ST 260, M32
Margin of Error
Lesson Objectives
Margin of Error q Learn the meaning of
“margin of error,” or “m.o.e.”
Whose error is it?
Why do we care? q Learn how to calculate the
m.o.e. for two situations: when
the true population std. dev., σ,
is known and when its unknown.
q Learn how to use the m.o.e. to
construct a confidence interval.
© Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 1 © Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 2
“Margin of Error” for estimating Illustration of “Margin of Error”
the True Mean of a µ = true population mean.
population. Find the interval around the mean where
95% of all possible sample means will lie.
m.o.e. at 95% confidence = .025 .95 .025
“The amount that when added and ? µ? X
subtracted to the true population mean -1.96 0 +1.96
will define a region that will include Z
the middle 95% of
all possible X-bar values.”
© Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 3 © Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 4
?-µ Margin of Error for 95%confidence:
Z= σ n standard error
? -µ σm.o.e. = 1.96 l of the mean
= σn
σ n σx
n
? =µ +
Margin of Error
© Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 5 © Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 6
© Department of ISM, University of Alabama, 1995-2003 1
ST 260, M32
Margin of Error
General form for “margin of error” Explanation of symbol:
when σ is known: Zα /2 cuts off the top tail at area = α/2
σm.o.e. = Zα l n α /2
2
where Zα is the 0 Zα /2 Z ~ N(0,1)
2
appropriate percentile from the
standard normal distribution,
i.e., the Z table .
© Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 7 © Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 8
Examples (1-α)100% Confidence Interval:
Amount of Area in Table value
confidence each tail
1-α α /2 Zα /2 Point estimate ± m.o.e.
.98 .0100 2.33 (1-α )100% is the amount of
confidence desired.
.95 .0250 1.96
.90 ? ? .95 confidence α = .05 risk
.80 ? ?
© Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 9 © Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 10
Example 1 Example 1, continued . . .
Estimate with 98% confidence the mean 98% confidence interval:
gallons of water used per shower for Point estimate ± m.o.e.
Dallas Cowboys after a game if the true
standard deviation is known to be 10 gallons.
The sample mean for 16 showers is 30.00 gal.
σm.o.e. = Zα l n
2
© Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 11 © Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 12
© Department of ISM, University of Alabama, 1995-2003 2
ST 260, M32
Margin of Error
Example 1, Statement in the L.O.P. What do we do if the
true population standard
I am 95% confident that the true mean gallons deviation σ is unknown?
of water used per shower for Dallas Cowboys
after a game will fall within the interval
24.175 to 35.825 gallons.
A statement in L.O.P. must contain four parts: • Replace σ with .
1. amount of confidence.
2. the parameter being estimated in L.O.P. • replace z with .
3. the population to which we generalize
in L.O.P.
4. the calculated interval. M32 Margin of Error 13 © Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 14
© Department of ISM, University of Alabama, 1995-2002
General form for “margin of error” Explanation of symbol:
when σ is UN-known: t α /2, n-1 cuts off the top tail at area = α/2
m.o.e. = t α 2 ,n–1 l s “estimated” α /2
standard
error of
n the
mean
where t α ,n–1 is the 0 tα / 2 , n–1 t-distribution
2 sx Use the t-table to find the value.
appropriate percentile from the
t-distribution.
© Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 15 © Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 16
Comparison of “Z” and “t” 0.1 0.05 0.025 0.01 0.005 0.1 0.05 0.025 0.01 0.005
Z t-dist. d.f. = 1 3.078 6.314 12.706 31.821 63.656 36 1.306 1.688 2.028 2.434 2.719
Yes Yes 2 1.886 2.920 4.303 6.965 9.925 37 1.305 1.687 2.026 2.431 2.715
3 1.638 2.353 3.182 4.541 5.841 38 1.304 1.686 2.024 2.429 2.712
39 1.304 1.685 2.023 2.426 2.708
4 1.533 2.132 2.776 3.747 4.604 40
5 1.476 2.015 2.571 3.365 4.032 41 1.303 1.684 2.021 2.423 2.704 0t
6 1.440 1.943 2.447 3.143 3.707 42 1.303 1.683 2.020 2.421 2.701
7 1.415 1.895 2.365 2.998 3.499 43 1.302 1.682 2.018 2.418 2.698
44
Bell shaped, 8 1.397 1.860 2.306 2.896 3.355 45 1.302 1.681 2.017 2.416 2.695 Table gives right-tail area.
symmetric 9 1.383 1.833 2.262 2.821 3.250 1.301 1.680 2.015 2.414 2.692 (e.g., for a right-tail area
10 1.372 1.812 2.228 2.764 3.169 97 1.301 1.679 2.014 2.412 2.690 of 0.025 and d.f. = 15,
98 the t value is 2.131.)
11 1.363 1.796 2.201 2.718 3.106 99 1.290 1.661 1.985 2.365 2.627
12 1.356 1.782 2.179 2.681 3.055 100 1.290 1.661 1.984 2.365 2.627
13 1.350 1.771 2.160 2.650 3.012 1.290 1.660 1.984 2.365 2.626
14 1.345 1.761 2.145 2.624 2.977 ∞
Mean =0 =0 1.290 1.660 1.984 2.364 2.626
Std. dev. =1 >1 15 1.341 1.753 2.131 2.602 2.947 1.282 1.645 1.960 2.326 2.576
16 1.337 1.746 2.120 2.583 2.921
17 1.333 1.740 2.110 2.567 2.898 Want 95% CI,
18 1.330 1.734 2.101 2.552 2.878 n = 20,
19 1.328 1.729 2.093 2.539 2.861
Degrees of freedom “∞” “n-1” 20 1.325 1.725 2.086 2.528 2.845 α/2 = .025
21 1.323 1.721 2.080 2.518 2.831
d.f. = 19
22 1.321 1.717 2.074 2.508 2.819
As “n–1” increases, “ tn–1” approaches “Z” 23 1.319 1.714 2.069 2.500 2.807 t.025, 19= 2.093
24 1.318 1.711 2.064 2.492 2.797
© Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 17 25 1.316 1.708 2.060 2.485 2.787
26 1.315 1.706 2.056 2.479 2.779
27 1.314 1.703 2.052 2.473 2.771
28 1.313 1.701 2.048 2.467 2.763
29 1.311 1.699 2.045 2.462 2.756
30 1.310 1.697 2.042 2.457 2.750
31 1.309 1.696 2.040 2.453 2.744
32 1.309 1.694 2.037 2.449 2.738
33 1.308 1.692 2.035 2.445 2.733
34 1.307 1.691 2.032 2.441 2.728
35 1.306 1.690 2.030 2.438 2.724
© Department of ISM, University of Alabama, 1995-2003 3
ST 260, M32
Margin of Error
0.1 0.05 0.025 0.01 0.005 0.1 0.05 0.025 0.01 0.005 0.1 0.05 0.025 0.01 0.005 0.1 0.05 0.025 0.01 0.005
d.f. = 1 3.078 6.314 12.706 31.821 63.656 36 1.306 1.688 2.028 2.434 2.719 0t d.f. = 1 3.078 6.314 12.706 31.821 63.656 36 1.306 1.688 2.028 2.434 2.719 0t
2 1.886 2.920 4.303 6.965 9.925 37 1.305 1.687 2.026 2.431 2.715 2 1.886 2.920 4.303 6.965 9.925 37 1.305 1.687 2.026 2.431 2.715
3 1.638 2.353 3.182 4.541 5.841 38 1.304 1.686 2.024 2.429 2.712 Table gives right-tail area. 3 1.638 2.353 3.182 4.541 5.841 38 1.304 1.686 2.024 2.429 2.712 Table gives right-tail area.
39 1.304 1.685 2.023 2.426 2.708 (e.g., for a right-tail area 2.132 2.776 3.747 4.604 39 1.304 1.685 2.023 2.426 2.708 (e.g., for a right-tail area
4 1.533 2.132 2.776 3.747 4.604 40 of 0.025 and d.f. = 15, 4 1.533 2.015 2.571 3.365 4.032 40 1.303 of 0.025 and d.f. = 15,
5 1.476 2.015 2.571 3.365 4.032 41 1.303 1.684 2.021 2.423 2.704 the t value is 2.131.) 5 1.476 1.943 2.447 3.143 3.707 41 1.303 1.684 2.021 2.423 2.704 the t value is 2.131.)
6 1.440 1.943 2.447 3.143 3.707 42 1.303 1.683 2.020 2.421 2.701 6 1.440 1.895 2.365 2.998 3.499 42 1.302 1.683 2.020 2.421 2.701
7 1.415 1.895 2.365 2.998 3.499 43 1.302 1.682 2.018 2.418 2.698 7 1.415 1.682 2.018 2.418 2.698
44 1.860 2.306 2.896 3.355 43 1.302
8 1.397 1.860 2.306 2.896 3.355 45 1.302 1.681 2.017 2.416 2.695 8 1.397 1.833 2.262 2.821 3.250 44 1.301 1.681 2.017 2.416 2.695
9 1.383 1.833 2.262 2.821 3.250 1.301 1.680 2.015 2.414 2.692 9 1.383 1.812 2.228 2.764 3.169 45 1.301 1.680 2.015 2.414 2.692
10 1.372 1.812 2.228 2.764 3.169 97 1.301 1.679 2.014 2.412 2.690 10 1.372 1.679 2.014 2.412 2.690
98
11 1.363 1.796 2.201 2.718 3.106 99 1.290 1.661 1.985 2.365 2.627 11 1.363 1.796 2.201 2.718 3.106 1.661 1.985 2.365 2.627
12 1.356 1.782 2.179 2.681 3.055 100 1.290 1.661 1.984 2.365 2.627 12 1.356 1.782 2.179 2.681 3.055 97 1.290 1.661 1.984 2.365 2.627
13 1.350 1.771 2.160 2.650 3.012 1.290 1.660 1.984 2.365 2.626 13 1.350 1.771 2.160 2.650 3.012 98 1.290 1.660 1.984 2.365 2.626
14 1.345 1.761 2.145 2.624 2.977 ∞ 14 1.345 1.761 2.145 2.624 2.977 99 1.290
1.290 1.660 1.984 2.364 2.626 1.660 1.984 2.364 2.626
15 1.341 1.753 2.131 2.602 2.947 1.282 1.645 1.960 2.326 2.576 15 1.341 1.753 2.131 2.602 2.947 100 1.290 1.645 1.960 2.326 2.576
16 1.337 1.746 2.120 2.583 2.921 16 1.337 1.746 2.120 2.583 2.921 1.282
17 1.333 1.740 2.110 2.567 2.898 17 1.333 1.740 2.110 2.567 2.898
18 1.330 1.734 2.101 2.552 2.878 Want 98% CI, 18 1.330 1.734 2.101 2.552 2.878 Want 90% CI,
19 1.328 1.729 2.093 2.539 2.861 n = 33, 19 1.328 1.729 2.093 2.539 2.861 n = 600,
20 1.325 1.725 2.086 2.528 2.845 20 1.325 1.725 2.086 2.528 2.845
21 1.323 1.721 2.080 2.518 2.831 α/2 = 21 1.323 1.721 2.080 2.518 2.831 α/2 =
22 1.321 1.717 2.074 2.508 2.819 22 1.321 1.717 2.074 2.508 2.819
23 1.319 1.714 2.069 2.500 2.807 23 1.319 1.714 2.069 2.500 2.807
24 1.318 1.711 2.064 2.492 2.797 24 1.318 1.711 2.064 2.492 2.797
25 1.316 1.708 2.060 2.485 2.787 25 1.316 1.708 2.060 2.485 2.787
26 1.315 1.706 2.056 2.479 2.779 d.f. = 26 1.315 1.706 2.056 2.479 2.779 d.f. =
27 1.314 1.703 2.052 2.473 2.771 27 1.314 1.703 2.052 2.473 2.771
28 1.313 1.701 2.048 2.467 2.763 t ,= 28 1.313 1.701 2.048 2.467 2.763
29 1.311 1.699 2.045 2.462 2.756 29 1.311 1.699 2.045 2.462 2.756
30 1.310 1.697 2.042 2.457 2.750 30 1.310
31 1.309 1.696 2.040 2.453 2.744 31 1.309 1.697 2.042 2.457 2.750
32 1.309 1.694 2.037 2.449 2.738 32 1.309
Same as Normal1.6962.0402.4532.744 t,=
33 1.308 1.692 2.035 2.445 2.733 33 1.308 2.037 2.449 2.738
34 1.307 1.691 2.032 2.441 2.728 34 1.307 1.694
35 1.306 1.690 2.030 2.438 2.724 35 1.306 1.692 2.035 2.445 2.733
1.691 2.032 2.441 2.728
1.690 2.030 2.438 2.724
Example 2 Example 2, continued . . .
Estimate with 98% confidence the mean 98% confidence interval:
gallons of water used per shower for Point estimate ± m.o.e.
Dallas Cowboys after a game. The sample
mean of 16 showers is 30.00 gallons and the
standard deviation is 10.4 gallons.
m.o.e. = t α , n-1l s n
2
© Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 21 © Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 22
Example 2, Statement in the L.O.P. Margin of Error for 95%confidence:
I am 95% confident that the true mean gallons = 1.96 l σ n
of water used per shower for Dallas Cowboys
after a game will fall within the interval To get a smaller Margin of Error:
23.235 to 36.765 gallons. q
q
A statement in L.O.P. must contain four parts:
1. amount of confidence.
2. the parameter being estimated in L.O.P.
3. the population to which we generalize
in L.O.P.
4. the calculated interval.
© Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 23 © Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 24
© Department of ISM, University of Alabama, 1995-2003 4
ST 260, M32
Margin of Error
Confidence vs. Probability What sample size is needed to
estimate the mean mpg of Toyota
q BEFORE a sample is collected, there is a Camrys with an m.o.e. of 0.2 mpg
95% probability that the future to be computed
sample mean, will fall within m.o.e. units of µ. at 90%confidence if the pop. std.
q AFTER the sample is collected, the computed dev. is 0.88 mpg?
sample mean either fell within m.o.e. units of µ,
or it did not. After the event, it does not σm.o.e. = Zα l n
2
make sense to talk about probability.
Analogy: Suppose you own 95 tickets in a
100-ticket lottery. The drawing was held
one hour ago, but you don’t know the result.
P(win) = 0 or 1, but you are very CONFIDENT
that you have won the lottery. M32 Margin of Error 25 M32 Margin of Error 26
© Department of ISM, University of Alabama, 1995-2002 © Department of ISM, University of Alabama, 1995-2002
Interpretations of the Example 3: A car rental agency wants to
Confidence Interval for µ estimate the average mileage driven by its
customers. A sample of 225 customer
• If we took many, many samples receipts, selected at random, yields an
average of 325 miles. Assuming that the
of size n, and calculated a population standard deviation is 120 miles,
confidence interval for each, construct a 95% confidence interval for the
then I would expect that 95% population average mileage.
of all these many intervals
would contain the true mean,
and 5% would not.
© Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 27 © Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 28
Example 4:
An insurance company collected data to
estimate the mean value of personal
property held by apartment renters in
Tuscaloosa. In a random sample of 45
renters, the average value of personal
property was $14,280 and the standard
deviation was $6,540.
© Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 29 © Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 30
© Department of ISM, University of Alabama, 1995-2003 5
Example 4: ST 260, M32
a. At 95% confidence, find the Margin of Error
margin of error for estimating the Example 4:
true population mean? b. Construct a 95% confidence interval for
the true mean value of personal property
owned by renters in Tuscaloosa.
© Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 31 © Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 32
Example 4: Example 4:
c. Three years ago the true mean value of d. Assuming that the true mean is
personal property owned by renters in unchanged from three years ago, find the
Tuscaloosa was $13,050. probability that a future sample of 45
renters would result in a mean that is
Based on your confidence interval, more extreme than the sample the
is there evidence that the true mean insurance company just took.
has changed?
P(X > 14,280 | true mean = 13,050)
What distribution must be used?
© Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 33 © Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 34
Example 4: e. Based on this probability,
would you conclude that the true mean
has changed from three years ago?
or, equivalently
Is there sufficient evidence to conclude
that the true mean has changed?
or, equivalently
Was the sample mean of $14,280
too close to $13,050 to call it unusual,
or was this value a rare event?
Had the sample mean been $15,000,
would your conclusion be different?
© Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 35 © Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 36
© Department of ISM, University of Alabama, 1995-2003 6