DPB 30063 / PROBABILITY 93 6.8 Tree Diagram • Tree diagram is a picture representation of outcomes or events. • It helps to organize calculations that involve several stages. • To construct a tree diagram, we start from a point. • From that point draw branches according to the number of events occur. • That will be the first stage of the tree diagram. • For the second stage, draw additional branches accordingly from each of the former branches. Example 4 Consider the flipping of a coin twice and recording the face up either head or tail. Solution:
DPB 30063 / PROBABILITY 94 Example 5 A bag contains three red pens, four green pens and six white pens. Two pens are picked at random one after another. a. List the sample space of this experiment. Draw a tree diagram b. If A denotes the event that two green pen will be selected, list the event in A. c. If B denote the event that at least one red pen will be selected, list the event in B. Solution :
DPB 30063 / PROBABILITY 95 Example 6 In a group of college students, some are in favor of studying in the library and others are against it. Two students are selected at random from the group and asked whether they are in favor of or against studying in the library. How many outcomes are possible? Draw a tree diagram for the experiment. From the tree diagram, list the event: a. A that both students are in favor of studying in the library. b. B that at most one student is against studying in the library. c. C that exactly one student is in favor of studying in the library. Solution :
DPB 30063 / PROBABILITY 96 Example 7 75% of PMS students walk to their classes, 20% motorcycles and the balance by bicycles. 5% of the students walking will be late for classes and 4% of those who ride motorcycles will be late. Only 1% of those riding bicycles will be late. Draw a suitable tree diagram. Solution :
DPB 30063 / PROBABILITY 97 6.9 Approaches to Probability Basically there are three conceptual approaches to probability. They are classical, relative frequency and subjective probabilities concept. a. Classical Probability The classical probability concept is applied to events on the assumptions that the outcomes of the experiment are equally likely to happen. For example, if we have h outcomes of an event E out of n possible outcomes in a sample space, then the probability of event E to happen is the ratio of h to n. That is : P(E) = n (E) n (S) , where E is the event and S is the sample space Example 8 A dice is tossed once. Find the probability getting a number greater than 4 Solution : b. Relative Frequency (Empirical Probability) Relative frequency probability is based on observation. The various outcomes of an event are likely to happen but the experiment is repeated a large number of times to generate a set of data. In such cases, the probability is calculated by using the frequency of the outcomes over the number of experiment done. P (E) = f / n where f is number of times event E occur and n is the total number of observations.
DPB 30063 / PROBABILITY 98 Example 9 A cosmetic company makes a survey on their new moisturizer. They took a sample of 500 respondents to answer a question on the performance of the moisturizer. The responses were classified as ‘satisfactory’, ‘not satisfied’, and ‘undecided’. The result is shown below : Response Frequency Satisfactory 250 Not Satisfied 150 Undecided 100 Total 500 Solution : The probability of a person who was satisfied with the moisturizer are 250/500 since 250 out of 500 answered satisfactory. c. Subjective Probability • There are situation that cannot be repeated. • This situation is subjected to personal assessment or judgment to determine probabilities. • Estimation obtained in such ways is known as subjective probability. • With this approach, the probability is measured using the ‘strength of belief’ that the event will occur when the experiment is performed. • Example of subjective probability can be seen in government or business decision making which are seldom repeatable. • For example, estimating the possibility that a company which was recently listed in KLSE will gain profit next year. Properties of Probability Assuming E is an event, 1. All probabilities are between 0 and 1, 0 < = P (E) < = 1 2. The sum of all the probabilities in the sample space is 1 3. The probability of an event that cannot occur is 0 4. The probability of an event that must occur is 1 5. Complement of probability : P (E’) = 1 - P (E)
DPB 30063 / PROBABILITY 99 6.10 Calculating Probability Probability Rules There are two rules which are the rule of addition and multiplication General Addition Rule Addition rule involve the union of two or more events. Union means that either A or/and B will occur. It also signify that at least one will occur. P ( A or B ) = P ( A U B ) = P ( A ) + P ( B ) - P ( A and B ) = P ( A ) + P ( B ) - P ( A n B ) Example 10: Given P ( A ) = 0.20, P ( B ) = 0.70, P ( A and B ) = 0.15, find P ( A U B ) Solution : Example 11 The probability that a student in a particular program is a member of the badminton team is 0.6 and in the football team is 0.5. The probability that the student is both a member of the badminton team as well as the football team is 0.3. What is the probability that a student chosen at random from the program is in the badminton or the football team ? Solution :
DPB 30063 / PROBABILITY 100 Example 12 S A 9 1 6 2 4, 5 7 3 B 8 Solution : 6.11 Probability Mutually Exclusive • Mutually exclusive events – event that cannot occur simultaneously (no overlap). • Events that do not have the same element. • Events that cannot happen at the same time The probability of two mutually exclusive events is found by adding the probability of each event.
DPB 30063 / PROBABILITY 101 Example 13. One card is drawn from a standard deck, what is the probability of drawing a 2 or an Ace ? Solution : Example 14 : A bag contains 4 green, 6 yellow and 8 blue marbles. One is drawn at random. Find : a. P( blue or green ) b. P( yellow or green ) c. P( purple or blue ) Solution: 6.12 Probability Mutually Inclusive Events : Inclusive events – events whose outcomes may occur simultaneously (overlap). Events that have outcomes in common. “or” probability for inclusive events :
DPB 30063 / PROBABILITY 102 Example 15 : One card is drawn from a standard deck of cards. What is the probability of drawing a queen or a diamond. Solution : Example 16 : A jar contains 5 green cubes, 6 red cubes, 8 green spheres and 10 red spheres. One object is pulled. Find: a. P (red or cubes) b. P (spheres or green) c. P (cubes or green) d. P (red or spheres) Solutions :
DPB 30063 / PROBABILITY 103 6.13 Probability of Independent Event • Event that has no connection to another events chances of happening. • The outcome of one event does not affect the outcome of the other events. • With the key word “with replacement” Compound events – two or more events linked by the word ‘and’ or the word ‘or’. ‘and’ probability for independent events : Example 17 : If you roll two six sided dice, what is the probability you will roll a 5 and then a 3. Solution : Example 18 : What is the probability that you will roll an even number on each ? Solution :
DPB 30063 / PROBABILITY 104 Example 19 : A bag contains 3 red, 5 blue and 2 orange marbles. If you pull two marbles one at a time with replacement, find : a. P (blue then/ and red) b. P (orange then red) Solution : Example 20 : A bag contains 2 red marbles and 1 blue marble. Nabilah draws a marble then return it to the bag. She draws a second marble. What is the probability that the first marble is red and the second is blue ? Solution : 6.14 Probability of Dependent Event • The first event influence / affect the probability of the next event. • The outcomes of one events does influence the outcomes of the other event. • With the key word “without replacement”. ‘and’ probability for dependent events :
DPB 30063 / PROBABILITY 105 Example 21 A jar contains 5 green, 10 red and 7 blue marbles. Two marbles are pulled one at a time without replacement. Find : a. p (green then red), b: p (blue then green) c: p (red then red) d: p (blue then blue) Solution: Example 22 A red and a blue dice are rolled. What is the probability of the red being an odd number and the blue being greater than four ? Solution :
DPB 30063 / PROBABILITY 106 Example 23 A box of chocolates contains 10 milk chocolate, 6 dark chocolate and 4 white chocolate candies. If selected at random, what is the probability that you eat a milk chocolate followed by a white chocolate ? Solution : Example 24 What is the probability of getting a heart, spade and another heart without replacing the previous card ? Solution :
DPB 30063 / PROBABILITY 107 SELF REVISION 1. Flipping a coin three times. What is the probability of getting exactly two heads. 2. A bags consists of 4 red marbles and 6 black marbles. What is the probability of getting red. A - (0.4) 3. A deck of 52 cards. What is the probability of getting (i) red cards, (ii) jack A – red cards (1/2), jack – 1/13 4. Differentiate dependent and independent probability. 5. Nabilah has a bag with 7 blues sweets and 3 red sweets in it. She picks a sweet at random from the bag, replaces it and then picks again at random. Draw a tree diagram to represent this situation and use it to calculate the probabilities that she picks : a. Two red sweets (9/100) b. No red sweets (49/100) c. At least 1 blue sweets (91/100) d. One sweets of each color. (42/100 @ 21/50) 6. There is a bag filled with 2 blue, 8 red and 2 yellow marbles. Haziq picks a marble out of the bag, notes its color and replaces it. He then takes another marble from the bag. Draw a probability tree. What is the probability of :- a. Getting 2 reds b. Getting 2 blues c. Getting 2 yellows d. Getting two different color 7. Nabihah is going to play one badminton match and one tennis match. The probability that she will win the badminton match is 9/10. The probability that she will win the tennis match is 2/5. Complete the probability tree diagram. (18/50)
DPB 30063 / PROBABILITY 108 8. There is a bag filled with 2 blue, 8 red and 2 yellow marbles. Haziq picks a marble out of the bag, notes its color and do not replaces it. He then takes another marble from the bag. Draw a probability tree. What is the probability of :- a. Getting 2 reds b. Getting 2 blues c. Getting 2 yellows d. Getting two different color 9. Aiman has a bag of 9 sweets. In the bag, there are 3 orange flavoured sweets, 4 strawberry flavoured sweets and 2 lemon flavoured sweets. He takes at random two of the sweets. He eats the sweets. Work out the probability that the two sweets Aiman eats are not the same flavoured. (52/72) 10. Afiq has 20 biscuits in a tin. He has 12 plain biscuits, 5 chocolate biscuits and 3 ginger biscuits. Afiq takes at random two biscuits from the tin. Work out the probability the two biscuits were not the same type. (111/190) 11. There are three different types of sandwiches on a shelf. There are 4 egg sandwiches, 5 cheese sandwiches and 2 chicken sandwiches. Amirah takes at random 2 of these sandwiches. Work out the probability that she takes 2 different types of sandwiches. (76/100) 12. In a supermarket, the probability that Atirah buys fruit is 0.7. In the same supermarket, the probability that Atirah independently buys vegetables is 0.4. Work out the probability that Atirah buys fruits or buys vegetables or buy both. (0.82) 13. Three companies LALA, MAMA and SASA are competing for a contract to build a house. The probabilities that companies LALA, MAMA, and SASA will win the contract are 0.3, 0.4 and 0.3 respectively. If company LALA, MAMA and SASA win the contract, the probability that they will make profits are 0.8, 0.9 and 0.7 respectively. Sketch a Tree Diagram based on the information given in the question.
DPB 30063 / PROBABILITY 109 14. The Venn diagram displays the result of a survey of 100 students in a high technology in their personal lives. P 2 3 1 1 6 T 17 C 68 • P – represent the number of students with smart phone • C – represent the number of students with a computer • T – represent the number of students with tablet a. How many students own only a smart phone. ( ) b. How many students own a smart phone. ( ) c. How many students own a computer and tablet. ( ) d. How many students own a computer and tablet only. ( ) e. How many students own a computer or an tablet. ( ) f. How many students own all three items. ( ) g. What does the 2 outside the three circles represent ? ( ) 15. 100 students are asked about which subjects they like :- 35 like English (f) 12 like English and Maths (c : ) 50 like Maths (e) 8 like English and Science (b : ) 29 like Science (d) 6 like Science and Maths only (a : ) 5 like all three subjects
DPB 30063 / PROBABILITY 110 Draw a Venn diagram. 16. 110 students were asked about which sport they like :- 40 like basketball (e) 11 like basketball and hockey (b : ) 45 like hockey (d) 8 like hockey and golf (a: ) 33 like golf (c) 5 like basketball and golf only 4 like all three sport
DPB 30063 / PROBABILITY 111 17. Table below shows the result of a survey that asked whether the person is involved in charity work. The person’s gender was recorded as well as their response frequently, sometimes or never. A person is randomly selected from the sample. Find the probability of each event occuring :- Frequently Sometimes Never Male 152 187 432 Female 127 204 445 Total ** Fill in the bold font a. Selecting a male given that they sometimes volunteer. b. Selecting someone and a female who never volunteer. c. Selecting someone who didn’t respond never.
DPB 30063 / PROBABILITY 112 d. Selecting a male or someone who responded sometimes. e. Selecting someone who responded sometimes or never. f. Selecting a female or someone who responded frequently. g. Selecting a person who respond never given they were a female.
DPB 30063 / PROBABILITY 113 18. Student were asked in a survey whether they had ever posted a video on You Tube. Results were broken down by gender. Gender No Yes Total Female 390 158 548 Male 192 134 326 Total 582 292 874 a. What is the probability of student had ever posted a video on You tube. Answer : b. What fraction of students surveyed were male. Answer : c. What fraction of students were male and had ever posted a video on You Tube. Answer : d. What is the probability that the student was male, given the student had ever posted a video on You Tube. Answer : e. What is the probability that the student had ever posted a video on You Tube given the student was male. Answer : f. What is the probability that the student had ever posted a video on You Tube, given the student was female. Answer :
DPB 30063 / PROBABILITY 114 g. What is the probability that the student had never posted a video on You Tube given the student was female. Answer : h. What is the probability that the student was female, given the student jad never posted a video on You Tube. Answer : *** Summarizing : “and” – both / total “ given” – both / (row or column total) 19. A group of 262 students was polled and asked the question, if you drink soda, do not prefer a diet version. The responses of those who did drink soda are shown in the table below : Prefer Diet ? Female Male All No 72 96 168 Yes 58 36 94 All 130 132 262 a. If you choose a person from the survey at random, what is the probability the person preferred a diet soda. Answer : b. If you choose a person from the survey at random, what is the probability that the person was a female who preferred a diet soda. Answer : ________ c. What is the probability that the person is female, given that the person preferred diet soda. Answer : ____ d. What is the probability that the person preferred diet soda, given that the person is female. Answer :
DPB 30063 / PROBABILITY 115 20. Researchers asked students in Form 3 through form 5 in three school district in Pahang about what they thought was the most important thing in school; making good grades, being popular or being good in sports. There were rural, sub-urban and urban schools surveyed. Here is the breakdown of the results. Rural Suburban Urban All Good Grades 57 87 103 247 Popular 50 42 49 141 Sports 42 22 26 90 All 149 151 178 478 a. Find the probability the student were from suburban school and thought sports were most important. Answer : b. Find the probability student was from a suburban school given the student thought sports were most important. Answer : c. Find the probability student thought sports were most important, given the student was from a suburban school. Answer : 21. A researcher studied the way grocery stores displayed their cereal boxes. Different types of cereal were classified as being targeted to adults or children and it was noted whether the box was displayed at bottom, middle or top shelf. Bottom Middle Top All Adult 6 2 8 16 Child 2 6 0 8 All 8 8 8 24 a. What is the probability that the cereal was on the middle shelf and targeted to children. Answer : b. What is the probability the cereal was on the middle shelf, given that it was targeted to children. Answer :
DPB 30063 / PROBABILITY 116 22. Refer to table below : a. How many boys are in the class. ( ) b. How many girls are in the class. ( ) c. How many students who can swim are in the class. ( ) d. How many students who cannot swim are in the class. ( ) e. How many student are in the class. ( ) f. How many students are boys or swimmers. ( ) g. How many students are boys and swimmers. ( ) h. Find the probability that at randomly chosen student is a girl. ( ) i. Find the probability that a randomly chosen student is a non swimmer. ( ) j. Find the probability that a randomly chosen student is a girl who can swim. ( ) Swimmer None Total Boy 11 10 21 Girl 13 16 29 Total 24 26 50
DPB 30063 / ESTIMATION AND HYPOTHESIS TESTING 117 CHAPTER 7 : ESTIMATION AND HYPOTHESIS TESTING
DPB 30063 / ESTIMATION AND HYPOTHESIS TESTING 118 CHAPTER 7 : ESTIMATION AND HYPOTHESIS TESTING Learning Objective End of chapter, student be able to : 1. Explain the concept of estimation theory. 2. Initiate hypotesis testing. Introduction • Statistical estimation is the procedure of using a sample statistics to estimate a population parameter. • A statistics is used to estimate a parameter is called an estimator. • The value taken by the estimator is called an estimate. • For example, the sample mean (say 7.65) is an estimator of the population mean. • Statistical estimation is divided into two major categories :- a. Point estimation b. Interval Estimation a. Point Estimation • Used of number • In point estimation a single statistics is used to provide an estimate of the population parameter • Change in sample will cause deviation is estimate b. Interval Estimation (range in value) • An interval estimate is a range of value within which a researcher can say with some confidence that the population parameter fall. • This range is called confidence interval.
DPB 30063 / ESTIMATION AND HYPOTHESIS TESTING 119 7.1 What is Hypothesis • Is a statement about the value of a population parameter. • Or a premise or claim that we want to test. • Below are examples of hypotesis or statements made about population parameter :- i. The mean monthly income from all sources for senior citizens is RM 950.00. ii. Twenty percent of juvenile offenders ultimately are caught and sentences to prison. iii. The mean outside diameter of ball bearings produced during the day is 1.00 milimeter. iv. Ninety percent of the federal income taxs forms are filled out correnctly. 7.2 Hypothesis Testing Is a procedure based on sample evidence and probability theory used to determine whether the hypothesis is a reasonable statement and should not be rejected or is unreasonable and should be rejected. 7.3 Five Steps Procedure For Testing A Hypothesis There is a five step procedure that systematizes hypothesis testing; when we get to step 5, we are ready to make a decision to reject or not to reject the hypothesis. Step 1 : State the null and alternate hypothesis (Ho and H1 or Ha). Step 2 : Select the level of significance (the risk we assume of rejecting the null hypothesis when it is actually true. Step 3 : The test statistic Step 4 : Formulate the decision rule Step 5 : Making a decision (Accept Ho or Reject Ho and Accept H1)
DPB 30063 / ESTIMATION AND HYPOTHESIS TESTING 120 7.4 Terms in Hypothesis Null Hypothesis - Ho – currently accepted value for a parameter Alternative Hypothesis - Ha or H1 - also called research hypothesis. Involves the claim to be tested Level of confidence - How confident we are in our decision. C - 95% , 99% Level of significance - alpha (α) = 1 - C If LOC = 95% α = 1 - 0.95 = 0.05 7.5 Writing an Hypothesis Example 1 A company has stated that their straw mechine makes straws that are 4 mm diameter. A worker believes the machine after maintenance no longer makes straws of this size and samples 100 straws to perform the hypothesis test with 99% confident.
DPB 30063 / ESTIMATION AND HYPOTHESIS TESTING 121 Example 2 Doctors believe that the average teen sleeps on average no longer than 10 hours per day. A researcher believe that teens on average sleep longer. Write null hypothesis and alternative hypothesis. Example 3 The school board claims that at least 60% of students bring a phone to school. A teacher believes this number is too high and randomly samples 25 students to test at a level of significance of 0.02. Write null hypothesis and alternative hypothesis.
DPB 30063 / ESTIMATION AND HYPOTHESIS TESTING 122 7.6 Basic of Hypothesis Testing Null Hypothesis : Ho : ρ = # µ = # parameter proportion σ = # Alternative Hypothesis : H1 : ρ < # µ > # parameter proportion σ ≠ # Level of Significance ( α ) - aplha (0.01 , 0.05, 0.10) 7.7 Alternative Hypothesis Left tailed : H1 : < Right Tailed : H1 : > Two Tailed : H1 : ≠
DPB 30063 / ESTIMATION AND HYPOTHESIS TESTING 123 7.8 Test of Hypothesis for Large Sample Assumption : A sample of size n (large) from a normal population with unknown mean and unknown variance. Example 4 From 100 male and female students majoring in Law in the faculty of Laws, a random sample of 40 students has a mean grade point average of 2.70 with a sample standard deviation of 0.50. The gradepoint average for male and female students is assumed to be normally distributed. The officer also is required to test the hypothesis that the overall grade-point average for all students majoring in Law is different from 3.00, using 5 percent level of significance. Solution :
DPB 30063 / ESTIMATION AND HYPOTHESIS TESTING 124 Example 5 It is claimed that a car is driven on the average more than 20,000 km per year. To test this claim, a random sample of 30 cars owners has been selected. The owners were asked to keep record of the kilometres they travel. Would you agree with this claim if the random sample showed an average of 22,500 kilometres and a standard deviation of 5500 kilometres ? Use a 0.05 level of significance to justify your answer.
DPB 30063 / ESTIMATION AND HYPOTHESIS TESTING 125 7.9 Test of Hypothesis for Small Sample Assumption : A sample of size n (small) from a normal population with unknown mean and unknown variance Test Statistics : Example 6 Tenaga Nasional Berhad (TNB) has listed a variety of home appliances each with annual total electricity consumption in kilowatt-hours. For a medium-sized refrigerator, the annual electricity consumption is said to be an average of 46 kilowatt-hours. A random sample of 12 homes included in the study showed that medium sized refrigerator consume the electricty at an average of 42 kilowatt-hours per year with a standard deviation of 11.9 kilowatt hours. A a significant level of 0.05 and assuming that the population is normal, does this sample show that medium sized refrigerator consume on the average, less than 46 kilowatt-hours per year ? Solution :
DPB 30063 / ESTIMATION AND HYPOTHESIS TESTING 126 SELF REVISION Question 1 : From a random sample of 100 students who have passes a statistics course, the average score was 71.8. Assuming that the population standard deviation is 8.9, with a significance level of 0.05, does it seem to signify that the average score is more than 70 ? Solution :
DPB 30063 / ESTIMATION AND HYPOTHESIS TESTING 127 Question 2 : A flour mill produces flour in small bags before distributing them to wholesalers. The average weight of each bag is 8 kg with a standard deviation of 0.5kg. A random sample of 50 bags was taken and found that the average weight is 7.8kg. Using a significance level of 0.01, test the hypothesis that µ = 8kg against the alternative where µ ≠ 8kg.
DPB 30063 / PAST SEMESTER QUESTION 128 PAST SEMESTER QUESTION
DPB 30063 / PAST SEMESTER QUESTION 129 PAST SEMESTER QUESTION (SET 1) QUESTION 1 a. State the types of data (qualitative or quantitative) for each following characteristics : i. Number of family ii. Blood glucose level iii. Eye color iv. Marital status v. Ethnicity b. Affan has been working on programing and updating a Website for his company for the past 7 months. The following data represent the number of hours that Affan worked for each month : 24, 25, 31, 50, 53, 66, 78 Based on the above data, you are required to find : i. Mean ii. Median iii. Mode c. A salesman keeps a record of the number of shops he visits each day. Shops visited Frequency 0 – 9 3 10 – 19 8 20 – 29 21 30 – 39 60 40 - 49 21 Based on the frequency Table above, you are required to calculate : i. Mean ii. Median iii. Mode iv. Determine the form of data dispersion.
DPB 30063 / PAST SEMESTER QUESTION 130 QUESTION 2 a. Find the mean deviation from mean of the data given (given mean = 64) 45, 55, 63, 76, 67, 84, 75, 48, 62, 65 b. The quality of lightbulbs, estimated life span (burning hours) for 100 bulbs for brand A are stated as below : Life span of bulbs (in hours) Brand A 0 – 50 15 50 – 100 20 100 – 150 18 150 – 200 25 200 - 250 22 100 Based on the table above, you have to find : i. Mean ii. Standard Deviation c. Using information in (b), compare two brands of lightbulbs if Brand B has standard deviation of 37.51 and mean 136.5. Which brand has more life span consistency, by calculating coefficient of variation, CV ?
DPB 30063 / PAST SEMESTER QUESTION 131 QUESTION 3 a. The survey result of the average monthly rents (in RM) for one-bedroom apartments and twobedroom apartments in randomly selected metropolitan areas are shown in Table 3. Determine if there is a relationship between the rents by using spearman Rank correlation coefficient. One bedroom , x Two-bendrrom, y 782 1223 486 902 451 739 529 954 618 1055 520 875 845 1455 b. A doctor wishes to know whether there is a relationship between a mother’s weight (in kg) and her newborn baby’s weight (in kg). Mothers’s weight , x Baby’s weight, y 79.8 3.0 72.6 3.7 84.8 4.2 95.3 3.2 88.9 4.0 64.4 4.2 93.0 3.4 97.5 3.9 From the table above, you are required to : i. Draw a scatter plot. ii. Identify the regression equation, y = a + bx using least square method.
DPB 30063 / PAST SEMESTER QUESTION 132 QUESTION 4 a. The data below shows the number of calories listed per serving for selected ready to eat cereals. 130 190 140 80 100 120 220 220 110 110 210 130 100 90 210 120 200 120 180 120 190 210 120 200 130 180 260 270 100 160 190 240 80 120 90 190 200 210 190 180 115 210 110 225 190 130 From the above table, you are required to : i. Construct a frequency distribution that consist of class interval, frequency, class boundaries and midpoint. ii. Draw a frequency polygon. b. In winter, the probability that it rains on any one day is 5/7. i. Using a tree diagram, show all the possible combinations for two consecutive days. Write the probabilities for each of the branches. ii. Calculate the probability that it will rain on both days. iii. Calculate the probability that it will rain on at least once day.
DPB 30063 / PAST SEMESTER QUESTION 133 PAST SEMESTER QUESTION (SET 2) QUESTION 1 a. State the type of variable (qualitative or quantitative) for the following statements :- i. The numbers on the jerseys of volleyball players ii. A marital status of respondents iii. A social class of residents iv. An income level of respondents v. An academic qualification of respondents b. Identify the median, mode and mean. 95, 103 105 110 104 105 112 90 c. The following table shows the heights of 50 students. Height Number of students 145 – 149 1 150 – 154 2 155 – 159 16 160 – 164 19 165 – 169 5 170 – 174 6 175 - 179 1 Calculate : i. Mean ii. Median iii. Mode
DPB 30063 / PAST SEMESTER QUESTION 134 QUESTION 2 a. Determine the mean deviation for the following data : 4, 7, 11, 3, 1 b. A sample of the monthly amount invested in the HARITH Company’s profit-sharing plan by employees was organizes into a frequency distribution table for futher study. Amount Invested (RM) Number of Employees 30 – 34 3 35 – 39 7 40 – 44 11 45 – 49 22 50 – 54 40 55 – 59 24 60 – 64 9 65 - 69 4 Calculate Variance and Standard Deviation. c. Based on the answer in (b), calculate Pearson’s Coefficient of Skewness 1 (PCS 1) when mean = 29.9 and mode = 33.83 and interpret a conclusion.
DPB 30063 / PAST SEMESTER QUESTION 135 QUESTION 3 a. The table below shows the interest rates for car loans and the number of customers who apply for the loans in a month from a finance company. Interest rate in % Number of applicants 6.0 80 6.2 80 6.5 78 6.8 75 7.0 70 7.2 60 7.5 60 7.8 55 8.0 50 8.2 48 8.4 45 8.7 40 Draw the Scatter Diagram for the above data. b. A production manager collected the data below on production cost and the quantity produced for 10 consecutive days. These data are given below : Day Quantity (‘000) units Cost (RM’000) 1 10 20 2 13 28 3 20 38 4 18 35 5 17 33 6 15 30 7 16 34 8 14 29 9 11 23 10 12 25 By using the Least Squares Method, calculate the regression equation for cost.
DPB 30063 / PAST SEMESTER QUESTION 136 QUESTION 4 a. The data below show the distance (in km) covered by 24 cars within 2 hours. 140 128 125 149 96 108 136 84 112 123 120 130 89 103 103 65 97 145 87 102 78 98 126 67 i. Illustrate frequency distribution table consist of tally, relative frequency, cumulative frequency, midpoint and class boundaries by using 60 as the lower limit of class interval and all classes having a uniform class size of 15. ii. Draw a “less than” ogive for the frequency distribution above. b. Three companies, A, B and C are competing for a contract to build a condominium. The probabilities that companies A, B and C will win the contract are 0.25, 0.45 and 0.3 respectively. If company A, B and C win the contract, the probability that they will make profits are 0.8, 0.9 and 0.7 respectively. i. Illustrate a tree diagram based on the information given in the above statement. ii. Calculate the probability that the companies will make profits. iii. If the contract is found to be profitable, calculate the probability that the contract is given to Company A.
DPB 30063 / PAST SEMESTER QUESTION 137 PAST SEMESTER QUESTION (SET 3) QUESTION 1 a. List five (5) types of data collection method. b. Define the following terms :- i. Mean ii. Mode c. Identify THREE types of average measures of central tendency. d. The table below shows the working experience in months of 120 employees at Karu Company. Working experience in months Number of employees 11 – 14 16 15 – 18 20 19 – 22 28 23 – 26 24 27 – 30 16 31 – 34 11 35 - 38 5 Calculate : i. Mean ii. Standard Deviation iii. Pearson’s coefficient of skewness 1.
DPB 30063 / PAST SEMESTER QUESTION 138 QUESTION 2 a. A market survey on a sample 150 customers purchasing a brand of perfume reveals the following age distribution. Age (in years) Number of customers 15 – 19 9 20 – 24 16 25 – 29 27 30 – 34 44 35 – 39 42 40 – 44 10 45 – 49 2 Total 150 i. Draw a histogram to represent the above data. ii. Schedule a ‘more than’ ogive table to represent the above data. iii. Sketch a ‘more than’ ogive on a graph paper to represent the above data. b. In a small village, one bus arrives a day. The probability of rain in the village is 0.3. If it rains, the probability of a bus being late is 0.4. If it does not rain, the probability of a bus being late is 0.15. i. Sketch a tree diagram to summarize the above information ii. Identify the probability, if that bus being late, the probability that rain in the village.
DPB 30063 / PAST SEMESTER QUESTION 139 QUESTION 3 (a) Define the following terms : i. Statistics ii. Population iii. Sample iv. Face to face interview v. Telephone interview (b) The table below shows the Mathematics and Science test scores for five students : Students Mathematics Science AA 96 98 BB 89 75 CC 88 90 DD 66 50 EE 72 60 i. Draw a scatter for the above data and determine the type of relationship between the Mahematics test scores and the Science test scored. ii. Calculate Spearman’s rank correlation. QUESTION 4 (a) The following data shows the salary of eight workers at Farid Inc. RM 550 RM 950 RM 600 RM 850 RM 700 RM 610 RM 750 RM 600 Find the value of :- i. Mean ii. Median iii. Mode iv. Range
140 REFERENCES Faizah Omar, Lau Too Kya, Phang Yook Ngor and Zainudin Awang. (2019). Statistics. Fourth Edition. Oxford Fajar Muhammad Rozi Malim, Faridah Abdul Halim. (2011). Business Statistics. Oxford Fajar Zuraini Zainal Abidin, Hamidah Abd Latif and Mohd Rakimi Shaffai. (2021). Statistics Workbook. First Edition
T-table Lampiran A
Terbitan