is the standard form for reporting observed correlations between two v ariables. correlation matrix
Although any number of variables can be displayed in a correlation matrix, each entry represents
the bivariate relationship between a pair of variables. Exhibit 23.4 shows a correlation matrix that The standard form for reporting
relates some measures of salesperson job performance to characteristics of the sales force.6 correlation coefficients for more
than two variables.
Note that the main diagonal consists of correlations of 1.00. Why is this? Simply put, any
variable is correlated with itself perfectly. Had this been a covariance matrix, the diagonal would
display the variance for any given variable.
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•566 PART SIX Data Analysis and Presentation
EXHIBIT 23.4
Pearson Product-Moment Correlation Matrix for Salesperson Examplea
Variables S JS GE SE OD VI JT RA TP WL
Performance (S) 1.00 1.00 1.00 1.00 1.00 1.00
2.18d 2.02 2.44b 2.26b .49b 1.00
Job satisfaction (JS) .45b 1.00 2.05 2.38b 2.22d
.26b 2.09 2.27c
Generalized self-esteem (GE) .31b .10 1.00 .38b 2.12
.09
Specific self-esteem (SE) .61b .28b .36b 1.00 2.04
Other-directedness (OD) .05 2.03 2.44b 2.24c
Verbal intelligence (VI) 2.36b 2.13 2.14 2.11
Job-related tension (JT) 2.48b 2.56b 2.32b 2.34b
Role ambiguity (RA) 2.26c 2.24c 2.32b 2.39b
Territory potential (TP) .49b .31b .04 .29b
Workload (WL) .45b .11 .29c .29c © Cengage Learning 2013
aNumbers below the diagonal are for the sample; those above the diagonal are omitted.
bp , .001
cp , .01
dp , .05
Performance (S) was measured by identifying the salesperson’s actual annual sales volume
in dollars. Notice that the performance variable has a 0.45 correlation with the workload vari-
able (WL), which was measured by recording the number of accounts in a sales territory. Notice
also that the salesperson’s perception of job-related tension ( JT) as measured by an attitude scale
has a 20.48 correlation with performance (S).Thus, when perceived job tension is high, perfor-
mance is low.
Just as in hypothesis testing, researchers are also concerned with statistical significance in corre-
lation analysis.The procedure for determining statistical significance is the t-test of the significance
of a correlation coefficient.Typically it is hypothesized that r 5 0, and then a t-test is performed.
The logic behind the test is similar to that for the significance tests already considered. Statistical
programs usually indicate the p-value associated with each correlation and/or star significant cor-
relations using asterisks. The Research Snapshot “What Makes Attractiveness?” displays the way
correlation matrices are often reported.
Regression Analysis
Regression analysis is another technique for measuring the linear association between a depen-
dent and an independent variable. Although simple regression and correlation are mathemati-
cally equivalent in most respects, regression is a dependence technique where correlation is an
interdependence technique.A dependence technique makes a distinction between dependent and
independent variables, specifying the cause and the effect. An interdependence technique does not
make this distinction and simply is concerned with how variables relate to one another.
Thus, with simple regression, a dependent (or criterion) variable,Y, is linked to an independent
(or predictor) variable, X. Regression analysis attempts to predict the values of Y, a continuous,
interval-scaled dependent variable from specific values of the independent variable.
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
RESEARCH
SNAPSHOT
What Makes Attractiveness? © Jill Wendell/Jupiter Images
What are the things that make someone attractive? Many someone appears), age, man-
people are interested in this question. Among these are com- ner of dress (how modern),
panies that hire people to sell fashion. The correlation matrix and personality (warm versus
below was computed with SPSS. The correlations show how cold). Thus, a sample of con-
different characteristics related to each other. Variables include sumers rated a model shown
a measure of fit, meaning how well the person matches a in a photog raph on those
fashion retail concept, attractiveness, weight (how overweight characteristics. The results reveal the following:
Correlations
Fit Attract Weight Age Modern Cold
Fit Pearson Correlation 1 0.831** 20.267* 0.108 20.447** 20.583
Attract Sig. (two-tailed) 0.000 0.036 0.404 0.000 0.000
Weight N 62 62
Age Pearson Correlation 20.831** 62 62 62 62
Modern Sig. (two-tailed) 1 20.275 0.039 20.428** 20.610**
Cold N 0.000 0.766 0.001 0.000
Pearson Correlation 62 62 0.030 62
Sig. (two-tailed) 20.275* 62 62 62 0.058
N 20.267* 1 0.082 0.262* 0.653
Pearson Correlation 0.036 0.030 0.528 0.040 62
Sig. (two-tailed) 62 62 62 0.104
N 0.108 0.082 62 62 0.423
Pearson Correlation 0.404 0.039 0.528 1 20.019 62
Sig. (two-tailed) 62 0.755 0.603**
N 62 62 0.082 0.000
Pearson Correlation 20.447 62 0.262* 20.019 62 62
Sig. (two-tailed) 0.000 20.428** 0.040 1 1
N 62 0.882
0.001 62 62 62 62
20.583** 62 0.058 0.603**
0.000 0.653 0.104 0.000
62 20.610** 0.423
0.000 62 62
62 62
*Correlation is significant at the 0.05 level (two-tailed).
**Correlation is significant at the 0.01 level (two-tailed).
Thus, if the model seems to “fit” the store concept, she seems lower attractiveness. Using these correlations, a retailer can help
attractive. If she is judged as being overweight, she is seen determine what employees should look like!
as less attractive. Age is unrelated to attractiveness or fit.
Modernness and perceived coldness also are associated with Correlations can be found using SPSS by navigating as
shown below:
Courtesy of SPSS Statistics 17.0
Courtesy of SPSS Statistics 17.0
© Cengage Learning 2013
567
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•568 PART SIX Data Analysis and Presentation
The Regression Equation
simple (bivariate) linear The discussion here concerns simple (bivariate) linear regression. Simple regression investigates a
regression straight-line relationship of the type
A measure of linear association Y 5 a 1 bX,
that investigates straight-line
relationships between a con- where Y is a continuous dependent variable and X is an independent variable that is usually
continuous, although dichotomous nominal or ordinal variables can be included in the form of
tinuous dependent variable and a dummy variable. Alpha (a) and beta (b) are two parameters that must be estimated so that the
an independent variable that is equation best represents a given set of data. These two parameters determine the height of the
usually continuous, but can be regression line and the angle of the line relative to horizontal. When these parameters change,
a categorical dummy variable. the line changes. Regression techniques have the job of estimating values for these parameters that
make the line fit the observations the best.
The result is simply a linear equation, or the equation for a line, where a represents the Y inter-
cept (where the line crosses the Y-axis) and b is the slope coefficient.The slope is the change in
Y associated with a change of one unit in X. Slope may also be thought of as rise over run, that is,
how much Y rises (or falls if negative) for every one unit change in the X-axis.
Parameter Estimate Choices
The estimates for α and are the key to regression analysis. In most business research, the estimate
of is more important. The explanatory power of regression rests with because this is where
the direction and strength of the relationship between the independent and dependent variable is
explained.
A Y-intercept term is sometimes referred to as a constant because a represents a fixed point.
An estimated slope coefficient is sometimes referred to as a regression weight, regression coefficient,
parameter estimate, or sometimes even as a path estimate. The term path estimate is a d escriptive
term adapted because of the way hypothesized causal relationships are often represented in diagrams:
X b1 Y
For all practical purposes, these terms are used interchangeably.
Parameter estimates can be presented in either raw or standardized form. One potential prob-
lem with raw parameter estimates is due to the fact that they reflect the measurement scale range.
So, if a simple regression involved distance measured with miles, very small parameter estimates
may indicate a strong relationship. In contrast, if the very same distance is measured with centime-
ters, a very large parameter estimate would be needed to indicate a strong relationship.
Exhibit 23.5 provides an illustration. Suppose a researcher was interested in how much space
was allocated to a specific snack food on a shelf and how it related to sales. Fifteen observations
are taken from 15 different stores.The upper line represents a typical distance showing shelf space
measured in centimeters.The lower line is the same distance shown in miles.The top frame shows
hypothetical regression results if the independent variable is measured in centimeters. The bot-
tom frame shows the very same regression results if the independent variable is measured in miles.
Even though these two regression lines are the same, the parameter coefficients do not seem
comparable.
EXHIBIT 23.5 12.5 cm © Cengage Learning 2013
The Advantage of Estimated Regression Line: Yˆ 5 22 1 .07X1
Standardized Regression
Weights 0.000078 miles
Estimated Regression Line: Yˆ 5 0.00014 1 .0000004X1
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•CHAPTER 23 Bivariate Statistical Analysis: Measures of Association 569
Thus, researchers often explain regression results by referring to a standardized regression standardized regression
coefficient (). A standardized regression coefficient provides a common metric allowing regres- coefficient (b)
sion results to be compared to one another no matter what the original scale range may have been.
Due to the mathematics involved in standardization, the standardized Y-intercept term is always 0.7 The estimated coefficient
The regression equation for the shelf space example would then become: indicating the strength of
relationship between an
YN 5 0 1 0.16X1 independent variable and
dependent variable expressed
Even if the distance measures for the 15 observations were converted to some other metric (feet, on a standardized scale where
meters, and so on), the standardized regression weight would still be 0.16. higher absolute values indicate
stronger relationships (range is
Researchers use shorthand to label regression coefficients as either “raw” or “standardized.” from 21 to 1).
The most common shorthand is as follows:
■■ B0 or b0 5 raw (unstandardized) Y-intercept term; what was referred to as a above
■■ B1 or b1 5 raw regression coefficient or estimate
■■ b1 5 standardized regression coefficients
Raw Regression Estimates (b1)
Raw regression weights have the advantage of retaining the scale metric—which is also their key
disadvantage.Where should the researcher focus then? Should the standardized or unstandardized
coefficients be interpreted? The answer to this question is fairly simple.
■■ If the purpose of the regression analysis is forecasting, then raw parameter estimates must be
used. This is another way of saying that the researcher is interested primarily in prediction.
Thus, when the researcher above wants to predict how much will be sold based on the amount of
shelf space, raw regression coefficients must be used. For instance, the forecast for 14 centimeter of
shelf space can be found as follows:
YN 5 22 1 0.07(14) 5 23.0
The same result can be found by using the equation representing the distance in miles.
Standardized Regression Estimates (b)
Standardized regression estimates have the advantage of a constant scale. No matter what range of
values the independent variables take on, will not be affected.When should standardized regres-
sion estimates be used?
■■ Standardized regression estimates should be used when the researcher is testing explanatory
hypotheses. In other words, standardized estimates are appropriate when the researcher’s pur-
pose is to explain the relationship rather than to make an actual prediction.
Visual Estimation of a Simple Regression Model
As mentioned above, simple regression involves finding a best-fit line, given a set of observations
plotted in two-dimensional space. Many ways exist to estimate where this line should go. Estima-
tion techniques involve terms such as instrumental variables, maximum likelihood, visual estima-
tion, and ordinary least squares (OLS).We focus on the latter two in this text.
Suppose a researcher is interested in forecasting sales for a construction distributor (wholesaler)
in Florida.The distributor believes a reasonable association exists between sales and building permits
issued by counties. Using bivariate linear regression on the data in Exhibit 23.6, the researcher will be
able to explain sales potential (Y ) in various counties based on the number of building permits (X ).
The data are plotted in a scatter diagram in Exhibit 23.7. In the diagram the vertical axis
indicates the value of the dependent variable, Y, and the horizontal axis indicates the value of the
independent variable, X. Each single point in the diagram represents an observation of X and Y at
a given point in time.The values are simply points in a Cartesian plane.
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
•570 PART SIX Data Analysis and Presentation
EXHIBIT 23.6 Dealer Y X
Dealer’s Sales Volume Building
Relationship of Sales 1 Permits
Potential to Building Permits 2 (Thousands)
Issued 3 86
4 77 93
5 79 95 © Cengage Learning 2013
6 80 104
7 83 139
8 101 180
9 117 165
10 129 147
11 120 119
12 97 132
13 106 126
14 99 156
15 121 129
103 96
86 108
99
EXHIBIT 23.7 140 Actual Y
130 for Dealer 7
The Best-Fit Line or Knocking 120
Out the Pins 110 Y for Dealer 7
100
Sales ($000) 90 Y = 99.8 Y for Dealer 3 140 160 180 200
80
70 Actual Y
60 for Dealer 3
50
40 b1 (raw parameter/slope
30 b0 (Y intercept) ϭ 31.5 coefficient) ϭ 0.546
20
10 60 80 100 120 © Cengage Learning 2013
BP
0 0 20 40
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•CHAPTER 23 Bivariate Statistical Analysis: Measures of Association 571
One way to determine the relationship between X and Y is to simply visually draw the best-fit
straight line through the points in the figure.That is, try to draw a line that goes through the center
of the plot of points. If the points are thought of as bowling pins, the best-fit line can be thought
of as the path that would on average knock over the most bowling pins. For any given value of the
independent variable, a prediction can be made by selecting the dependent variable that goes along
with that value. For example, if we want to forecast sales if building permits are 150, we simply
follow the dotted lines shown in the exhibit to yield a prediction of about 112.The better one can
estimate where the best-fit line should be, the smaller will be the error in prediction.
Errors in Prediction
Any method of drawing a line can be used to perform regression. However, some methods will
obviously have more errors than others. Consider our bowling ball line above. One person may be
better at guessing where it should be than another.We would know who was better by determin-
ing the total error of prediction.
Let’s consider error by first thinking about what value of sales would be the best guess if we had no
information about any other variable. In that case, our univariate best guess would be the mean sales of
99.8. If the spot corresponding to 156 building permits (X 5 156) were predicted with the mean, the
resulting error in prediction would be represented by the distance of the gray and light blue vertical line.
Once information about the independent variable is provided, we can then use the predic-
tion provided by the best-fit line. In this case, our best-fit line is the “bowling ball” line shown
in Exhibit 23.7. The error in prediction using this line would be indicated by the vertical line
extending up from the regression line to the actual observation. Thus, it appears that at least for
this observation, our prediction using the regression line has reduced the error in prediction that
would result from guessing with the mean. Statistically, this is the goal of regression analysis. We
would like an estimation technique that would place our line so that the total sum of all errors
over all observations is minimized. In other words, no line fits better overall. Although with good
guess work, visual estimation may prove somewhat accurate, we have more accurate methods of
establishing a regression line.
Ordinary Least-Squares (OLS) Method
of Regression Analysis
The researcher’s task is to find the best means for fitting a straight line to the data. Ordinary least
squares (OLS) is a relatively straightforward mathematical technique that guarantees that the result-
ing straight line will produce the least possible total error in using X to predict Y. The logic is based
on how much better a regression line can predict values of Y compared to simply using the mean
as a prediction for all observations no matter what the value of X may be.
Unless the dependent and independent variables are perfectly related, no straight line can con-
nect all observations. More technically, the procedure used in the least-squares method generates a
straight line that minimizes the sum of squared deviations of the actual values from this predicted
regression line.With the symbol e representing the deviations of the observations from the regres-
sion line, no other line can produce less error.The deviations are squared so that positive and nega-
tive misses do not cancel each other out.The OLS criterion is as follows:
On
e 2 is minimum
i
i51
where
ei 5 Yi 2 YN i (the residual)
Yi 5 actual observed value of the dependent variable
YNi 5 estimated value of the dependent variable (pronounced “Y-hat”)
n 5 number of observations
i 5 number of the particular observation
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
•572 PART SIX Data Analysis and Presentation
The general equation for any straight line can be represented as Y 5 b0 1 b1X. If we think of
this as the true hypothetical line that we try to estimate with sample observations, the regression
equation will represent this with a slightly different equation:
Yi 5 b0 1 b1X1 1 ei
The equation means that the predicted value for any value of X (Xi) is determined as a function of
the estimated slope coefficient, plus the estimated intercept coefficient plus some error.
The raw parameter estimates can be found using the following formulas:
na
O O OO Ob1 5
XiYi b 2 a Xi b a Yi b
2
n a Xi2b 2 a Xi b
and
b0 5 Y 2 b1X
where
Yi 5 ith observed value of the dependent variable
Xi 5 ith observed value of the independent variable
Y 5 mean of the dependent variable
X 5 independent variable
X 5 mean of the independent variable
n 5 number of observations
b0 5 intercept estimate
b1 5 slope estimate (regression weight)
The careful reader may notice some similarity between the correlation calculation and the
equation for b1. In fact, the standardized regression coefficient from a simple regression equals the
Pearson correlation coefficient for the two variables. Once the estimates are obtained, a predicted
value for the dependent variable can be found for any value of Xi with this equation:
YNi 5 b0 1 bi Xi
Appendix 23A demonstrates the arithmetic necessary to calculate the parameter estimates.
Statistical Significance of Regression Model
As with ANOVA, the researcher needs a way of testing the statistical significance of the regression
model.Also like ANOVA, an F-test provides the answer to this question.
The overall F-test for regression can be illustrated with Exhibit 23.7. Once again, examine
the colored line showing the predicted value for X 5 156, which represents the small vertical line
located at the upper right of the exhibit.
1. The total vertical line including the gray and light blue segments represents the total deviation
of the observation from the mean:
Yi 2 Y
2. The gray portion represents how much of the total deviation is explained by the regression line:
YNi 2 Y
3. The light blue portion represents how much of the total deviation is not explained by the
regression line (also equal to ei):
Yi 2 YNi
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.