Engineering Probability and Statistics Problem Set

Statistics and Probability

Exercises

De La Salle University - Dasmari˜ nas College of Science and Computer Studies MATHEMATICS AND STATISTICS DEPARTMENT Dasmari˜ nas City Exercises in Statistics and Probability

Contents 1 Basic Concepts and Terms in Statistics

3

2 Sampling Techniques

5

3 Methods of Data Presentation

6

4 Measures of Central Tendency

10

5 Measures of Location

12

6 Measures of Dispersion

13

7 Random Experiment and Related Terms

15

8 Counting 8.1 Fundamental Counting Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Permutation and Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16 16 18

9 Probability

20

10 Bayes’ Theorem

23

11 Random Variables and Probability Distributions

25

12 Discrete Probability Distribution 12.1 Binomial Distribution . . . . . . . 12.2 Negative Binomial Distribution . 12.3 Geometric Distribution . . . . . . 12.4 Poisson Distribution . . . . . . . 12.5 Hypergeometric Distribution . . .

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27 27 27 27 28 28

13 Continuous Probability Distribution 13.1 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 30

14 Estimation 14.1 Estimation of Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Estimation of Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Prepared by: Celine Sarmiento

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Page 1 of 37

Statistics and Probability

Exercises

15 Hypothesis Testing 15.1 Test on the Mean of a Single Population . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Test on the Difference of Means of Two Population . . . . . . . . . . . . . . . . . . . .

34 34 34

16 Correlation and Regression

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Prepared by: Celine Sarmiento

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Statistics and Probability

1

Exercises

Basic Concepts and Terms in Statistics

Identify whether the specified study would be descriptive or inferential statistics. 1. A tire manufacturer wants to estimate the average life of a new type of steel-belted radial. 2. A sportswriter plans to list the winning times for all swimming events in the 2,000 Olympics. 3. A politician obtains the exact number of votes casted for his opponent in 1992. 4. A medical researcher tests an anticancer drug that may have harmful side effects. 5. A 1984 study by the an organization concluded that an estimated 83% of households in Cavite are involved in at least one form gardening. 6. A bowler wants to find his bowling average for the past 12 games. 7. A basketball player wants to estimate his chance of winning the most valuable player award based on his current season averages and the averages of his opponents. 8. As of 2004, actor Tom Cruise had received a total of 3 academy award (Oscar) nominations. 9. In 2000, the population of Nome, Alaska, was 3,505 residents. 10. Psychology students with a undergraduate GPA 0f 3.5 would be expected to attain a graduate school GPA of at least 3.0. Identify the population and the variable of interest in the following: 1. A statistics teacher, would like to know the effect of using computer-aided instruction in the performance of Statistics classes. 2. The head librarian would like to identify the journals commonly borrowed by DLSU-D students. 3. Some parents would like to determine whether social networking is good or bad to the habits of their children. 4. A group of students taking Statistics conducted a study on the effect of boy-girl relationship to the academic performance of the students. 5. A study to be conducted by an NGO would determine the Filipinos awareness about the laws governing taxation. Identify the following variables as either qualitative or quantitative. 1. The number of people in a jury. 2. The color of your house. 3. A persons height in feet. 4. The speed of a car in miles per hour.

Prepared by: Celine Sarmiento

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Statistics and Probability

Exercises

5. Outcome of tossing a coin. 6. Classify the colors of automobiles on a used car lot. Identify whether the variables are continuous or discrete. 1. Country in North America. 2. Volume of a parallelepiped. 3. Product of the numbers shown when a pair of dice is tossed. 4. Temperatures recorded at intervals during a day. 5. Weight of each bunch of grapes sold at a supermarket yesterday. 6. Temperatures recorded every half hour at a weather bureau. 7. Lifetimes of television tubes produced by a company. 8. Yearly incomes of college professors. 9. The driving distance from San Diego to Boston is 3,043.83 miles. 10. According to the 2,000 U.S. Census, the population of Chicago was 2,896,016 persons. Select the level of measurement attained by the following sets of observations or data. 1. A students city of residence 2. Average height of men on a basketball team 3. A viewers excellent rating of a movie on a scale of excellent, good, fair, or poor 4. Weights recorded for two women participating in a weight management program 5. Student number 6. Basketball players jersey number 7. Weighs of a sample of candies 8. Zip codes 9. SSS number 10. Final course grades 11. Movies on a certain TV show are classified as 2 thumbs up, 1 thumb up, or 0 thumbs up 12. Voters are classified as low-income, middle-income, or high-income 13. The colors of MM candies 14. Percentage scores on a Math exam 15. Letter grades on an English essay

Prepared by: Celine Sarmiento

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Statistics and Probability

2

Exercises

Sampling Techniques

Identify the type of sampling used in the following statements. 1. An architect selects every 13th paint sample from the assembly line for careful testing and analysis. 2. A magazine editor obtains sample data from readers who decide to send through fax the questionnaire printed in the latest issue. 3. A professional photographer selects 18 men and 18 women from each of four groups of models. 4. A Ph.D. student surveyed all students from 4 randomly selected universities in Cavite. 5. A reporter interviews a man on the street regarding the 2010 elections. 6. Victor is studying environmental engineers but can only find five. He asks these engineers if they know any more. They give him several referrals, which in turn provide additional contacts. In this way, he manages to contact sufficient engineers. 7. Noemi wants 100 opinions about a new style of cheese. She sets up a stall and canvasses passers-by until she has got 100 people to taste the cheese and complete the questionnaire. 8. Casey Cartwright, ENT student, would like to know the views of the street vendors about the price increase of commodities. She interviews every third vendor she encounters along the street. 9. An actress is preparing for a role as a person who has recovered from severe drug addiction. She decided to get information from people with experience in such a case. To determine the persons to interview, she searched for one person and asked this person to refer someone he knows who has been through the same experience.

Prepared by: Celine Sarmiento

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Statistics and Probability

3

Exercises

Methods of Data Presentation

Read, analyze and solve each problem carefully. 1. Every New Year’s Day, members of the Jacksonport Polar Bear Club gather at the shore of Lake Michigan, strip down to their bathing suits, and plunge into the icy water! Look at the table for details on this zany event. Then answer the questions.

(a) In which two years did the Polar Bears have the coldest temperature for their plunge? (b) In what year did the greatest number of people take part in this event, and in what year did the fewest take the plunge? (c) In general how has the popularity of this even changed over the years since 1993? Tell how you know. (d) What do you notice about the changes in the number of participants and the air temperatures from 1998 to 1999? Do you think these two factors are related? 2. Ms. Hearditall keeps track of hard-to-believe excuses her students give for not doing their homework. Check out this circle graph to find out what kinds of excuses Ms. Hearditall gets. Then answer the questions.

(a) Which kind of excuse do Ms. Hearditall’s students give most often? Prepared by: Celine Sarmiento

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Statistics and Probability

Exercises

(b) What percentage of the excuses are pet problems? (c) Which kind of excuse makes up 17% of the total? (d) Which kind of excuse is used least? 3. Have you ever thought about collecting something odd - such as chamber pots or bags of potato chips or refrigerator magnets? People collect all of these things and many more. Take a look at this bar graph of odd stuff. Then answer the questions.

(a) Which collection has the greatest number of items? (b) How many rubber ducks are in the world’s largest rubber-duck collection? (c) Which collection has about the same number of items as the world’s largest collection of clothing tags? (d) About how many more piggy banks are there than yo-yos in the record-setting collections? (e) How many items are there in the ”airplane sickness bags” collection? (Let’s hope those bags are empty!) 4. Does it sometimes seem that everywhere you look, people are talking on cell phones or instant messaging friends on the internet? Well, they are - and in huge numbers! Look at the line graphs below to see just how many people are on the grid. Then answer the questions.

Prepared by: Celine Sarmiento

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Statistics and Probability

Exercises

(a) About how many Americans used cell phones in 1990? (b) In what year did the number of people in the United States using cell phones reach 50 million? (c) How many people were using the Internet in 1996? (d) How many people were using the Internet in 1998? (e) Which one-year period saw the greatest increase in the number of Internet users? (f) Describe the rate of increase in cell-phone use from 1990 through 1996. (g) How many Americans used cell phones in 2002? (h) Based on the information shown in the graphs, estimate the number of cell-phone users in the U.S. in 2005. (i) Estimate the number of Internet users in 2005. (j) What similarity do you notice in these two graphs? 5. From 1982 to 2002, the average minimum April temperature (Celsius) was recorded as follows: 6, 1, 8, 9, 6, 9, 7, 2, 7, 0, 6, 2, 5, 7, 6, 2, 6, 8, 6, 4, 6, 8, 6, 4, 7, 6, 7, 8, 7, 3, 6, 8, 8, 8, 7, 8, 8, 1, 8, 1, 7, 9 (a) Prepare a complete frequency distribution table for the given quantitative data. (b) Draw the histogram that represents the frequency distribution. (c) Draw the frequency polygon that represents the frequency distribution. 6. Assume the annual numbers of road fatalities from 1960 to 1992 were as follows: 10, 7, 8, 8, 17, 15, 17, 23, 14, 26, 31, 20, 32, 29, 31, 32, 38, 29, 30, 24, 30, 29, 26, 28, 37, 33, 32, 36, 32, 32, 26, 17, 20 (a) Prepare a complete frequency distribution table for the given quantitative data. Prepared by: Celine Sarmiento

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Statistics and Probability

Exercises

(b) Draw the histogram that represents the frequency distribution. (c) Draw the frequency polygon that represents the frequency distribution. 7. Fifty staff members of a construction company were surveyed to find out what their weekly salary was to the nearest dollar. The results are as follows: 514, 476, 497, 511, 484, 513, 471, 470, 441, 466, 443, 481, 502, 528, 459, 548, 521, 517, 463, 478, 473, 514, 542, 519, 522, 523, 546, 487, 486, 473, 527, 470, 440, 564, 499, 523, 484, 463, 461, 437, 555, 525, 461, 539, 466, 470, 486, 490, 543, 519 (a) Prepare a complete frequency distribution table for the given quantitative data. (b) Draw the histogram that represents the frequency distribution. (c) Draw the frequency polygon that represents the frequency distribution. 8. Complete the FDT below by entering the missing information.

Prepared by: Celine Sarmiento

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Statistics and Probability

4

Exercises

Measures of Central Tendency

Read, analyze and solve each problem carefully. 1. Determine the mean, median, and mode of the following sets of ungrouped data. (a) 21, 19, 16, 18, 19, 22, 25, 26, 35, 35, 40, 42, 41, 26, 27 (b) 5, 6, 6, 9, 12, 12, 14, 15, 12, 57, 45, 35, 28, 12 2. A sample of 20 families gave the following data on the number of children per family. 0, 1, 3, 4, 4, 4, 2, 2, 3, 5, 6, 7, 5, 6, 5, 3, 3, 4, 5, 0 Find the mean, median, and mode. 3. Caily is working part time at a restaurant in Makati. If she works for 30 hours in one week on the average, how many hours a day does she work? (Consider 6 days a week work.) 4. The average of four numbers is 40. The third is 12 more than the second. The second is five times the first. The fourth is 8 less than the first. Find the numbers. 5. The mean height of Dan, Charles, Steve, and Sean is 66 inches. What is the height of Billy if the mean height of the five boys is 70 inches? 6. The six departments of a company, consisting of 22, 32, 18, 16, 10, and 12 employees have an equal monthly salary of Php 7,200, Php 7,600, Php 6,900, Php 8,200, Php 7,800, and Php 7,200, respectively. What is the mean monthly salary of all the employees of the company? 7. The distribution of scores of 50 students in a Statistics test is given in the table below. Scores Frequency 49-55 3 6 56-62 63-69 11 15 70-76 77-83 10 3 84-90 91-97 2 N = 50 Find the mean, median, and mode of the given grouped data. 8. The following are the daily salaries of a group of employees. Daily Salary in Peso Frequency 300-399 15 400-499 21 500-599 45 600-699 39 700-799 24 800-899 6 N = 150 Prepared by: Celine Sarmiento

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Statistics and Probability

Exercises

Find the mean, median, and mode of the given grouped data. 9. Find the mean, median, and mode of the grouped data below. Class Interval Frequency 16-22 13 16 23-29 30-36 24 37-43 19 44-50 8

Prepared by: Celine Sarmiento

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Statistics and Probability

5

Exercises

Measures of Location

Read, analyze and solve each problem carefully. 1. Determine the P60 , P27 , D4 , D9 , and Q3 of the following sets of ungrouped data. (a) 21, 19, 16, 18, 19, 22, 25, 26, 35, 35, 40, 42, 41, 26, 27 (b) 5, 6, 6, 9, 12, 12, 14, 15, 12, 57, 45, 35, 28, 12 2. A sample of 20 families gave the following data on the number of children per family. 0, 1, 3, 4, 4, 4, 2, 2, 3, 5, 6, 7, 5, 6, 5, 3, 3, 4, 5, 0 Find P88 , P78 , D2 , D1 , and Q1 . 3. The distribution of scores of 50 students in a Statistics test is given in the table below. Scores Frequency 49-55 3 56-62 6 63-69 11 15 70-76 77-83 10 84-90 3 2 91-97 N = 50 Find P45 , D8 , and Q1 of the given grouped data. 4. The following are the daily salaries of a group of employees. Daily Salary in Peso Frequency 300-399 15 400-499 21 500-599 45 600-699 39 24 700-799 800-899 6 N = 150 Find P95 , D4 , and Q3 of the given grouped data. 5. Find P32 , D2 , and Q2 of the grouped data below. Class Interval Frequency 16-22 13 16 23-29 30-36 24 37-43 19 44-50 8

Prepared by: Celine Sarmiento

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Statistics and Probability

6

Exercises

Measures of Dispersion

Read, analyze and solve each problem carefully. 1. Find the range, variance, standard deviation, and coefficient of variation of the following ungrouped data. (a) 0, 2, 4, 6, 8 (b) 1, 3, 5, 7, 9 (c) 36, 49, 75, 89, 95, 99 (d) 510, 607, 705, 810, 849, 999 (e) 10.5, 6.75, 4.15, 9.18, 8.25, 12.05 2. The following scores were obtained by a class of boys and girls in a 20 item test in mathematics. • Boys: 6, 7, 8, 9, 10, 10, 16, 18 • Girls: 4, 5, 6, 7, 7, 8, 8, 10, 10, 12, 15, 19 Find the range, variance, standard deviation, and coefficient of variation for: (a) the boys. (b) the girls. (c) the whole class. 3. Three basketball players are being considered for selection for an international competition. Their latest records showed that the points they made in several games played. • Player A: 17, 30, 24, 26, 28, 10, 16, 17, 18 • Player B: 29, 34, 14, 16, 26, 10, 25, 20, 30 • Player C: 22, 27, 12, 16, 20, 25, 25, 26, 22 Who among they would you select? Why? 4. The distribution of scores of 50 students in a Statistics test is given in the table below. Scores Frequency 49-55 3 56-62 6 63-69 11 70-76 15 77-83 10 84-90 3 91-97 2 N = 50 Find the variance and standard deviation of the given grouped data. 5. The following are the daily salaries of a group of employees. Prepared by: Celine Sarmiento

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Statistics and Probability

Exercises

Daily Salary in Peso Frequency 300-399 15 400-499 21 500-599 45 39 600-699 24 700-799 800-899 6 N = 150 Find the variance and standard deviation of the given grouped data. 6. Find the variance and standard deviation of the grouped data below. Class Interval Frequency 16-22 13 23-29 16 24 30-36 37-43 19 8 44-50 7. Shown below are Squidward’s scores, the mean, and standard deviation of each three tests given to 500 applicants. Test Squidward’s Score Mean Standard Deviation Math 95 90.5 9.5 Science 88 85 10 English 90 88.6 5.5 Compute the standard score of Squidward for each test.

Prepared by: Celine Sarmiento

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Statistics and Probability

7

Exercises

Random Experiment and Related Terms

Read, analyze and solve each problem carefully. 1. An experiment involves tossing a fair coin twice. List down the elements of the sample space Ω of this experiment. 2. If the possible blood types are A, B, AB, and O, and each type can be (+) or (-), draw a tree diagram for all the possible classifications of blood types and create the sample space Ω for selecting a blood sample. 3. Students in a certain university are classified as male (M) or female (F); freshmen (1), sophomore (2), junior (3) or senior (4); and regular (R) or irregular (I). Draw a tree diagram for all the possible classification of students and create the sample space Ω for randomly selecting student from this university. 4. A box contains a black marble (B), a red marble (R), and a yellow marble (Y). Two marbles are selected without replacement. Draw a tree diagram and determine the sample space of this experiment. 5. Determine the sample space for the genders of children in a family consisting of three children. List down the elements of the following events and indicate what type of events it is. (a) A = {The children are all boys.} (b) B = {The children are 2 girls and a boy in any order.} (c) C = {The children consist of at least 2 girls.} 6. A single die is rolled. List the outcomes in each event and indicate what type of event it is. (a) A = {Getting an odd number.} (b) B = {Getting a number greater than four.} (c) C = {Getting a number less than one.} 7. Consider rolling a fair die twice. List the elements of the following events. (a) A = {The sum of the numbers on top is 5.} (b) B = {The sum of the numbers on top is at least 8.} (c) C = {The sum of the numbers on top is at most 4.} (d) D = {The sum of the numbers on top is equal to 7 or 10.} (e) E = {The sum of the numbers on top is equal to 7 and the product is 12} (f) F = {The number on top of the first is smaller than the number on top of the second.}

Prepared by: Celine Sarmiento

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Statistics and Probability

8

Exercises

Counting

Read, analyze and solve each problem carefully.

8.1

Fundamental Counting Principle

1. Say the only clean clothes Clint has are 2 shirts and 3 pairs of jeans. How many different ways can Clint dress up with a shirt and a pair of jeans? 2. In a class of 7 students, four are to be chosen to sit in a row for a picture. How many linear arrangements are possible? 3. A newborn male child may be given one (such as John) or two (such as John Patrick) names. Assuming that the name will not be repeated (that is, no child may be named John John), if there are 150 names to choose from, in how many ways can a child be named? 4. How many different 4-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 if: (a) repetition is not allowed? How many of these numbers are even? How many are these numbers are odd? (b) repetition is allowed? 5. A test consists of 15 multiple choice questions, with each question having four possible answers. In how many different ways can a student check off one answer to each question? 6. Three-digit numbers are formed using the digits 2, 4, 5, and 9, without repetition of digits. How many such numbers can be formed if the numbers are divisible by 5? 7. Suppose a code consists of two letters followed by a digit. Find the number of: (a) codes. (b) codes with distinct letters. (c) codes with the same letters. 8. Three cards are chosen in in succession from a deck with 52 cards. Find the number of ways this can be done: (a) with replacement. (b) without replacement. 9. Michael will draw two cards successively from a standard deck of playing cards. (a) With replacement, how many ways can he choose: i. 2 kings. ii. a red card and a spade. iii. 2 queens or 2 aces. (b) Without replacement, how many ways can he choose: i. 2 kings. Prepared by: Celine Sarmiento

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Statistics and Probability

Exercises

ii. a red card and a spade. iii. 2 queens or 2 aces. 10. There are four bus lines from city A to city B and three bus lines from city B to city C. Find the number of ways a person can travel by bus: (a) from A to C by way of B. (b) round-trip from A to C by way of B. (c) round-trip from A to C by way of B, without using a bus line more than once. 11. Suppose 50 science students are polled to see whether or not they have studied French or German yielding the following data: 25 studied French, 20 studied German, 5 studied both. Find the number of student who studied (Hint: Use Venn Diagram): (a) only French. (b) only German. (c) neither language. 12. In a survey of 60 people, it was found out that: 25 read Newsweek magazine, 26 read Time, 26 read Fortune, 9 read both Newsweek and Fortune, 11 read both Newsweek and Time, 8 read both Time and Fortune, 3 read all three magazines (Hint: Use Venn Diagram). (a) Find the number of people who read at least one of the three magazines. (b) Find the number of people who read exactly one magazine. 13. Suppose a bookcase shelf has 6 mathematics texts, 3 physics texts, 4 chemistry texts, and 5 computer science texts. Find the number of ways a student can choose: (a) one of the texts. (b) one of each type of text. 14. A history class contains 7 male students and 5 female students. Find the number of ways that the class can elect: (a) a class representative. (b) two class representatives, one male and one female. (c) any two class representatives. (d) a president and a vice-president. 15. Five cards are selected from a 52-card deck for a poker hand. (a) How many different hands are there? (b) How many hands contain all four aces? (c) How many hands contain 3 face cards and an even numbered card? (d) A royal flush is a hand that contains the ace, king, queen, jack, and 10 cards of the same suit. How many ways are there to get a royal flush? 16. How many 9 letter (using letters of the English alphabet) palindromes are possible? (A palindrome is a string of letters that reads the same backward and forward, such as the string XCZCX.) Prepared by: Celine Sarmiento

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Statistics and Probability

8.2

Exercises

Permutation and Combination

1. Four mathematics books and three chemistry books are to be placed on a shelf. In how many ways can this be done if the mathematics books are next to each other and the chemistry books are next to each other? 2. From a group of 8 men and 5 women a committee of 4 people is to be formed. In how many ways can this be done if the committee must contain at least 1 woman? 3. In how many ways can 5 people line up for a group picture if: (a) two want to stand next to each other? (b) two refuse to stand next to each other? 4. In how many ways can 5 girls and 3 boys be arranged in a row without restriction? such that the 3 boys are together? such that all the girls are together and all the boys are together? 5. How many ways can 3 blue, 4 red and 2 green identical bulbs be arranged in a string of Christmas tree lights with 9 sockets? 6. Eight persons, consisting of 4 married couples are to be seated in a row of 8 chairs. How many sitting arrangements are there if there are no restrictions? the women must sit together? then men must sit together and the women must sit together? each married couple must sit together? 7. From a bag containing 7 black balls and 5 white balls, how many sets of 5 balls, of which 3 are black and 2 are white, can be drawn? 8. During a JS Prom Night, 10 boys and 8 girls were nominated for the Prince and Princess of Hearts. How many ways can a Prince, a Princess, and a Princesss court of two girls can be selected? 9. Blues Pizza Palace sells plain pizza or with one or more of the following toppings: pepperoni, sausage, mushroom, or pineapple. How many different pizzas can be made? 10. Find the number of permutations that can be formed from all the letters of each word: (a) BASIC (b) STATISTICS (c) MATHEMATICS (d) PROBABILITY (e) PHILIPPINES 11. A railway coach has 12 seats facing backwards and 12 seats facing forwards. In how many ways can 10 passengers be seated if 2 refuse to face forwards and 4 refuse to face backwards? 12. Find the number of ways 12 students can be partitioned in to three teams: A, B, C so that each team contains four students. 13. A box contains 7 blue socks and 5 red socks. Find the number of ways two socks can be drawn from the box if: (a) they can be any color. Prepared by: Celine Sarmiento

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Statistics and Probability

Exercises

(b) they must be the same color. 14. A class contain 10 students with 6 men and 4 women. Find the number of ways: (a) a 4-member committee can be selected from the students. (b) a 4-member committee with 2 men and 2 women can be selected. (c) the class can elect a president, vice-president, treasurer, and secretary. 15. A carton of 12 transistor batteries includes one that is defective. In how many different ways can an inspector choose three of the batteries and: (a) get one that is defective. (b) not get the one that is defective.

Prepared by: Celine Sarmiento

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Statistics and Probability

9

Exercises

Probability

Read, analyze and solve each problem carefully. Express your answers as fractions in simplest form or as decimals rounded to four significant digits. 1. A spinner has 5 equal sectors colored orange, red, green, blue and white. What is the probability of landing on green after spinning the spinner? 2. A single 6-sided fair die is rolled. What is the probability that the number on top is an odd number? 3. A glass jar contains 7 red, 5 orange, 6 pink and 4 violet marbles. If a single marble is chosen at random from the jar, what is the probability of choosing a violet marble? 4. A coin is tossed twice. What is the probability that at most one tail occurs? 5. A glass jar contains 7 red, 5 orange, 6 pink and 4 violet marbles. If a single marble is chosen at random from the jar, what is the probability of choosing a marble that is NOT violet? 6. In a group of 110 engineering students, 27 are taking up BS Electronics and Communications Engineering (ECE) and 31 are taking up BS Civil Engineering (CE). Find the probability that a student picked from this group at random is either an ECE OR CE student. 7. If a card is drawn at random from an ordinary deck of 52 cards, find the probability that it is a flower or an odd numbered card. 8. The probability that a patient entering St. Lukes Hospital will consult a physician in 0.7, that he/she will consult a dentist is 0.5 and that he/she will consult a physician or a dentist or both is 0.9. What is the probability that a patient entering the hospital will consult both a physician and a dentist? 9. Suppose you roll a pair of dice, a blue one and a green one. What is the probability that the sum is 9 given that the blue die shows the number 5?

Prepared by: Celine Sarmiento

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Statistics and Probability

Exercises

10. A certain airplane has two independent alternators to provide electrical power. The probability that a given alternator will work on a 1-hour flight is 0.89. What is the probability that on a 1-hour flight: (a) both will work? (b) both will not work? (c) one will work and the other will not? 11. From the integers from 1 to 40 inclusive, one integer is selected at random. What is the probability that it is: (a) a perfect square. (b) a prime number. (c) a multiple of 9. (d) a perfect square or composite. (e) a perfect square or not a multiple of 2. 12. If two cards are dealt from a standard 52-card deck successively, without replacement, what is the probability that both are diamonds? 13. Four radar sets, operating independently, are set to detect any aircraft flying through a certain area. Each set has a probability of 0.032 of failing to detect a plane. If an aircraft enters the area, what is the probability that it goes undetected? 14. Suppose we have a box containing 25 flashlight bulbs, of which 8 are defective. If three bulbs are selected at random and removed from the box in succession without replacing the first, what is the probability that the first 2 are working properly and the third one is defective? 15. The table below shows the distribution of 350 engineering students of a particular university according to their specialization and their gender. Specialization Electrical Engineering Civil Engineering Electronics and Communications Engineering Mechanical Engineering Industrial Engineering Chemical Engineering

Male 43 56 49 39 34 43

Female 12 20 18 5 9 22

If an engineering student is chosen at random, (a) what is the probability that he is a male? (b) what is the probability that the student is a mechanical engineering student? (c) what is the probability that the student is taking civil engineering or electrical engineering? (d) what is the probability that the student is a female or taking industrial engineering? (e) what is the probability that the student is a male or taking chemical engineering? Prepared by: Celine Sarmiento

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Statistics and Probability

Exercises

(f) what is the probability that the students is a male or not taking electrical engineering nor electronics and communications engineering? (g) what is probability that the student is a female and taking electrical engineering? (h) what is the probability that the students is a female and taking civil engineering or a male and taking industrial engineering? 16. A community swim team has 170 members. Eighty two (82) of the members are advanced swimmers. Thirty eight (38) of the members are intermediate swimmers. The remainder are novice swimmers. Fifty six (56) of the advanced swimmers practice 5 times a week. Twenty three (23) of the intermediate swimmers practice 5 times a week.Twelve (12) of the novice swimmers practice 5 times a week. Suppose one member of the swim team is randomly chosen. (a) What is the probability that the member is a novice swimmer? (b) What is the probability that the member practices 5 times a week? (c) What is the probability that the member is an advanced swimmer and practices 5 times a week? 17. A bag contains 180 balls that are either pink or purple and either dull or shiny. There are 45 shiny pink balls, 70 shiny balls, and 62 pink balls. If a ball is chosen at random, what is the probability that it is either a shiny ball or a pink ball? What is the probability that it is a dull purple ball? 18. A jar contains 19 green marbles numbered 1 to 19, and 35 white marbles numbered 1 to 35. A marble is drawn at random from the jar. Find the probability that the marble is white or even-numbered. 19. Urn A contains seven blue chips and six yellow chips; Urn B contains five blue and seven yellow chips. Two chips are drawn simultaneously from Urn A and placed in Urn B. Then a single chip is drawn from Urn B. What is the probability that the chip drawn from Urn B is blue?

Prepared by: Celine Sarmiento

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Statistics and Probability

10

Exercises

Bayes’ Theorem

Read, analyze and solve each problem carefully. Express your answers as fractions in simplest form or as decimals rounded to four significant digits. 1. In a study of pleas and prison sentences, it is found that 45% of the subjects studied were sent to prison. Among those sent to prison, 40% chose to plead guilty. Among those not sent to prison, 60% chose to plead guilty. (a) If one of the study subjects is randomly selected, find the probability of getting someone who was sent to prison. (b) If one of the study subjects is randomly selected, find the probability of getting someone who was NOT sent to prison. (c) If a study is randomly selected and it is then found that the subject entered a guilty plea, find the probability that this person was sent to prison. (d) If a study is randomly selected and it is then found that the subject entered a guilty plea, find the probability that this person was NOT sent to prison. 2. Small cars get better mileage, but they are not as safe as bigger cars. Small cars accounted for 18% of the vehicles on the road, but accidents involving small cars led to 11,898 fatalities during the recent year (Reader’s Digest, May 2000). Assume the probability a small car is involved in an accident is 0.18. The probability of an accident involving a small car leading to fatality is 0.128 and the probability of an accident not involving a small car leading to fatality is 0.05. Use the following notations: • S = a small car is involved in an accident • S 0 = NOT a small car involved in an accident • F = the accident lead to fatality Determine the following: (a) What is the probability that a small car was involved in an accident? (b) What is the probability that NOT a small car was involved in an accident? (c) If a small car was involved, what is the probability that the accident lead to fatality? (d) If a small car was NOT involved, what is the probability that the accident lead to fatality? (e) Suppose you learn of an accident involving a fatality, what is the probability that a small car was involved? 3. A student answers a multiple choice examination question that has 4 possible answers. Suppose that the probability that the student knows the answer to the question is 0.80 and the probability that the student guesses is 0.20. If the student guesses, the probability of getting the correct answer is 0.25. If it is answered correctly, what is the probability that the student really knew the correct answer?

Prepared by: Celine Sarmiento

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Statistics and Probability

Exercises

4. Two masked robbers try to rob a bank but the teller presses a button that sets off an alarm and locks all the doors. The robbers realizing they are trapped, throw away their masks and disappear into the chaotic crowd. Confronted with 53 people claiming they are innocent, the police gives everyone a lie detector test. Suppose that guilty people are detected with probability 0.85 and innocent people appear to be guilty with probability 0.08. What is the probability that Mr. Santos was one of the robbers given that the lie detector says he is?

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Statistics and Probability

11

Exercises

Random Variables and Probability Distributions

Read, analyze and solve each problem carefully. 1. Determine whether the random variable in each situation is discrete or continuous. (a) The number of newspapers sold by New York Times each month. (b) The amount of ink used in printing the Sunday edition of the New York Times. (c) The actual number of ounces in a gallon bottle of laundry detergent. (d) The number of defective parts in a shipment of nuts and bolts. (e) The number of people collecting unemployment insurance each month. (f) The closing price of a particular stock on the New York Stock Exchange. (g) The number of shares of a particular stock that are traded on a particular day. (h) The quarterly earnings of a particular firm. (i) The percentage change in yearly earnings between 2008 and 2009 for a particular firm. (j) The number of new products introduced per year by a firm. (k) The time until a pharmaceutical company gains approval from the U.S. Food and Drug Administration to market a new drug. 2. Two dice are rolled and we define the familiar sample space Ω = {(1, 1), (1, 2), . . . , (6, 6)} containing 36 elements. Let X denote the random variable whose value for any element of Ω is the sum of the numbers on the two dice. Enumerate the elements of X. 3. Three cards are drawn in succession from a deck without replacement. Find the probability distribution for the number of spades. 4. Find the probability distribution for the number of jazz CDs when 4 CDs are selected at random from a collection consisting of 5 jazz CDs, 2 classical CDs, and 3 rock CDs. 5. From a box containing 4 dimes and 2 nickels, 3 coins are selected at random without replacement. Find the probability distribution for the total T of the 3 coins. 6. From a box containing 4 black balls and 2 green balls, 3 balls are drawn in succession, each ball being replaced in the box before the next draw is made. Find the probability distribution for the number of green balls. 7. A shipment of 8 similar microcomputers to a retail outlet contains 3 that are defective. If a school makes a random purchase of 2 of these computers, find the probability distribution for the number of defectives. x 8. Does f (x) = , where x can take on values 0, 1, 2, and 3 determine a probability distribution? 5 x 9. Does f (x) = , where x can take on the values 0, 1, 2, 3, and 4 determine a probability 10 distribution? Prepared by: Celine Sarmiento

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Statistics and Probability

Exercises

10. Determine the value of c so that the function f (x) = c(x2 + 4) for x = {0, 1, 2, 3} can serve a probability distribution of the discrete random variable X. 11. Repair costs for a particular machine are represented by the following probability distribution: $50 x f (x) 0.30

$200 0.20

$350 0.50

What is the expected value of the repairs? 12. A coin is biased so that the head is three times as likely to occur as a tail. Find the expected number of tails when this coin is tossed twice. 13. The probability distribution of x, the number of imperfections per 10 meters of a synthetic fabric is continuous roll of uniform width. x f (x)

0 0.41

1 0.37

2 0.16

3 0.05

4 0.01

Find the average number of imperfections per 10 meters of this fabric.

Prepared by: Celine Sarmiento

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Statistics and Probability

12

Exercises

Discrete Probability Distribution

Read, analyze and solve each problem carefully. Express your answers as fractions in simplest form or as decimals rounded to four significant digits.

12.1

Binomial Distribution

1. The probability that a certain kind of component will survive a given shock test is 3/4. Find the probability that exactly 2 of the next 4 components tested survive. 2. The probability that a log-on to the network is successful is 0.87. Ten users attempt to log on independently. Find the probability that between 4 and 8 log-ons are successful. 3. The probability that a patient recovers from a rare blood disease is 0.40. If 15 people are known to have contracted this disease, what is the probability that (a) at least 10 survive? (b) from 3 to 8 survive? (c) exactly 5 survive?

12.2

Negative Binomial Distribution

1. An oil company conducts a geological study that indicates that an exploratory oil well should have a 20% chance of striking oil. What is the probability that the third strike comes on the seventh well drilled? 2. It is reported that 10% of the apples from the Apple Farm are bad. If apples are randomly selected from this farm one after another, what is the probability that the 10th apple selected will be the 2nd bad apple selected? 3. In an NBA championship series, the team which wins four games out of seven will be the winner. Suppose that team A has probability 0.55 of winning over team B and both teams A and B face each other in the championship games. (a) What is the probability that team A will win the series in six games? (b) What is the probability that team A will win the series?

12.3

Geometric Distribution

1. When taping a TV commercial, the probability that a certain actor will get his lines straight on any one take is 0.40. What is the probability that this actor will get his lines straight for the first time on the fourth take? 2. Find the probability that a person flipping a balanced coin requires 4 tosses to get a head. 3. The probability that any given person will believe a rumor about the private life of a certain politician is 0.25. What is the probability that the fifth person to hear the rumor will be the first one to believe it?

Prepared by: Celine Sarmiento

Page 27 of 37

Statistics and Probability

Exercises

4. Find the probability that a person drawing a single card with replacement from a standard deck requires 6 draws to get a face card.

12.4

Poisson Distribution

1. The average number of traffic accidents on a certain section of highway is two per week. Assume that the number of accidents follow a Poisson distribution with µ = 2. (a) Find the probability of no accidents on this section of highway during a 1-week period. (b) Find the probability of at most three accidents on this section of highway during a week period. 2. The average number of monthly breakdowns of a computer is 1.8. Find the probabilities that this computer will function for a month: (a) without a breakdown. (b) with only one breakdown. 3. On the average, 8 people per hour use an express teller machine situated inside a commercial complex every afternoon. What is the probability that, during a selected Friday afternoon, (a) exactly 6 people will use the teller machine? (b) at least 4 people will use the machine?

12.5

Hypergeometric Distribution

1. A carton contains 24 light bulbs, three of which are defective. What is the probability that, if a sample of six is chosen at random from the carton of bulbs, 4 will not be defective? 2. Find the probability of getting 4 face cards if a hand of 10 cards is drawn from a standard deck of cards without replacement. 3. A customs inspector decides to inspect 3 of 16 shipments that arrive from Madrid by plane. If the selection is random and 5 of the shipments contain contraband, find the probabilities that the customs inspector will catch: (a) none of the shipments with contraband; (b) one of the shipments with contraband; (c) two of the shipments with contraband; (d) three of the shipments with contraband.

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Statistics and Probability

13

Exercises

Continuous Probability Distribution

Read, analyze and solve each problem carefully. Express your answers as fractions in simplest form or as decimals rounded to four significant digits.

13.1

Normal Distribution

1. Find the area under a normal curve: (a) below z = 1.54 (b) to the left of z = −0.17 (c) above z = 1.13 (d) to the right of z = −0.29 (e) between the mean and a point z = 1.32. (f) between z = −2.0 and z = −1.56. (g) between z = −1.5 and z = 2.25. 2. Determine the value of the following probabilities: (a) P (Z < 1.42) (b) P (Z < −0.11) (c) P (Z > −1.88) (d) P (Z > 2.12) (e) P (−0.34 < Z < 1.14) (f) P (0.79 < Z < 1.75) 3. Given a normal distribution with mean 40 and standard deviation 6, find the value of x that has: (a) 45% of the area to the left, (b) 14% of the area to the right. 4. Given a test with a mean of 84 and a standard deviation of 12. (a) What is the probability of an individual obtaining a score of 100 or above in this test? (b) If 654 students took the examination, then how many students got a score below 60? 5. The time to complete a construction project is normally distributed with a mean of 60 weeks and a standard deviation of 4 weeks. What is the probability that the project will take longer than 65 weeks? 6. In the November 1990 issue of Chemical Engineering Progress a study discussed the percent purity of oxygen from a certain supplier. Assume that the mean was 99.61 with a standard deviation of 0.08. Assume that the distribution of percent purity was approximately normal. (a) What percentage of the purity values would you expect to be between 99.5 and 99.7? (b) What purity value would you expect to exceed exactly 5% of the population? Prepared by: Celine Sarmiento

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Statistics and Probability

Exercises

7. The tensile strength of a certain metal component is normally distributed with mean of 10,000 kilogram per square centimeter and a standard deviation of 100 kilograms per square centimeter. Measurements are recorded to the nearest 50 kilograms per square meter. (a) What proportion of these components exceeds 10,150 per square centimeter in tensile strength? (b) If specifications require that all components have tensile strength between 9,800 and 10,200 kilograms per square centimeter inclusive, what proportion of pieces would you expect to scrap? 8. Anna Lincoln has been the production manager of Medical Suppliers Inc., for the past 17 years. Medical Suppliers Inc., is a producer of bandages and arm slings. During the past 5 years, the demand of No-Stick bandages has been fairly constant. On the average, sales have been about 87,000 packages of No-Stick. Susan has reason to believe that the distribution of No-Stick following a normal curve, with a standard deviation of 4,000 packages. What is the probability that sales will be less than 81,000 packages?

13.2

Exponential Distribution

1. The time between arrivals of cars at Als full-service gas pump follows an exponential probability distribution with a mean time between arrivals of 3 minutes. Al would like to know the probability that the time between two successive arrivals will be 2 minutes or less. 2. A pumping station operator observes that the demand for water at a certain hour of the day can be modelled as an exponential random variable with a mean of 100 cfs (cubic feet per second). Find the probability that the demand will exceed 200 cfs on a randomly selected day. 3. Assume that the time required to download a file from the Internet is exponentially distributed with mean equal to 4 minutes. What is the probability that a download will require at least 2 but not more than 4 minutes? 4. Jobs are sent to a printer at an average of 3 jobs per hour. (a) What is the expected time between jobs? (b) What is the probability that the next job is sent within 5 minutes? 5. Suppose that the amount of time one spends in a bank is exponentially distributed with mean 10 minutes. What is the probability that a customer will spend more than 15 minutes in the bank? 6. Suppose the mean time between failures (MTBF) for a certain electronic component is 2,400 hours. (a) What is the probability that an electronic component will exceed 4,000 before failing? (b) What is the probability that an electronic component will fail before 1,800 hours? (c) Determine the interval of such time that the probability of at least one failure is 50%. (d) Determine the interval of such time that the probability of no failures is 25%. 7. The length of life of a certain type of electronic tube is exponentially distributed with a mean life of 500 hours. (a) Find the probability that a tube will last more than 800 hours. Prepared by: Celine Sarmiento

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Statistics and Probability

Exercises

(b) Find the probability that a tube will fail within the first 200 hours. (c) Find the probability that the length of life of a tube will be between 400 and 700 hours.

Prepared by: Celine Sarmiento

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Statistics and Probability

14

Exercises

Estimation

Read, analyze and solve each problem carefully.

14.1

Estimation of Means

1. An electrical firm manufactures light bulbs that have a length life that is approximately normally distributed with a standard deviation of 40 hours. (a) If a sample of 30 bulbs has an average life of 780 hours, find a 96% confidence interval for the population mean of all bulbs produced by this firm. (b) How large sample is needed if we wish to be 96% confident that our sample mean will be within 10 hours of the true mean? 2. A survey of the delivery time of 100 orders worth Php 20,000 from SHK Pizza yielded a mean of 55 minutes with a standard deviation of 12 minutes. Assuming that the delivery time follows a normal distribution, construct a 95% confidence interval for the true mean. 3. The heights of a random sample of 50 college students showed a mean of 174.5 cm and a standard deviation of 6.9 cm. What can we assert with 98% confidence about the possible size of our error if we estimate the mean height of all college students to be 174.5? 4. The monthly wage of new employees at a certain broadcasting company is said to follow a normal distribution with a standard deviation of Php 1,000. How large sample be needed to be 99% confident that the sample mean will be within Php 300 of the true mean. 5. A random sample of 100 automobile owners shows that, in the state of Virginia, an automobile is driven on the average 23,500 kilometers per year with standard deviation of 3,900 kilometers. (a) What can we assert with 99% confidence about the possible size of our error if we estimate the average number of kilometers driven by car owners in Virginia to be 23,500 kilometers per year? (b) Construct a 99% confidence interval for the average number of kilometers an automobile is driven in Virginia. 6. A random sample of 8 cigarettes of a certain brand has average nicotine content of 3.6 milligrams and a standard deviation of 0.9 milligrams. Construct a 99% confidence interval for the true average nicotine content of this particular brand of cigarettes, assuming an approximate normal distribution. 7. The mean and standard deviation for the quality grade point average of a random sample of 38 college seniors are calculated to be 2.7 and 0.3, respectively. Find the 95% confidence interval for the mean of the entire senior class. 8. In a study of the use of hypnosis to relieve pain, sensory ratings were measured for 1% subjects, with the results given below. Construct a 95% confidence interval for the true mean sensory ratings for all subjects. 8.8

6.2

7.7

Prepared by: Celine Sarmiento

6.4

6.1

6.8

9.8

8.2

8.5

7.3

7.1

6.0

6.2

5.9

9.4 Page 32 of 37

Statistics and Probability

Exercises

9. A machine is producing metal pieces that are cylindrical in shape. A sample of pieces is taken and the diameters are 1.01, 0.97, 1.03, 1.04, 0.99, 0.98, 0.99, 1.01, and 1.03 centimeters. Find a 99% confidence interval for the mean diameter of pieces from this machine, assuming an approximate normal distribution.

14.2

Estimation of Proportion

1. In a random sample of n = 500 families owning television sets in the city of Hamilton, Canada, it is found that x = 340 subscribed to HBO. (a) Find a 95% confidence interval for the actual proportion of families in this city who subscribe to HBO. (b) How large sample is required if we want 95% confident that our estimate of p is within 0.02? (c) How large sample is required if we want to be at least 95% confident that our estimate of p is within 0.02? 2. A manufacturer of compact disk players uses a set of comprehensive tests to access the electrical function of its product. All compact disk players must pass all tests prior to being sold. A random sample of 500 disk players resulted in 15 failing one or more tests. Find a 90% confidence interval for the proportion of compact disk players from the population that pass all tests. 3. A study is to be made to estimate to percentage of citizens in a town who favor having their water fluoridated. How large a sample is needed if one wishes to be at least 95% confident that our estimate is within 1% of the true percentage? 4. A sample of 81 college students finds that 27 attend 3 or more fun games each summer. Find a 95% confidence interval for the true population proportion of KSU students that attend 3 or more Braves games each summer. 5. A random sample of 140 college students finds that 113 of those students polled avoid classes that start before 9:30 AM. Construct a 99% confidence interval for the true population proportion of students who avoid classes that start before 9:30 AM. 6. In a random sample of 1,000 homes in a certain city, it is found that 228 are heated by oil. Find the 99% confidence interval for the proportion of homes in this city that are heated by oil. 7. Compute a 98% confidence interval for the proportion of defective items in a process when it is found that a sample of 100 yields 8 defectives. 8. A national electronics chain wishes to estimate the percentage of its customers who would pay a yearly membership fee in order to receive a 15% discount on all purchases of books, CD’s, DVD’s, games and software. Find the sample size needed to ensure that the sample estimate differs from the true population percentage by no more than 2.5%. Test at 95% confidence. 9. A study is to be made to estimate the proportion of residents in a certain city and its suburbs who favor the construction of a nuclear power plant. How large a sample is needed if one wishes to be at least 95% confident that the estimate is within 0.04 of the true proportion of residents in this city and its suburbs that favor the construction of the nuclear power plant?

Prepared by: Celine Sarmiento

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Statistics and Probability

15

Exercises

Hypothesis Testing

Read and analyze each problem carefully. Perform a complete hypothesis testing for each item.

15.1

Test on the Mean of a Single Population

1. A researcher reports that the average salary of assistant professors is more than $42,000. A sample of 30 assistant professors has a mean salary of $43,260. At α = 0.05, test the claim that assistant professors earn more than $42,000 a year. The standard deviation of the population is $5230. 2. A random sample of 100 recorded deaths in the United States during the past year showed an average life span of 71.8 years. Assuming a population standard deviation of 8.9 years, does this seem to indicate that the mean life span today is greater than 70 years? Use 0.05 level of significance. 3. The mean weight of the baggage carried into an airplane by individual passengers at Roxas City Airport is 19.8 kilograms. A statistician takes a random sample of 110 passengers and obtains a sample mean weight of 18.5 kilograms with standard deviation of 8.5 kilograms. Test the claim at α = 0.01 level of significance. 4. A national magazine claims that the average college student watches less television than the general public. The national average is 29.4 hours per week, with a standard deviation of 2 hours. A sample of 20 college students has a mean of 27 hours. Is there enough evidence to support the claim at α = 0.01? 5. A job placement director claims that the average starting salary for nurses is $24,000. A sample of 10 nurses has a mean of $23,450 and a standard deviation of $400. Is there enough evidence to reject the director’s claim at α = 0.05? 6. According to the Department of Education, high school teachers work an average of 40 hours per week during the school year. A district supervisor of a certain schools surveyed 28 randomly selected teachers and found that they work an average of 42.6 hours a week and the standard deviation was 3.75 hours. Test if the mean number of hours worked by teachers in the supervisor’s school district differs from the national average. Use α = 0.01.

15.2

Test on the Difference of Means of Two Population

1. An agronomist randomly selected 20 matured calamansi trees of one variety and have a mean height of 10.8 feet with standard deviation of 1.25 feet, while 12 randomly selected calamansi trees of another variety have a mean height of 9.6 feet with standard deviation of 1.45 feet. Test whether the difference between the two sample means is significant. Use α = 0.05. 2. To compare freshmen’s knowledge of mathematics in two departments of the College of Business Administration, a certain professor in Statistics got a sample of economics and accountancy students and gave them special examination. A sample of 25 economics major students had a mean score of 85.85 with standard deviation of 7.5. A sample of 28 accounting major students had a mean score of 90.5 with a standard deviation of 10.3. Is there a significant difference between the two sample means? Use 0.05 level of significance.

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Statistics and Probability

Exercises

3. The daily sales of two newspaper vendors were recorded on a random basis. The result of two samples are as follows: • Vendor I: Php 108, Php 125, Php 130, Php 116, Php 120, Php 119 • Vendor II: Php 113, Php 120, Php 120, Php 110, Php 125, Php 120 Is there a significant difference between the mean sales of the two newspaper vendors? Test at 0.01 significance level.

Prepared by: Celine Sarmiento

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Statistics and Probability

16

Exercises

Correlation and Regression

Read, analyze and solve each problem carefully. 1. The data on yearly consumption of cigarettes in the Philippines and the percentage of the country’s population admitted to mental institutions as psychiatric cases were collected for 8 years. The correlation coefficient r = 0.61. What can we conclude about the data? 2. The data below consists of age and the income in thousands of dollars. Find the value of r and interpret the result. Age 60 63 51 25 47 56 19 24 25 20 66 19 48 52 27 Income 43.4 18.8 14.4 29.4 19.4 83 10.4 12.6 36.4 29.6 17.2 17.2 67 33 37.4 3. A teacher is interested in knowing whether or not two IQ tests produce linearly related scores. A sample of 10 students was taken randomly. Five students took Test 1 and 5 students took Test 2 in the morning. In the afternoon, those who took Test 1 took Test 2 and vice versa. The results are shown in the table below: B C D E F Student A Test 1 125 145 110 120 124 110 Test 2 114 127 126 116 108 100

G H I J 121 142 100 126 129 131 96 113

(a) Plot a scatter diagram for these data. (b) Solve for the value of r. 4. The grades of a class of 9 students on a midterm report (x) and on the final examination (y) are as follows x y

77 82

50 66

71 78

72 34

81 47

94 85

96 99

99 99

67 68

(a) Find the equation of the regression line. (b) Estimate the final examination grade of a student who received a grade of 85 on the midterm report but was ill at the time of the final examination. (c) Estimate the midterm mark of the a student who received a grade of 65 on the final examination. 5. A student counted the number of words in an essays she had written, recording the total every 10 lines. No. of lines x No. of words y

10 75

20 136

30 210

40 291

50 368

60 441

70 519

80 588

(a) Draw the scatter diagram to show the data. (b) Calculate the equation of the regression lie. (c) How many words (approximately) has she written if she writes: Prepared by: Celine Sarmiento

Page 36 of 37

Statistics and Probability

Exercises

i. 65 lines ii. 100 lines iii. 1,000 lines

References: • Altares, P., et. al. (2013). Elementary Statistics with Computer Applications (2nd Edition). Rex Printing Company, Inc., Quezon City. • Mendenhall, W., et. al. (1999). Introduction to Probability and Statistics. Brooks/Cole Publishing Company, USA. • Reyes, C. and Saren, L. (2003). Elementary Statistics. National Bookstore, Mandaluyong City. • Walpole, R., et. al. (2005). Probability and Statistics for Engineers and Scientists (7th Edition). Pearson Education, Inc., New Jersey.

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