Test of Two Variances Another of the uses of the F distribution is testing two variances. It is often desirable to compare two variances rather than two averages. For instance, college administrators would like two college professors grading exams to have the same variation in their grading. In order for a lid to fit a container, the variation in the lid and the container should be the same. A supermarket might be interested in the variability of check-out times for two checkers. In order to perform a F test of two variances, it is important that the following are true: The populations from which the two samples are drawn are normally distributed. The two populations are independent of each other. Unlike most other tests in this book, the F test for equality of two variances is very sensitive to deviations from normality. If the two distributions are not normal, the test can give higher p -values than it should, or lower ones, in ways that are unpredictable. Many texts suggest that students not use this test at all, but in the interest of completeness we include it here. Suppose we sample randomly from two independent normal populations. Let σ 1 2 and σ 2 2 be the population variances and s 1 2 and s 2 2 be the sample variances. Let the sample sizes be n 1 and n 2 . Since we are interested in comparing the two sample variances, we use the F ratio: F = [ ( s 1 ) 2 ( σ 1 ) 2 ] [ ( s 2 ) 2 ( σ 2 ) 2 ] F has the distribution F ~ F ( n 1 – 1, n 2 – 1) where n 1 – 1 are the degrees of freedom for the numerator and n 2 – 1 are the degrees of freedom for the denominator. If the null hypothesis is σ 1 2 = σ 2 2 , then the F Ratio becomes F = [ ( s 1 ) 2 ( σ 1 ) 2 ] [ ( s 2 ) 2 ( σ 2 ) 2 ] = ( s 1 ) 2 ( s 2 ) 2 . NOTE The F ratio could also be ( s 2 ) 2 ( s 1 ) 2 . It depends on H a and on which sample variance is larger. If the two populations have equal variances, then s 1 2 and s 2 2 are close in value and F = ( s 1 ) 2 ( s 2 ) 2 is close to one. But if the two population variances are very different, s 1 2 and s 2 2 tend to be very different, too. Choosing s 1 2 as the larger sample variance causes the ratio ( s 1 ) 2 ( s 2 ) 2 to be greater than one. If s 1 2 and s 2 2 are far apart, then F = ( s 1 ) 2 ( s 2 ) 2 is a large number. Therefore, if F is close to one, the evidence favors the null hypothesis (the two population variances are equal). But if F is much larger than one, then the evidence is against the null hypothesis. A test of two variances may be left, right, or two-tailed. Two college instructors are interested in whether or not there is any variation in the way they grade math exams. They each grade the same set of 30 exams. The first instructor's grades have a variance of 52.3. The second instructor's grades have a variance of 89.9. Test the claim that the first instructor's variance is smaller. (In most colleges, it is desirable for the variances of exam grades to be nearly the same among instructors.) The level of significance is 10%. Let 1 and 2 be the subscripts that indicate the first and second instructor, respectively. n 1 = n 2 = 30. H 0 : σ 1 2 = σ 2 2 and H a : σ 1 2 < σ 2 2 Calculate the test statistic: By the null hypothesis ( σ 1 2 = σ 2 2 ) , the F statistic is: F = [ ( s 1 ) 2 ( σ 1 ) 2 ] [ ( s 2 ) 2 ( σ 2 ) 2 ] = ( s 1 ) 2 ( s 2 ) 2 = 52.3 89.9 = 0.5818 Distribution for the test: F 29,29 where n 1 – 1 = 29 and n 2 – 1 = 29. Graph: This test is left tailed. Draw the graph labeling and shading appropriately. Probability statement: p -value = P ( F < 0.5818) = 0.0753 Compare α and the p -value: α = 0.10 α > p -value. Make a decision: Since α > p -value, reject H 0 . Conclusion: With a 10% level of significance, from the data, there is sufficient evidence to conclude that the variance in grades for the first instructor is smaller. Press STAT and arrow over to TESTS . Arrow down to D:2-SampFTest . Press ENTER . Arrow to Stats and press ENTER . For Sx1 , n1 , Sx2 , and n2 , enter ( 52.3 ) , 30 , ( 89.9 ) , and 30 . Press ENTER after each. Arrow to σ1: and σ2 . Press ENTER . Arrow down to Calculate and press ENTER . F = 0.5818 and p -value = 0.0753. Do the procedure again and try Draw instead of Calculate . Try It The New York Choral Society divides its singers who are men up into four categories from highest voices to lowest: Tenor1, Tenor2, Bass1, Bass2. In the table are heights of the men in the Tenor1 and Bass2 groups. One suspects that taller men will have lower voices, and that the variance of height may go up with the lower voices as well. Do we have good evidence that the variance of the heights of singers in each of these two groups (Tenor1 and Bass2) are different? Tenor1 Bass2 Tenor 1 Bass 2 Tenor 1 Bass 2 69 72 67 72 68 67 72 75 70 74 67 70 71 67 65 70 64 70 66 75 72 66 69 76 74 70 68 72 74 72 68 75 71 71 72 64 68 74 66 74 73 70 75 68 72 66 72 References “MLB Vs. Division Standings – 2012.” Available online at http://espn.go.com/mlb/standings/_/year/2012/type/vs-division/order/true. Chapter Review The F test for the equality of two variances rests heavily on the assumption of normal distributions. The test is unreliable if this assumption is not met. If both distributions are normal, then the ratio of the two sample variances is distributed as an F statistic, with numerator and denominator degrees of freedom that are one less than the samples sizes of the corresponding two groups. A test of two variances hypothesis test determines if two variances are the same. The distribution for the hypothesis test is the F distribution with two different degrees of freedom. Assumptions: The populations from which the two samples are drawn are normally distributed. The two populations are independent of each other. Formula Review F has the distribution F ~ F ( n 1 – 1, n 2 – 1) F = s 1 2 σ 1 2 s 2 2 σ 2 2 If σ 1 = σ 2 , then F = s 1 2 s 2 2 Use the following information to answer the next two exercises. There are two assumptions that must be true in order to perform an F test of two variances. Name one assumption that must be true. The populations from which the two samples are drawn are normally distributed. What is the other assumption that must be true? Use the following information to answer the next five exercises. Two coworkers commute from the same building. They are interested in whether or not there is any variation in the time it takes them to drive to work. They each record their times for 20 commutes. The first worker’s times have a variance of 12.1. The second worker’s times have a variance of 16.9. The first worker thinks that he is more consistent with his commute times. Test the claim at the 10% level. Assume that commute times are normally distributed. State the null and alternative hypotheses. H 0 : σ 1 = σ 2 H a : σ 1 < σ 2 or H 0 : σ 1 2 = σ 2 2 H a : σ 1 2 < σ 2 2 What is s 1 in this problem? What is s 2 in this problem? 4.11 What is n ? What is the F statistic? 0.7159 What is the p -value? Is the claim accurate? No, at the 10% level of significance, we do not reject the null hypothesis and state that the data do not show that the variation in drive times for the first worker is less than the variation in drive times for the second worker. Use the following information to answer the next four exercises. Two students are interested in whether or not there is variation in their test scores for math class. There are 15 total math tests they have taken so far. The first student’s grades have a standard deviation of 38.1. The second student’s grades have a standard deviation of 22.5. The second student thinks his scores are more consistent. State the null and alternative hypotheses. What is the F Statistic? 2.8674 What is the p -value? At the 5% significance level, do we reject the null hypothesis? Reject the null hypothesis. There is enough evidence to say that the variance of the grades for the first student is higher than the variance in the grades for the second student. Use the following information to answer the next three exercises. Two cyclists are comparing the variances of their overall paces going uphill. Each cyclist records their speeds going up 35 hills. The first cyclist has a variance of 23.8 and the second cyclist has a variance of 32.1. The cyclists want to see if their variances are the same or different. Assume that commute times are normally distributed. State the null and alternative hypotheses. What is the F Statistic? 0.7414 At the 5% significance level, what can we say about the cyclists’ variances? Homework Three students, Linda, Tuan, and Javier, are given five laboratory rats each for a nutritional experiment. Each rat’s weight is recorded in grams. Linda feeds her rats Formula A, Tuan feeds his rats Formula B, and Javier feeds his rats Formula C. At the end of a specified time period, each rat is weighed again and the net gain in grams is recorded. Linda's rats Tuan's rats Javier's rats 43.5 47.0 51.2 39.4 40.5 40.9 41.3 38.9 37.9 46.0 46.3 45.0 38.2 44.2 48.6 Determine whether or not the variance in weight gain is statistically the same among Javier’s and Linda’s rats. Test at a significance level of 10%. H 0 : σ 1 2 = σ 2 2 H a : σ 1 2 ≠ σ 1 2 df ( num ) = 4; df ( denom ) = 4 F 4, 4 3.00 2(0.1563) = 0.3126. Using the TI-83+/84+ function 2-SampFtest, you get the test statistic as 2.9986 and p -value directly as 0.3127. If you input the lists in a different order, you get a test statistic of 0.3335 but the p -value is the same because this is a two-tailed test. Answers may vary. Decision: Do not reject the null hypothesis; Conclusion: There is insufficient evidence to conclude that the variances are different. A grassroots group opposed to a proposed increase in the gas tax claimed that the increase would hurt working-class people the most, since they commute the farthest to work. Suppose that the group randomly surveyed 24 individuals and asked them their daily one-way commuting mileage. The results are as follows. working-class professional (middle incomes) professional (wealthy) 17.8 16.5 8.5 26.7 17.4 6.3 49.4 22.0 4.6 9.4 7.4 12.6 65.4 9.4 11.0 47.1 2.1 28.6 19.5 6.4 15.4 51.2 13.9 9.3 Determine whether or not the variance in mileage driven is statistically the same among the working class and professional (middle income) groups. Use a 5% significance level. Use the following information to answer the next two exercises. lists the number of pages in four different types of magazines. home decorating news health computer 172 87 82 104 286 94 153 136 163 123 87 98 205 106 103 207 197 101 96 146 Which two magazine types do you think have the same variance in length? Which two magazine types do you think have different variances in length? The answers may vary. Sample answer: Home decorating magazines and news magazines have different variances. Is the variance for the amount of money, in dollars, that shoppers spend on Saturdays at the mall the same as the variance for the amount of money that shoppers spend on Sundays at the mall? Suppose that the shows the results of a study. Saturday Sunday Saturday Sunday 75 44 62 137 18 58 0 82 150 61 124 39 94 19 50 127 62 99 31 141 73 60 118 73 89 Are the variances for incomes on the East Coast and the West Coast the same? Suppose that shows the results of a study. Income is shown in thousands of dollars. Assume that both distributions are normal. Use a level of significance of 0.05. East West 38 71 47 126 30 42 82 51 75 44 52 90 115 88 67 H 0 : = σ 1 2 = σ 2 2 H a : σ 1 2 ≠ σ 1 2 df ( n ) = 7, df ( d ) = 6 F 7,6 0.8117 0.7825 Answers may vary. Alpha: 0.05 Decision: Do not reject the null hypothesis. Reason for decision: p -value > alpha Conclusion: There is not sufficient evidence to conclude that the variances are different. Thirty men in college were taught a method of finger tapping. They were randomly assigned to three groups of ten, with each receiving one of three doses of caffeine: 0 mg, 100 mg, 200 mg. This is approximately the amount in no, one, or two cups of coffee. Two hours after ingesting the caffeine, the men had the rate of finger tapping per minute recorded. The experiment was double blind, so neither the recorders nor the students knew which group they were in. Does caffeine affect the rate of tapping, and if so how? Here are the data: 0 mg 100 mg 200 mg 0 mg 100 mg 200 mg 242 248 246 245 246 248 244 245 250 248 247 252 247 248 248 248 250 250 242 247 246 244 246 248 246 243 245 242 244 250 King Manuel I, Komnenus ruled the Byzantine Empire from Constantinople (Istanbul) during the years 1145 to 1180 A.D. The empire was very powerful during his reign, but declined significantly afterwards. Coins minted during his era were found in Cyprus, an island in the eastern Mediterranean Sea. Nine coins were from his first coinage, seven from the second, four from the third, and seven from a fourth. These spanned most of his reign. We have data on the silver content of the coins: First Coinage Second Coinage Third Coinage Fourth Coinage 5.9 6.9 4.9 5.3 6.8 9.0 5.5 5.6 6.4 6.6 4.6 5.5 7.0 8.1 4.5 5.1 6.6 9.3 6.2 7.7 9.2 5.8 7.2 8.6 5.8 6.9 6.2 Did the silver content of the coins change over the course of Manuel’s reign? Here are the means and variances of each coinage. The data are unbalanced. First Second Third Fourth Mean 6.7444 8.2429 4.875 5.6143 Variance 0.2953 1.2095 0.2025 0.1314 Here is a strip chart of the silver content of the coins: While there are differences in spread, it is not unreasonable to use ANOVA techniques. Here is the completed ANOVA table: Source of Variation Sum of Squares ( SS ) Degrees of Freedom ( df ) Mean Square ( MS ) F Factor (Between) 37.748 4 – 1 = 3 12.5825 26.272 Error (Within) 11.015 27 – 4 = 23 0.4789 Total 48.763 27 – 1 = 26 P ( F > 26.272) = 0; Reject the null hypothesis for any alpha. There is sufficient evidence to conclude that the mean silver content among the four coinages are different. From the strip chart, it appears that the first and second coinages had higher silver contents than the third and fourth. The American League and the National League of Major League Baseball are each divided into three divisions: East, Central, and West. Many years, fans talk about some divisions being stronger (having better teams) than other divisions. This may have consequences for the postseason. For instance, in a certain year Tampa Bay won 90 games and did not play in the postseason, while Detroit won only 88 and did play in the postseason. This may have been an oddity, but is there good evidence that in this year, the American League divisions were significantly different in overall records? Use the following data to test whether the mean number of wins per team in the three American League divisions were the same or not. Note that the data are not balanced, as two divisions had five teams, while one had only four. Division Team Wins East NY Yankees 95 East Baltimore 93 East Tampa Bay 90 East Toronto 73 East Boston 69 Division Team Wins Central Detroit 88 Central Chicago Sox 85 Central Kansas City 72 Central Cleveland 68 Central Minnesota 66 Division Team Wins West Oakland 94 West Texas 93 West LA Angels 89 West Seattle 75 Here is a stripchart of the number of wins for the 14 teams in the AL for the 2012 season. While the spread seems similar, there may be some question about the normality of the data, given the wide gaps in the middle near the 0.500 mark of 82 games (teams play 162 games each season in MLB). However, one-way ANOVA is robust. Here is the ANOVA table for the data: Source of Variation Sum of Squares ( SS ) Degrees of Freedom ( df ) Mean Square ( MS ) F Factor (Between) 344.16 3 – 1 = 2 172.08 Error (Within) 1,219.55 14 – 3 = 11 110.87 1.5521 Total 1,563.71 14 – 1 = 13 P ( F > 1.5521) = 0.2548 Since the p -value is so large, there is not good evidence against the null hypothesis of equal means. We decline to reject the null hypothesis. Thus, for 2012, there is not any have any good evidence of a significant difference in mean number of wins between the divisions of the American League.

Math


Lab: One-Way ANOVA One-Way ANOVA Class Time: Names: Student Learning Outcome The student will conduct a simple one-way ANOVA test involving three variables. Collect the Data Record the price per pound of eight fruits, eight vegetables, and eight breads in your local supermarket. Fruits Vegetables Breads Explain how you could try to collect the data randomly. Analyze the Data and Conduct a Hypothesis Test State the null hypothesis and the alternative hypothesis. Compute the following: Fruit: x ¯ = ______ s x = ______ n = ______ Vegetables: x ¯ = ______ s x = ______ n = ______ Bread: x ¯ = ______ s x = ______ n = ______ Find the following: df ( num ) = ______ df ( denom ) = ______ State the approximate distribution for the test. Test statistic: F = ______ Sketch a graph of this situation. CLEARLY, label and scale the horizontal axis and shade the region(s) corresponding to the p -value. p -value = ______ Test at α = 0.05. State your decision and conclusion. Decision: Why did you make this decision? Conclusion (write a complete sentence). Based on the results of your study, is there a need to investigate any of the food groups’ prices? Why or why not?

Math


Review Exercises (Ch 3-13) These review exercises are designed to provide extra practice on concepts learned before a particular chapter. For example, the review exercises for Chapter 3, cover material learned in chapters 1 and 2. Chapter 3 Use the following information to answer the next six exercises: In a survey of 100 stocks on NASDAQ, the average percent increase for the past year was 9% for NASDAQ stocks. 1 . The “average increase” for all NASDAQ stocks is the: population statistic parameter sample variable 2. All of the NASDAQ stocks are the: population statistics parameter sample variable 3. Nine percent is the: population statistics parameter sample variable 4. The 100 NASDAQ stocks in the survey are the: population statistic parameter sample variable 5. The percent increase for one stock in the survey is the: population statistic parameter sample variable 6. Would the data collected by qualitative, quantitative discrete, or quantitative continuous? Use the following information to answer the next two exercises: Thirty people spent two weeks around Mardi Gras in New Orleans. Their two-week weight gain is below. (Note: a loss is shown by a negative weight gain.) Weight Gain Frequency –2 3 –1 5 0 2 1 4 4 13 6 2 11 1 7. Calculate the following values: the average weight gain for the two weeks the standard deviation the first, second, and third quartiles 8. Construct a histogram and box plot of the data. Chapter 4 Use the following information to answer the next two exercises: A recent poll concerning credit cards found that 35 percent of respondents use a credit card that gives them a mile of air travel for every dollar they charge. Thirty percent of the respondents charge more than $2,000 per month. Of those respondents who charge more than $2,000, 80 percent use a credit card that gives them a mile of air travel for every dollar they charge. 9. What is the probability that a randomly selected respondent will spend more than $2,000 AND use a credit card that gives them a mile of air travel for every dollar they charge? (0.30)(0.35) (0.80)(0.35) (0.80)(0.30) (0.80) 10. Are using a credit card that gives a mile of air travel for each dollar spent AND charging more than $2,000 per month independent events? Yes No, and they are not mutually exclusive either. No, but they are mutually exclusive. Not enough information given to determine the answer 11. A sociologist wants to know the opinions of employed adult women about government funding for day care. She obtains a list of 520 members of a local business and professional women’s club and mails a questionnaire to 100 of these women selected at random. Sixty-eight questionnaires are returned. What is the population in this study? all employed adult women all the members of a local business and professional women’s club the 100 women who received the questionnaire all employed women with children Use the following information to answer the next two exercises: An article from The San Jose Mercury News was concerned with the racial mix of the 1500 students at Prospect High School in Saratoga, CA. The table summarizes the results. (Male and female values are approximate.) Suppose one Prospect High School student is randomly selected. Gender/Ethnic group White Asian Hispanic Black American Indian Male 400 168 115 35 16 Female 440 132 140 40 14 12. Find the probability that a student is Asian or Male. 13. Find the probability that a student is Black given that the student is female. 14. A sample of pounds lost, in a certain month, by individual members of a weight reducing clinic produced the following statistics: Mean = 5 lbs. Median = 4.5 lbs. Mode = 4 lbs. Standard deviation = 3.8 lbs. First quartile = 2 lbs. Third quartile = 8.5 lbs. The correct statement is: One fourth of the members lost exactly two pounds. The middle fifty percent of the members lost from two to 8.5 lbs. Most people lost 3.5 to 4.5 lbs. All of the choices above are correct. 15. What does it mean when a data set has a standard deviation equal to zero? All values of the data appear with the same frequency. The mean of the data is also zero. All of the data have the same value. There are no data to begin with. 16. The statement that describe the illustration is: the mean is equal to the median. There is no first quartile. The lowest data value is the median. The median equals Q 1 + Q 3 2 . 17. According to a recent article in the San Jose Mercury News the average number of babies born with significant hearing loss (deafness) is approximately 2 per 1000 babies in a healthy baby nursery. The number climbs to an average of 30 per 1000 babies in an intensive care nursery. Suppose that 1,000 babies from healthy baby nurseries were randomly surveyed. Find the probability that exactly two babies were born deaf. 18. A “friend” offers you the following “deal.” For a $10 fee, you may pick an envelope from a box containing 100 seemingly identical envelopes. However, each envelope contains a coupon for a free gift. Ten of the coupons are for a free gift worth $6. Eighty of the coupons are for a free gift worth $8. Six of the coupons are for a free gift worth $12. Four of the coupons are for a free gift worth $40. Based upon the financial gain or loss over the long run, should you play the game? Yes, I expect to come out ahead in money. No, I expect to come out behind in money. It doesn’t matter. I expect to break even. Use the following information to answer the next four exercises: Recently, a nurse commented that when a patient calls the medical advice line claiming to have the flu, the chance that they truly have the flu (and not just a nasty cold) is only about 4%. Of the next 25 patients calling in claiming to have the flu, we are interested in how many actually have the flu. 19. Define the random variable and list its possible values. 20. State the distribution of X . 21. Find the probability that at least four of the 25 patients actually have the flu. 22. On average, for every 25 patients calling in, how many do you expect to have the flu? Use the following information to answer the next two exercises: Different types of writing can sometimes be distinguished by the number of letters in the words used. A student interested in this fact wants to study the number of letters of words used by Jhumpa Lahiri in her novels. She opens a novel at random and records the number of letters of the first 250 words on the page. 23. What kind of data was collected? qualitative quantitative continuous quantitative discrete 24. What is the population under study? Chapter 5 Use the following information to answer the next seven exercises: A recent study of mothers of junior high school children in Santa Clara County reported that 76% of the mothers are employed in paid positions. Of those mothers who are employed, 64% work full-time (over 35 hours per week), and 36% work part-time. However, out of all of the mothers in the population, 49% work full-time. The population under study is made up of mothers of junior high school children in Santa Clara County. Let E = employed and F = full-time employment. 25. Find the percent of all mothers in the population that are NOT employed. Find the percent of mothers in the population that are employed part-time. 26. The “type of employment” is considered to be what type of data? 27. Find the probability that a randomly selected mother works part-time given that she is employed. 28. Find the probability that a randomly selected person from the population will be employed or work full-time. 29. Being employed and working part-time: mutually exclusive events? Why or why not? independent events? Why or why not? Use the following additional information to answer the next two exercises: We randomly pick ten mothers from the above population. We are interested in the number of the mothers that are employed. Let X = number of mothers that are employed. 30. State the distribution for X . 31. Find the probability that at least six are employed. 32. We expect the statistics discussion board to have, on average, 14 questions posted to it per week. We are interested in the number of questions posted to it per day. Define X . What are the values that the random variable may take on? State the distribution for X . Find the probability that from ten to 14 (inclusive) questions are posted to the listserv on a randomly picked day. 33. A person invests $1,000 into stock of a company that hopes to go public in one year. The probability that the person will lose all his money after one year (i.e. his stock will be worthless) is 35%. The probability that the person’s stock will still have a value of $1,000 after one year (i.e. no profit and no loss) is 60%. The probability that the person’s stock will increase in value by $10,000 after one year (i.e. will be worth $11,000) is 5%. Find the expected profit after one year. 34. Rachel’s piano cost $3,000. The average cost for a piano is $4,000 with a standard deviation of $2,500. Becca’s guitar cost $550. The average cost for a guitar is $500 with a standard deviation of $200. Matt’s drums cost $600. The average cost for drums is $700 with a standard deviation of $100. Whose cost was lowest when compared to their own instrument? 35. Explain why each statement is either true or false given the box plot in . Twenty-five percent of the data re at most five. There is the same amount of data from 4–5 as there is from 5–7. There are no data values of three. Fifty percent of the data are four. Using the following information to answer the next two exercises: 64 faculty members were asked the number of cars they owned (including spouse and children’s cars). The results are given in the following graph: 36. Find the approximate number of responses that were three. 37. Find the first, second and third quartiles. Use them to construct a box plot of the data. Use the following information to answer the next three exercises: shows data gathered from 15 girls on the Snow Leopard soccer team when they were asked how they liked to wear their hair. Supposed one girl from the team is randomly selected. Hair Style/Hair Color Blond Brown Black Ponytail 3 2 5 Plain 2 2 1 38. Find the probability that the girl has black hair GIVEN that she wears a ponytail. 39. Find the probability that the girl wears her hair plain OR has brown hair. 40. Find the probability that the girl has blond hair AND that she wears her hair plain. Chapter 6 Use the following information to answer the next two exercises: X ~ U (3, 13) 41. Explain which of the following are false and which are true. f ( x ) = 1 10 , 3 ≤ x ≤ 13 There is no mode The median is less than the mean. P ( x > 10) = P ( x ≤ 6) 42. Calculate: the mean. the median. the 65 th percentile. 43. Which of the following is true for the box plot in ? Twenty-five percent of the data are at most five. There is about the same amount of data from 4–5 as there is from 5–7. There are no data values of three. Fifty percent of the data are four. 44. If P ( G | H ) = P ( G ), then which of the following is correct? G and H are mutually exclusive events. P ( G ) = P ( H ) Knowing that H has occurred will affect the chance that G will happen. G and H are independent events. 45. If P ( J ) = 0.3, P ( K ) = 0.63, and J and K are independent events, then explain which are correct and which are incorrect. P ( J AND K ) = 0 P ( J OR K ) = 0.9 P ( J OR K ) = 0.72 P ( J ) ≠ P ( J | K ) 46. On average, five students from each high school class get full scholarships to four-year colleges. Assume that most high school classes have about 500 students. X = the number of students from a high school class that get full scholarships to four-year schools. Which of the following is the distribution of X ? P (5) B (500, 5) Exp ( 1 5 ) N ( 5 , ( 0.01 ) ( 0.99 ) 500 ) Chapter 7 Use the following information to answer the next three exercises: Richard’s Furniture Company delivers furniture from 10 A.M. to 2 P.M. continuously and uniformly. We are interested in how long (in hours) past the 10 A.M. start time that individuals wait for their delivery. 47. X ~ _________ U (0, 4) U (10, 20) Exp (2) N (2, 1) 48. The average wait time is: 1 hour. 2 hours. 2.5 hours. 4 hours. 49. Suppose that it is now past noon on a delivery day. The probability that a person must wait at least 1.5 more hours is: 1 4 1 2 3 4 3 8 50. Given: X ~ Exp ( 1 3 ) Find P ( x > 1). Calculate the minimum value for the upper quartile. Find P ( x = 1 3 ) 51. 40% of full-time students took 4 years to graduate 30% of full-time students took 5 years to graduate 20% of full-time students took 6 years to graduate 10% of full-time students took 7 years to graduate The expected time for full-time students to graduate is: 4 years 4.5 years 5 years 5.5 years 52. Which of the following distributions is described by the following example? Many people can run a short distance of under two miles, but as the distance increases, fewer people can run that far. binomial uniform exponential normal 53. The length of time to brush one’s teeth is generally thought to be exponentially distributed with a mean of 3 4 minutes. Find the probability that a randomly selected person brushes their teeth less than 3 4 minutes. 0.5 3 4 0.43 0.63 54. Which distribution accurately describes the following situation? The chance that a teenage boy regularly gives his mother a kiss goodnight is about 20%. Fourteen teenage boys are randomly surveyed. Let X = the number of teenage boys that regularly give their mother a kiss goodnight. B (14,0.20) P (2.8) N (2.8,2.24) Exp ( 1 0.20 ) 55. A 2008 report on technology use states that approximately 20% of U.S. households have never sent an e-mail. Suppose that we select a random sample of fourteen U.S. households. Let X = the number of households in a 2008 sample of 14 households that have never sent an email B (14,0.20) P (2.8) N (2.8,2.24) Exp ( 1 0.20 ) Chapter 8 Use the following information to answer the next three exercises: Suppose that a sample of 15 randomly chosen people were put on a special weight loss diet. The amount of weight lost, in pounds, follows an unknown distribution with mean equal to 12 pounds and standard deviation equal to three pounds. Assume that the distribution for the weight loss is normal. 56. To find the probability that the mean amount of weight lost by 15 people is no more than 14 pounds, the random variable should be: number of people who lost weight on the special weight loss diet. the number of people who were on the diet. the mean amount of weight lost by 15 people on the special weight loss diet. the total amount of weight lost by 15 people on the special weight loss diet. 57. Find the probability asked for in Question 56 . 58. Find the 90 th percentile for the mean amount of weight lost by 15 people. Using the following information to answer the next three exercises: The time of occurrence of the first accident during rush-hour traffic at a major intersection is uniformly distributed between the three hour interval 4 p.m. to 7 p.m. Let X = the amount of time (hours) it takes for the first accident to occur. 59. What is the probability that the time of occurrence is within the first half-hour or the last hour of the period from 4 to 7 p.m.? cannot be determined from the information given 1 6 1 2 1 3 60. The 20 th percentile occurs after how many hours? 0.20 0.60 0.50 1 61. Assume Ramon has kept track of the times for the first accidents to occur for 40 different days. Let C = the total cumulative time. Then C follows which distribution? U (0,3) Exp (13) N (60, 5.477) N (1.5, 0.01875) 62. Using the information in Question 61 , find the probability that the total time for all first accidents to occur is more than 43 hours. Use the following information to answer the next two exercises: The length of time a parent must wait for his children to clean their rooms is uniformly distributed in the time interval from one to 15 days. 63. How long must a parent expect to wait for his children to clean their rooms? eight days three days 14 days six days 64. What is the probability that a parent will wait more than six days given that the parent has already waited more than three days? 0.5174 0.0174 0.7500 0.2143 Use the following information to answer the next five exercises: Twenty percent of the students at a local community college live in within five miles of the campus. Thirty percent of the students at the same community college receive some kind of financial aid. Of those who live within five miles of the campus, 75% receive some kind of financial aid. 65. Find the probability that a randomly chosen student at the local community college does not live within five miles of the campus. 80% 20% 30% cannot be determined 66. Find the probability that a randomly chosen student at the local community college lives within five miles of the campus or receives some kind of financial aid. 50% 35% 27.5% 75% 67. Are living in student housing within five miles of the campus and receiving some kind of financial aid mutually exclusive? yes no cannot be determined 68. The interest rate charged on the financial aid is _______ data. quantitative discrete quantitative continuous qualitative discrete qualitative 69. The following information is about the students who receive financial aid at the local community college. 1st quartile = $250 2nd quartile = $700 3rd quartile = $1200 These amounts are for the school year. If a sample of 200 students is taken, how many are expected to receive $250 or more? 50 250 150 cannot be determined Use the following information to answer the next two exercises: P ( A ) = 0.2, P ( B ) = 0.3; A and B are independent events. 70. P ( A AND B ) = ______ 0.5 0.6 0 0.06 71. P ( A OR B ) = _______ 0.56 0.5 0.44 1 72. If H and D are mutually exclusive events, P ( H ) = 0.25, P ( D ) = 0.15, then P ( H | D ). 1 0 0.40 0.0375 Chapter 9 73. Rebecca and Matt are 14 year old twins. Matt’s height is two standard deviations below the mean for 14 year old boys’ height. Rebecca’s height is 0.10 standard deviations above the mean for 14 year old girls’ height. Interpret this. Matt is 2.1 inches shorter than Rebecca. Rebecca is very tall compared to other 14 year old girls. Rebecca is taller than Matt. Matt is shorter than the average 14 year old boy. 74. Construct a histogram of the IPO data (see ). Use the following information to answer the next three exercises: Ninety homeowners were asked the number of estimates they obtained before having their homes fumigated. Let X = the number of estimates. x Relative Frequency Cumulative Relative Frequency 1 0.3 2 0.2 4 0.4 5 0.1 75. Complete the cumulative frequency column. 76. Calculate the sample mean (a), the sample standard deviation (b) and the percent of the estimates that fall at or below four (c). 77. Calculate the median, M , the first quartile, Q 1 , the third quartile, Q 3 . Then construct a box plot of the data. 78. The middle 50% of the data are between _____ and _____. Use the following information to answer the next three exercises: Seventy 5 th and 6 th graders were asked their favorite dinner. Pizza Hamburgers Spaghetti Fried shrimp 5th grader 15 6 9 0 6th grader 15 7 10 8 79. Find the probability that one randomly chosen child is in the 6th grade and prefers fried shrimp. 32 70 8 32 8 8 8 70 80. Find the probability that a child does not prefer pizza. 30 70 30 40 40 70 1 81. Find the probability a child is in the 5 th grade given that the child prefers spaghetti. 9 19 9 70 9 30 19 70 82. A sample of convenience is a random sample. true false 83. A statistic is a number that is a property of the population. true false 84. You should always throw out any data that are outliers. true false 85. Lee bakes pies for a small restaurant in Felton, CA. She generally bakes 20 pies in a day, on average. Of interest is the number of pies she bakes each day. Define the random variable X . State the distribution for X . Find the probability that Lee bakes more than 25 pies in any given day. 86. Six different brands of Italian salad dressing were randomly selected at a supermarket. The grams of fat per serving are 7, 7, 9, 6, 8, 5. Assume that the underlying distribution is normal. Calculate a 95% confidence interval for the population mean grams of fat per serving of Italian salad dressing sold in supermarkets. 87. Given: uniform, exponential, normal distributions. Match each to a statement below. mean = median ≠ mode mean > median > mode mean = median = mode Chapter 10 Use the following information to answer the next three exercises: In a survey at Kirkwood Ski Resort the following information was recorded: 0–10 11–20 21–40 40+ Ski 10 12 30 8 Snowboard 6 17 12 5 Suppose that one person from was randomly selected. 88. Find the probability that the person was a skier or was age 11–20. 89. Find the probability that the person was a snowboarder given they were age 21–40. 90. Explain which of the following are true and which are false. Sport and age are independent events. Ski and age 11–20 are mutually exclusive events. P (Ski AND age 21–40) < P (Ski|age 21–40) P (Snowboard OR age 0–10) < P (Snowboard|age 0–10) 91. The average length of time a person with a broken leg wears a cast is approximately six weeks. The standard deviation is about three weeks. Thirty people who had recently healed from broken legs were interviewed. State the distribution that most accurately reflects total time to heal for the thirty people. 92. The distribution for X is uniform. What can we say for certain about the distribution for X ¯ when n = 1? The distribution for X ¯ is still uniform with the same mean and standard deviation as the distribution for X . The distribution for X ¯ is normal with the different mean and a different standard deviation as the distribution for X . The distribution for X ¯ is normal with the same mean but a larger standard deviation than the distribution for X . The distribution for X ¯ is normal with the same mean but a smaller standard deviation than the distribution for X . 93. The distribution for X is uniform. What can we say for certain about the distribution for ∑ X when n = 50? distribution for ∑ X is still uniform with the same mean and standard deviation as the distribution for X . The distribution for ∑ X is normal with the same mean but a larger standard deviation as the distribution for X . The distribution for ∑ X is normal with a larger mean and a larger standard deviation than the distribution for X . The distribution for ∑ X is normal with the same mean but a smaller standard deviation than the distribution for X . Use the following information to answer the next three exercises: A group of students measured the lengths of all the carrots in a five-pound bag of baby carrots. They calculated the average length of baby carrots to be 2.0 inches with a standard deviation of 0.25 inches. Suppose we randomly survey 16 five-pound bags of baby carrots. 94. State the approximate distribution for X ¯ , the distribution for the average lengths of baby carrots in 16 five-pound bags. X ¯ ~ ______ 95. Explain why we cannot find the probability that one individual randomly chosen carrot is greater than 2.25 inches. 96. Find the probability that x ¯ is between two and 2.25 inches. Use the following information to answer the next three exercises: At the beginning of the term, the amount of time a student waits in line at the campus store is normally distributed with a mean of five minutes and a standard deviation of two minutes. 97. Find the 90 th percentile of waiting time in minutes. 98. Find the median waiting time for one student. 99. Find the probability that the average waiting time for 40 students is at least 4.5 minutes. Chapter 11 Use the following information to answer the next four exercises: Suppose that the time that owners keep their cars (purchased new) is normally distributed with a mean of seven years and a standard deviation of two years. We are interested in how long an individual keeps his car (purchased new). Our population is people who buy their cars new. 100. Sixty percent of individuals keep their cars at most how many years? 101. Suppose that we randomly survey one person. Find the probability that person keeps their car less than 2.5 years. 102. If we are to pick individuals ten at a time, find the distribution for the mean car length ownership. 103. If we are to pick ten individuals, find the probability that the sum of their ownership time is more than 55 years. 104. For which distribution is the median not equal to the mean? Uniform Exponential Normal Student t 105. Compare the standard normal distribution to the Student’s t -distribution, centered at zero. Explain which of the following are true and which are false. As the number surveyed increases, the area to the left of –1 for the Student’s t -distribution approaches the area for the standard normal distribution. As the degrees of freedom decrease, the graph of the Student’s t -distribution looks more like the graph of the standard normal distribution. If the number surveyed is 15, the normal distribution should never be used. Use the following information to answer the next five exercises: We are interested in the checking account balance of twenty-year-old college students. We randomly survey 16 twenty-year-old college students. We obtain a sample mean of $640 and a sample standard deviation of $150. Let X = checking account balance of an individual twenty year old college student. 106. Explain why we cannot determine the distribution of X . 107. If you were to create a confidence interval or perform a hypothesis test for the population mean checking account balance of twenty-year-old college students, what distribution would you use? 108. Find the 95% confidence interval for the true mean checking account balance of a twenty-year-old college student. 109. What type of data is the balance of the checking account considered to be? 110. What type of data is the number of twenty-year-olds considered to be? 111. On average, a busy emergency room gets a patient with a shotgun wound about once per week. We are interested in the number of patients with a shotgun wound the emergency room gets per 28 days. Define the random variable X . State the distribution for X . Find the probability that the emergency room gets no patients with shotgun wounds in the next 28 days. Use the following information to answer the next two exercises: The probability that a certain slot machine will pay back money when a quarter is inserted is 0.30. Assume that each play of the slot machine is independent from each other. A person puts in 15 quarters for 15 plays. 112. Is the expected number of plays of the slot machine that will pay back money greater than, less than or the same as the median? Explain your answer. 113. Is it likely that exactly eight of the 15 plays would pay back money? Justify your answer numerically. 114. A game is played with the following rules: it costs $10 to enter. a fair coin is tossed four times. if you do not get four heads or four tails, you lose your $10. if you get four heads or four tails, you get back your $10, plus $30 more. Over the long run of playing this game, what are your expected earnings? 115. The mean grade on a math exam in Rachel’s class was 74, with a standard deviation of five. Rachel earned an 80. The mean grade on a math exam in Becca’s class was 47, with a standard deviation of two. Becca earned a 51. The mean grade on a math exam in Matt’s class was 70, with a standard deviation of eight. Matt earned an 83. Find whose score was the best, compared to their own class. Justify your answer numerically. Use the following information to answer the next two exercises: A random sample of 70 compulsive gamblers were asked the number of days they go to casinos per week. The results are given in the following graph: 116. Find the number of responses that were five. 117. Find the mean, standard deviation, the median, the first quartile, the third quartile and the IQR . 118. Based upon research at De Anza College, it is believed that about 19% of the student population speaks a language other than English at home. Suppose that a study was done this year to see if that percent has decreased. Ninety-eight students were randomly surveyed with the following results. Fourteen said that they speak a language other than English at home. State an appropriate null hypothesis. State an appropriate alternative hypothesis. Define the random variable, P ′. Calculate the test statistic. Calculate the p -value. At the 5% level of decision, what is your decision about the null hypothesis? What is the Type I error? What is the Type II error? 119. Assume that you are an emergency paramedic called in to rescue victims of an accident. You need to help a patient who is bleeding profusely. The patient is also considered to be a high risk for contracting hepatitis. Assume that the null hypothesis is that the patient does not have hepatitis. What is a Type I error? 120. It is often said that Californians are more casual than the rest of Americans. Suppose that a survey was done to see if the proportion of Californian professionals that wear jeans to work is greater than the proportion of non-Californian professionals. Fifty of each was surveyed with the following results. Fifteen Californians wear jeans to work and six non-Californians wear jeans to work. Let C = Californian professional; NC = non-Californian professional State appropriate null and alternate hypotheses. Define the random variable. Calculate the test statistic and p -value. At the 5% significance level, what is your decision? What is the Type I error? What is the Type II error? Use the following information to answer the next two exercises: A group of Statistics students have developed a technique that they feel will lower their anxiety level on statistics exams. They measured their anxiety level at the start of the quarter and again at the end of the quarter. Recorded is the paired data in that order: (1000, 900); (1200, 1050); (600, 700); (1300, 1100); (1000, 900); (900, 900). 121. This is a test of (pick the best answer): large samples, independent means small samples, independent means dependent means 122. State the distribution to use for the test. Chapter 12 Use the following information to answer the next two exercises: A recent survey of U.S. teenage pregnancy was answered by 720 girls, age 12–19. Six percent of the girls surveyed said they have been pregnant. We are interested in the true proportion of U.S. girls, age 12–19, who have been pregnant. 123. Find the 95% confidence interval for the true proportion of U.S. girls, age 12–19, who have been pregnant. 124. The report also stated that the results of the survey are accurate to within ±3.7% at the 95% confidence level. Suppose that a new study is to be done. It is desired to be accurate to within 2% of the 95% confidence level. What is the minimum number that should be surveyed? 125. Given: X ~ Exp ( 1 3 ) . Sketch the graph that depicts: P ( x > 1). Use the following information to answer the next three exercises: The amount of money a customer spends in one trip to the supermarket is known to have an exponential distribution. Suppose the mean amount of money a customer spends in one trip to the supermarket is $72. 126. Find the probability that one customer spends less than $72 in one trip to the supermarket? 127. Suppose five customers pool their money. How much money altogether would you expect the five customers to spend in one trip to the supermarket (in dollars)? 128. State the distribution to use if you want to find the probability that the mean amount spent by five customers in one trip to the supermarket is less than $60. Chapter 13 Use the following information to answer the next two exercises: Suppose that the probability of a drought in any independent year is 20%. Out of those years in which a drought occurs, the probability of water rationing is 10%. However, in any year, the probability of water rationing is 5%. 129. What is the probability of both a drought and water rationing occurring? 130. Out of the years with water rationing, find the probability that there is a drought. Use the following information to answer the next three exercises: Apple Pumpkin Pecan Female 40 10 30 Male 20 30 10 131. Suppose that one individual is randomly chosen. Find the probability that the person’s favorite pie is apple or the person is male. 132. Suppose that one male is randomly chosen. Find the probability his favorite pie is pecan. 133. Conduct a hypothesis test to determine if favorite pie type and gender are independent. Use the following information to answer the next two exercises: Let’s say that the probability that an adult watches the news at least once per week is 0.60. 134. We randomly survey 14 people. On average, how many people do we expect to watch the news at least once per week? 135. We randomly survey 14 people. Of interest is the number that watch the news at least once per week. State the distribution of X . X ~ _____ 136. The following histogram is most likely to be a result of sampling from which distribution? Chi-Square Geometric Uniform Binomial 137. The ages of De Anza evening students is known to be normally distributed with a population mean of 40 and a population standard deviation of six. A sample of six De Anza evening students reported their ages (in years) as: 28; 35; 47; 45; 30; 50. Find the probability that the mean of six ages of randomly chosen students is less than 35 years. Hint: Find the sample mean. 138. A math exam was given to all the fifth grade children attending Country School. Two random samples of scores were taken. The null hypothesis is that the mean math scores for boys and girls in fifth grade are the same. Conduct a hypothesis test. n x ¯ s 2 Boys 55 82 29 Girls 60 86 46 139. In a survey of 80 males, 55 had played an organized sport growing up. Of the 70 females surveyed, 25 had played an organized sport growing up. We are interested in whether the proportion for males is higher than the proportion for females. Conduct a hypothesis test. 140. Which of the following is preferable when designing a hypothesis test? Maximize α and minimize β Minimize α and maximize β Maximize α and β Minimize α and β Use the following information to answer the next three exercises: 120 people were surveyed as to their favorite beverage (non-alcoholic). The results are below. Beverage/Age 0–9 10–19 20–29 30+ Totals Milk 14 10 6 0 30 Soda 3 8 26 15 52 Juice 7 12 12 7 38 Totals 24 330 44 22 120 141. Are the events of milk and 30+: independent events? Justify your answer. mutually exclusive events? Justify your answer. 142. Suppose that one person is randomly chosen. Find the probability that person is 10–19 given that they prefer juice. 143. Are “Preferred Beverage” and “Age” independent events? Conduct a hypothesis test. 144. Given the following histogram, which distribution is the data most likely to come from? uniform exponential normal chi-square Solutions Chapter 3 1. c. parameter 2. a. population 3. b. statistic 4. d. sample 5. e. variable 6. quantitative continuous 7. 2.27 3.04 –1, 4, 4 8. Answers will vary. Chapter 4 9. c. (0.80)(0.30) 10. b. No, and they are not mutually exclusive either. 11. a. all employed adult women 12. 0.5773 13. 0.0522 14. b. The middle fifty percent of the members lost from 2 to 8.5 lbs. 15. c. All of the data have the same value. 16. c. The lowest data value is the median. 17. 0.279 18. b. No, I expect to come out behind in money. 19. X = the number of patients calling in claiming to have the flu, who actually have the flu. X = 0, 1, 2, …25 20. B (25, 0.04) 21. 0.0165 22. 1 23. c. quantitative discrete 24. all words used by Tom Clancy in his novels Chapter 5 25. 24% 27% 26. qualitative 27. 0.36 28. 0.7636 29. No No 30. B (10, 0.76) 31. 0.9330 32. X = the number of questions posted to the statistics listserv per day. X = 0, 1, 2,… X ~ P (2) 0 33. $150 34. Matt 35. false true false false 36. 16 37. first quartile: 2 second quartile: 2 third quartile: 3 38. 0.5 39. 7 15 40. 2 15 Chapter 6 41. true true False – the median and the mean are the same for this symmetric distribution. true 42. 8 8 P ( x < k ) = 0.65 = ( k – 3) ( 1 10 ) . k = 9.5 43. False – 3 4 of the data are at most five. True – each quartile has 25% of the data. False – that is unknown. False – 50% of the data are four or less. 44. d. G and H are independent events. 45. False – J and K are independent so they are not mutually exclusive which would imply dependency (meaning P ( J AND K ) is not 0). False – see answer c. True – P ( J OR K ) = P ( J ) + P ( K ) – P ( J AND K ) = P ( J ) + P ( K ) – P ( J ) P ( K ) = 0.3 + 0.6 – (0.3)(0.6) = 0.72. Note the P ( J AND K ) = P ( J ) P ( K ) because J and K are independent. False – J and K are independent so P ( J ) = P ( J | K ) 46. a. P (5) Chapter 7 47. a. U (0, 4) 48. b. 2 hour 49. a. 1 4 50. 0.7165 4.16 0 51. c. 5 years 52. c. exponential 53. 0.63 54. B (14, 0.20) 55. B (14, 0.20) Chapter 8 56. c. the mean amount of weight lost by 15 people on the special weight loss diet. 57. 0.9951 58. 12.99 59. c. 1 2 60. b. 0.60 61. c. N (60, 5.477) 62. 0.9990 63. a. eight days 64. c. 0.7500 65. a. 80% 66. b. 35% 67. b. no 68. b. quantitative continuous 69. c. 150 70. d. 0.06 71. c. 0.44 72. b. 0 Chapter 9 73. d. Matt is shorter than the average 14 year old boy. 74. Answers will vary. 75. x Relative Frequency Cumulative Relative Frequency 1 0.3 0.3 2 0.2 0.2 4 0.4 0.4 5 0.1 0.1 76. 2.8 1.48 90% 77. M = 3; Q 1 = 1; Q 3 = 4 78. 1 and 4 79. d. 8 70 80. c. 40 70 81. a. 9 19 82. b. false 83. b. false 84. b. false 85. X = the number of pies Lee bakes every day. P (20) 0.1122 86. CI: (5.25, 8.48) 87. uniform exponential normal Chapter 10 88. 77 100 89. 12 42 90. false false true false 91. N (180, 16.43) 92. a. The distribution for X ¯ is still uniform with the same mean and standard deviation as the distribution for X . 93. c. The distribution for ∑ X is normal with a larger mean and a larger standard deviation than the distribution for X . 94. N ( 2 , 0.25 16 ) 95. Answers will vary. 96. 0.5000 97. 7.6 98. 5 99. 0.9431 Chapter 11 100. 7.5 101. 0.0122 102. N (7, 0.63) 103. 0.9911 104. b. Exponential 105. true false false 106. Answers will vary. 107. Student’s t with df = 15 108. (560.07, 719.93) 109. quantitative continuous data 110. quantitative discrete data 111. X = the number of patients with a shotgun wound the emergency room gets per 28 days P (4) 0.0183 112. greater than 113. No; P ( x = 8) = 0.0348 114. You will lose $5. 115. Becca 116. 14 117. Sample mean = 3.2 Sample standard deviation = 1.85 Median = 3 Q 1 = 2 Q 3 = 5 IQR = 3 118. d. z = –1.19 e. 0.1171 f. Do not reject the null hypothesis. 119. We conclude that the patient does have the HIV virus when, in fact, the patient does not. 120. c. z = 2.21; p = 0.0136 d. Reject the null hypothesis. e. We conclude that the proportion of Californian professionals that wear jeans to work is greater than the proportion of non-Californian professionals when, in fact, it is not greater. f. We cannot conclude that the proportion of Californian professionals that wear jeans to work is greater than the proportion of non-Californian professionals when, in fact, it is greater. 121. c. dependent means 122. t 5 Chapter 12 123. (0.0424, 0.0770) 124. 2,401 125. Check student's solution. 126. 0.6321 127. $360 128. N ( 72 , 72 5 ) Chapter 13 129. 0.02 130. 0.40 131. 100 140 132. 10 60 133. p -value = 0; Reject the null hypothesis; conclude that they are dependent events 134. 8.4 135. B (14, 0.60) 136. d. Binomial 137. 0.3669 138. p -value = 0.0006; reject the null hypothesis; conclude that the averages are not equal 139. p -value = 0; reject the null hypothesis; conclude that the proportion of males is higher 140. Minimize α and β 141. No Yes, P ( M AND 30+) = 0 142. 12 38 143. No; p -value = 0 144. a. uniform References Data from the San Jose Mercury News . Baran, Daya. “20 Percent of Americans Have Never Used Email.” Webguild.org, 2010. Available online at: http://www.webguild.org/20080519/20-percent-of-americans-have-never-used-email (accessed October 17, 2013). Data from Parade Magazine .

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