AP Statistics Packet 12/13

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1 AP Statistics Packet 1/13 Inference for Proportions Inference for a Population Proportion Comparing Two Proportions Inference for Tables: Chi-Square Procedures Test for Goodness of Fit Inference for Two-Way Tables 1

2 HW #5 6 8, 1, 11, 13, EQUALITY FOR WOMEN? Have efforts to promote equality for women gone far enough in the United States? A poll on this issue by the cable network MSNBC contacted 119 adults. A newspaper article about the poll said, Results have a margin of sampling error of plus or minus 3 percentage points. (a) Overall, 54% of the sample (55 of 119 people) answered Yes. Find a 95% confidence interval for the proportion in the adult population who would say Yes if asked. Is the report s claim about the margin of error roughly right? (Assume that the sample is an SRS.) p = true proportion of adults who would say Yes if asked whether efforts to promote equality for women have gone far enough in the U.S. SRS 119 < 1% of adult population np ˆ = 55 > 1 and n(1 pˆ ) = 469 > 1 One-proportion z interval (.54)(.46) 95% CI =.54± = (.51,.57) We are 95% confident that the true proportion of adults who would say Yes if asked whether efforts to promote equality for women have gone far enough in the U.S. is between 51 and 57%. The margin of error is about 3% as stated. (b) The news article said that 65% of men, but only 43% of women, think that efforts to promote equality have gone far enough. Explain why we do not have enough information to give confidence intervals for men and women separately. We weren t given sample sizes for each gender. (c) Would a 95% confidence interval for women alone have a margin of error less than.3, about equal to.3, or greater than.3? Why? You see that the news article s statement about the margin of error for poll results is a bit misleading. The margin of error for women alone would be greater than.3 since the sample size is smaller.

3 1.7 TEENS AND THEIR TV SETS The New York Times and CBS News conducted a nationwide poll of 148 randomly selected 13- to 17-year-olds. Of these teenagers, 69 had a television in their room and 189 named Fox as their favorite television network. We will act as if the sample were an SRS. (a) Use your calculator to give 95% confidence intervals for the proportion of all people in this age group who have a TV in their room and the proportion who would choose Fox as their favorite network. Check that we can use our methods. (Check assumptions.) Check assumptions for TV in room: SRS 148 < 1% of all 13- to 17-year olds np ˆ = 69 > 1, n(1 pˆ ) = 356 >1 Check assumptions for prefer FO: SRS 148 < 1% of all 13- to 17-year olds np ˆ = 189 > 1, n(1 pˆ ) = 859 >1 95% CI fortv inroom =.66 ± 1.96 = (.631,.689) (.66)(.34) % CI for preferring FO =.18 ± 1.96 = (.157,.3) (.18)(.8) 148 (b) The news article says, In theory, in 19 cases out of, the poll results will differ by no more than three percentage points in either direction from what would have been obtained by seeking out all American teenagers. Explain how your results agree with this statement. In both cases, the margin of error for a 95% CI ( 19 cases out of ) was (no more than) 3%. (c) Is there good evidence that more than half of all teenagers have a TV in their room? p = proportion of teenagers with TV s in their rooms H : p=.5 vh : p>.5 A Assumptions checked above One-proportion z test z = = 1.36 (.5)(.5) 148 p-value = P(Z >1.36) <. since the p-value is so small, we reject H, and conclude that more than half of teenagers have TVs in their rooms. (Additionally, the interval from (a) does not include.5 or less) 3

4 1.8 WE WANT TO BE RICH In a recent year, 73% of first-year college students responding to a national survey identified being very well-off financially as an important personal goal. A state university finds that 13 of an SRS of of its first-year students say that this goal is important. (a) Give a 95% confidence interval for the proportion of all first-year students at the university who would identify being well-off as an important goal. p = true proportion of all first-year students at this university who identify being well-off as an important goal SRS < 1% of first-year students at this university np ˆ = 13 > 1 and n(1 pˆ ) = 68 > 1 One-proportion z interval (.66)(.34) 95% CI =.66± 1.96 = (.594,. 76) We are 95% confident that the true proportion of all first-year students at this university who identify being well-off as an important goal is between 59.4 and 7.6%. (b) Is there good evidence that the proportion of all first-year students at this university who think being very well-off is important differs from the national value. Yes; the 95% confidence interval for this university contains only values that are less than.73, so it is likely that for this particular population, p differs from.73 (specifically, is less than.73). 1.1 STARTING A NIGHT CLUB A college student organization wants to start a nightclub for students under the age of 1. To assess support for this proposal, they will select an SRS of students and ask each respondent if he or she would patronize this type of establishment. They expect that about 7% of the student body would respond favorably. What sample size is required to obtain a 9% confidence interval with an approximate margin of error of.4. Suppose that 5% of the sample responds favorably. Calculate the margin of error of the 9% confidence interval. Solving for n: (.7)(.3) =.4 we find n 356 n (.5)(.5) If p =.5, then the true margin of error = =

5 1.11 SCHOOL VOUCHERS A national opinion poll found that 44% of all American adults agree that parents should be given vouchers good for education at any public or private school of their choice. The result was based on a small sample. How large an SRS is required to obtain a margin of error of.3 (that is, ±3%) in a 95% confidence interval? (a) Answer this question using the previous poll s result as the guessed value p *. Solving for n: (.44)(.56) 1.96 =.3 we find n 15 n (b) Do the problem again using the conservative guess p * =.5. By how much do the two sample sizes differ. (.5)(.5) Solving for n: 1.96 =.3 we find n 168 n An additional 16 people are needed DRUNKEN CYCLISTS In the United States approximately 9 people die in bicycle accidents each year. One study examined the records of 1711 bicyclists aged 15 or older who were fatally injured in bicycle accidents between 1987 and 1991 and were tested for alcohol. Of these, 54 tested positive for alcohol (blood alcohol concentration of.1% or higher). (a) Find a 95% confidence interval for p. p = true proportion of all cyclists involved in fatal accidents who had alcohol in their systems SRS we do NOT know that the examined records come from as SRS, so we proceed with caution is probably not < 1% of all bicyclists fatally injured between 1987 and 1991, so again proceed with caution np ˆ = 54 > 1 and n(1 pˆ ) = 1169 > 1 One-proportion z interval % CI = ± = (.95,.339) If our assumptions had been satisfied we would be 95% confident that the true proportion of all cyclists involved in fatal accidents who had alcohol in their systems is between 9.5 and 33.9%. (b) Can you conclude from your statistical analysis of this study that alcohol causes fatal bicycle accidents? Explain. No: We do not know; for example, what percentage of cyclists who were not involved in fatal accidents had alcohol in their systems. 5

6 1.1 COLLEGE FOOD Tonya, Frank, and Sarah are investigating student attitudes toward college food for an assignment in their introductory statistics class. Based on comments overheard from other students, they believe that fewer than 1 in 3 students like college food. To test this hypothesis, each selects an SRS of students who regularly eat in the cafeteria, and asks them if they like college food. Fourteen in Tonya s SRS of 5 replied, Yes, while 98 in Frank s sample of 35, and 14 in Sarah s sample of 5 said they like college food. Use your calculator to perform a test of significance on all three results and fill in a table like this: n ˆp z P-value Describe your findings in a short narrative. Although Tonya, Frank, and Sarah all recorded the same sample proportion, ˆp =.8, the p-values were all quite different. Conclude: For a given sample proportion, the larger the sample size, the smaller the p-value. 6

7 HW #6, 3, 5 7, 9 1. IN-LINE SKATERS A study of injuries to in-line skaters used data from the National Electronic Injury Surveillance System, which collects data from a random sample of hospital emergency rooms. In the six-month study period, 6 people came to the sample hospitals with injuries from inline skating. We can think of these people as an SRS of all people injured while skating. Researchers were able to interview 161 of these people. Wrist injuries (mostly fractures) were the most common. (a) The interviews found that 53 people were wearing wrist guards and 6 of these had wrist injuries. Of the 18 who did not wear wrist guards, 45 had wrist injuries. What are the two sample proportions of wrist injuries? 6 45 pˆ ˆ 1= =.113 and p = = (b) Give a 95% confidence interval for the difference between the two population proportions of wrist injuries. State carefully what populations your inference compares. We would like to draw conclusions about all in-line skaters, but we have data only for injured skaters. p 1 = true proportion of all people who went to the ER with wrist injuries who were wearing wrist guards, p = true proportion of all people who went to the ER with wrist injuries who were NOT wearing wrist guards SRSs, given independent samples 53 < 1% of people wearing wrist guards who came to the ER with skating injuries, 18 < 1% of people NOT wearing wrist guards who came to the ER with skating injuries np= ˆ > 5, n ˆ 1(1 p1) = 47 > 5, np= ˆ 45 > 5, n ˆ (1 p) = 63 > 5 Two-proportion z interval (.113)(.8868) (.4167)(.5833) 95% CI = ( ) ± = (.43,.177) We are 95% confident that the true proportion of skaters with wrist injuries is lower by 43 to 18% for skaters who are wearing wrist guards. As the problem notes, this interval should at least apply to the two groups about whom we have information: skaters (with and without wrist guards) with severe enough injuries to go to the emergency room. (c) What was the percent of nonresponse among the original sample of 6 injured skaters? Explain why nonresponse may bias your conclusions. 45/6 =.18 did not respond. If those who did not respond were different in some way from those who did, then not having them represented in our sample makes our conclusions suspect. 7

8 1.3 LYME DISEASE Lyme disease is spread in the northeastern United States by infected ticks. The ticks were infected mainly by feeding on mice, so more mice result in more infected ticks. The mouse population in turn rises and falls with the abundance of acorns, their favored food. Experimenters studied two similar forest areas in a year when the acorn crop failed. They added hundreds of thousands of acorns to one area to imitate an abundant acorn crop, while leaving the other area untouched. The next spring, 54 of the 7 mice trapped in the first area were in breeding condition, versus 1 of the 17 mice trapped in the second area. Give a 9% confidence interval for the difference between the proportion of mice ready to breed in good acorn years and bad acorn years. p 1 = true proportion of mice ready to breed in a good acorn year, p = true proportion of mice ready to breed in a bad acorn year SRSs, assumed independent samples 7 < 1% of mice in good year, 17 < 1% of mice in bad years np= ˆ > 5, n ˆ 1(1 p1) = 18 > 5, np= ˆ 1 > 5, n ˆ (1 p) = 7 > 5 Two-proportion z interval (.75)(.5) (.588)(.4118) 9% CI = ( ) ± = (.518,.3753) We are 9% confident that the true proportion of mice who are ready to breed in good acorn years is between -5% and 38% greater than the proportion of mice are ready to breed in bad acorn years. 8

9 1.5 THE GOLD COAST A historian examining British colonial records for the Gold Coast in Africa suspects that the death rate was higher among African miners than among European miners. In the year 1936, there were 3 deaths among 33,89 African miners and 7 deaths among 1541 European miners on the Gold Coast. Consider this year as a sample from the pre-war era in Africa. Is there good evidence that the proportion of African miners who died was higher than the proportion of European miners who died? p A = true proportion of African miners who died in 1936 on the Gold Coast, p E = true proportion of European miners who died in 1936 on the Gold Coast H pa = p H : p > p : E A A E SRSs, assumed independent samples 33,89 < 1% of African miners, 1541 < 1% of European miners n p ˆ = 3 > 5, n (1 pˆ ) = > 5, n p ˆ = 7 > 5, n (1 pˆ ) = 1534 > 5 A A Two-proportion z test A A 3+ 7 pˆ = = z = = (.656)( ) E E E E p-value = P(Z >.985) =.1635 The p-value is large, so we fail to reject H, we could easily see a result like this by chance alone. We do not have evidence to conclude that the death rates for African and European miners are different. 9

10 1.6 PREVENTING STROKES Aspirin prevents blood from clotting and so helps prevent strokes. The Second European Stroke Prevention Study asked whether adding another anticlotting drug named diprydamole would be more effective for patients who had already had a stroke. Here are the data on strokes and deaths during the two years of the study: Number of patients Number of strokes Number of deaths Aspirin alone Aspirin + dipyridamole (a) The study was a randomized comparative experiment. Outline the design of the study. (b) Use your calculator to determine if there is a significant difference in the proportion of strokes in the two groups. p 1 = true proportion of strokes among patients who used aspirin alone, p = true proportion of strokes among patients who used aspirin + dipyridamole H : p = p H : p p 1 A 1 SRSs, assumed independent samples 1649 < 1% of patients using aspirin alone, 165 < 1% of patients using aspirin + dipyridamol pooling we have pˆ = = npˆ = 1649(.11) = 181.4> 5, nqˆ = 1649(.89) = > npˆ = 165(.11) = 181.5> 5, nqˆ = 165(.89) = > 5 Two-proportion z test pˆ1 pˆ ( p1 p) () z = = = pq ˆˆ + (.11)(.89) + n n p-value = P(z >.73) =.64 The p-value is small, so we reject H. We have strong evidence to conclude that the proportions for stroke rates are different. (c) Is there a significant difference in death rates for the two groups? p 1 = true proportion of deaths among patients who used aspirin alone, p = true proportion of deathhs among patients who used aspirin + dipyridamole H : p = p H : p p 1 A 1 Assumptions are satisfied analogous to above z = -.16 p-value = P(z < -.16) =.878 The p-value is very large, so we fail to reject H. We have no evidence that the proportions for death rates are different. 1

11 1.7 ACCESS TO COMPUTERS A sample survey by Nielsen Media Research looked at computer access and use of the Internet. Whites were significantly more likely than blacks to own a home computer, but the black-white difference in computer access at work was not significant. The study team then looked separately at the households with at least \$4, income. The sample contained 1916 white and 131 black households in this class. Here are the sample counts for these households. Own home computer Computer access at work Blacks 86 1 Whites Do higher-income blacks and whites differ significantly at the 5% level in the proportion who own home computers? Do they differ significantly in the proportion who have computer access at work? p b = true proportion of blacks with \$4, income with home computer, p w = true proportion of whites with \$4, income with home computer p b = true proportion of blacks with \$4, income with computer access at work, p w = true proportion of whites with \$4, income with computer access at work H : pb = pw H pb = pw H : p p H : p p A b w The p-value is large, so we fail to reject H. We do not have evidence to conclude there is a difference in the proportions of higher-income blacks and whites who have home computers A b w SRSs, assumed independent samples 131 < 1% of blacks with \$4, income, 1916 < 1% of whites with \$4, income SRSs, assumed independent samples 131 < 1% of blacks with \$4, income, 1916 < 1% of whites with \$4, income pooling pooling pˆ = =.615 pˆ = = npˆ 131(.615) 8.5 5, ˆ b = = > nq b = 131(.385) = 5.5> 5 npˆ ˆ b = 131(.6) = 79> 5, nq b = 131(.398) = 5> 5 n ˆ 1916(.615) , ˆ wp= = > nq w = 1916(.385) = 738> n5ˆ ˆ wp= 1916(.6) = 1153> 5, nq w = 1916(.398) = 763> 5 Two-proportion z test Two-proportion z test z = = z = = 1.1 (.619)(.3981) (.615)(.385) p- value = Pz ( > 1.1) =.314 p-value <.4 The p-value is small, so we reject H. We have strong evidence to conclude that there is a difference in the proportions of higher-income blacks and whites who have computer access at work. (Specifically, higher-income blacks have more access at work.) 11

12 1.9 TREATING AIDS The drug AZT was the first drug that seemed effective in delaying the onset of AIDS. Evidence for AZT s effectiveness came from a large randomized comparative experiment. The subjects were 13 volunteers who were infected with HIV, the virus that causes AIDS, but did not yet have AIDS. The study assigned 435 of the subjects at random to take 5 milligrams of AZT each day, and another 435 to take a placebo. (The others were assigned to a third treatment, a higher dose of AZT. We will compare only two groups.) At the end of the study, 38 of the placebo subjects and 17 of the AZT subjects had developed AIDS. We want to test the claim that taking AZT lowers the proportion of infected people who will develop AIDS in a given period of time. (a) State hypotheses, and check that you can safely use the z procedure. p 1 = true proportion of HIV infected patients who develop AIDS while taking AZT, p = true proportion of HIV infected patients who develop AIDS while taking a placebo H : p1 = p H : p < p A 1 SRSs, assumed independent samples 435 < 1% of HIV infected patients taking AZT, 435 < 1% of of HIV infected patients taking a placebo pooling pˆ = = npˆ = 435(.63) = 7.5> 5, 1 nqˆ = 435(.9368) = 47.5> 5 1 same for npˆ and nqˆ (b) How significant is the evidence that AZT is effective? ˆp =.63, pˆ pˆ ( p p ) () 1 1 z = = pq ˆˆ + (.63)(.9368) + n n p= Pz ( >.96) =.17 =.96 The difference is statistically significant. With such a small p-value we reject H and conclude that the proportions are different. (c) The experiment was double-blind. Explain what this means. Neither the subjects nor the researchers who had contact with them knew which subjects were getting which drug if anyone had known, they might have confounded the outcome by letting their expectations or biases affect the results. 1

13 HW #7 35, 37, 39, POLICE RADAR AND SPEEDING Do drivers reduce excessive speed when they encounter police radar? Researchers studied the behavior of drivers on a rural interstate highway in Maryland where the speed limit was 55 miles per hour. They measured speed with an electronic device hidden in the pavement and, to eliminate large trucks, considered only vehicles less than feet long. During some time periods, police radar was set up at the measurement location. Here are some of the data: No radar Radar Number of vehicles 1,931 3,85 Number over 65 mph 5,69 1,51 (a) Give a 95% confidence interval for the proportion of vehicles going faster than 65 miles per hour when no radar is present. p = true proportion of vehicles going faster than 65 miles per hour when no radar is present. SRS assumed 1931 < 1% of all motorists np ˆ = 569 > 1 and n(1 pˆ ) = 741 > 1 One-proportion z interval (.44)(.56) 95% CI =.44± ,931 = (.43,.449) We are 95% confident that the true proportion of vehicles going faster than 65 miles per hour when no radar is present is between 43. and 44.9%. (b) Give a 95% confidence interval for the effect of radar, as measured by the difference in proportions of vehicles going faster than 65 miles per hour with and without radar. p 1 = true proportion of vehicles going faster than 65 miles per hour when no radar is present, p = true proportion of vehicles going faster than 65 miles per hour when radar is present. SRSs, assumed independent samples?? 1,931 < 1% of all motorists in no radar zones 385 < 1% of all motorists in radar zones np= ˆ > 5, n ˆ 1(1 p1) = 741 > 5, np= ˆ 151 > 5, n ˆ (1 p) = 34 > 5 Two-proportion z interval (.44)(.56) (.3)(.68) 95% CI = (.44.3) ± , = (.1,.138) We are 95% confident that the 1. to 13.8% of drivers are more apt to speed when no radar is present, then when radar is present. 13

14 (c) The researchers chose a rural highway so that cars would be separated rather than in clusters where some cars might slow because they see other cars slowing. Explain why such clusters might make inference invalid. In a cluster of cars, where one driver s behavior might affect the others, we do not have independence one of the important properties of a random sample SMALL-BUSINESS FAILURES, I A study of the survival of small businesses chose an SRS from the telephone directory s Yellow Pages listings of food-and-drink businesses in 1 counties in central Indiana. For various reasons, they study got no response from 45% of the businesses chosen. Interviews were completed with 148 businesses. Three years later, of these businesses had failed. (a) Give a 95% confidence interval for the percent of all small businesses in this class that fail within three years. p = true proportion of small businesses in this class that fail within three years SRS assumed 148 < 1% of small food-and-drink businesses np ˆ = > 1 and n(1 pˆ ) = 16 > 1 One-proportion z interval (.1486)(.8514) 95% CI =.1486± = (.913,.59) We are 95% confident that the true proportion of small businesses in this class that fail within three years is between 9.1 and.6%. (b) Based on the results of this study, how large a sample would you need to reduce the margin of error to.4? (.1486)(.8514) Solving for n: 1.96 =.4 we find n 34 n (We should not use p* =.5 here since we have evidence that the true value of p is not in the range.3 to.7.) (c) The authors hope that their findings describe the population of all small businesses. What about the study makes this unlikely? What population do you think the study findings describe? Aside from the 45% nonresponse rate, the sample comes from a limited area in Indiana, focuses on only one kind of business, and leaves out any businesses not in the Yellow Pages (there might be a few of these; perhaps they are more likely to fail). It is more realistic to believe that this describes businesses that match the above profile; it might generalize to food-and-drink establishments elsewhere, but probably not to hardware stores and other types of business. 14

15 1.39 SIGNIFICANT DOES NOT MEAN IMPORTANT Never forget that even small effects can be statistically significant if the samples are large. To illustrate this fact, return to the study of 148 small businesses from the preceding exercise. Of these, 16 were headed by men and 4 were headed by women. During a three-year period, 15 of the men s businesses and 7 of the women s businesses failed. (a) Find the proportions of failures for businesses headed by women and businesses headed by men. These sample proportions are quite close to each other. Test the hypothesis that the sample proportions of women s and men s businesses fail. (Use the two-sided alternative.) The test is very far from being significant. pˆ =.1415, pˆ =.1667; p=.6981 m w (b) Now suppose that the same sample proportions came from a sample 3 times as large. That is, 1 out of 16 businesses headed by women and 45 out of 318 businesses headed by men fail. Verify that the proportions of failures are exactly the same as in (a). Repeat the z test for the new data, and show that it is now significant at the α =.5 level. z =.1, p =.336 (c) It is wise to use a confidence interval to estimate the size of an effect, rather than just giving a P- value. Give 95% confidence intervals for the difference between the proportions of women s and men s businesses that fail for the settings of both (a) and (b). What is the effect of larger samples on the confidence interval? From (a) (-.156,.1559) From (b) (.178,.4936) The larger samples make the margin of error (and thus the length of the confidence interval) smaller 1.41 MEN VERSUS WOMEN The National Assessment of Educational Progress (NAEP) Young Adult Literacy Assessment Survey interviewed a random sample of 1917 people 1 to 5 years old. The sample contained 84 men, of whom 775 were fully employed. There were 177 women, and 68 of them were fully employed. (a) Use your calculator to find a 99% confidence interval to describe the difference between the proportions of young men and young women who are fully employed. Is the difference statistically significant at the 1% significance level? 99 % CI is.465 to.3359 since is not in this interval, we would reject H : p 1 = p at the 1% level (in fact, P is practically ). (b) The mean and standard deviation of scores on the NAEP s test of quantitative skills were x = 7.4and s = 59. for the men in the sample. For the women, the results were 1 1 x = and s = 59.. Is the difference between the mean scores for men and women significant at the 1% level? No: t =.8658, which gives a p-value close to.4, not significant. 15

16 HW #8 1 4, 7, 8, FINDING P-VALUES (a) Find the p-value corresponding to = 1.41 for a chi-square distribution with 1 degree of freedom: (i) using Table E and (ii) with your graphing calculator. (i). < p <.5 (ii) p =.35 (b) Find the area to the right of = 19.6 under the chi-square curve with 9 degrees of freedom: (i) using Table E and (ii) with your graphing calculator. (i). < p <.5 (ii) p =.4 (c) Find the p-value corresponding to = 7.4 for a chi-square distribution with 6 degrees of freedom: (i) using Table E and (ii) with your graphing calculator. (i) p >.5 (ii) p = ARE YOU MARRIED? According to the March Current Population Survey, the marital status of the U.S. adult population is as follows: Marital Status: Percent: Never married 8.1 Married 56.3 Widowed 6.4 Divorced 9. A random sample of 5 U.S. males, aged 5 to 9 years old, yielded the following frequency distribution: Marital Status: Percent: Never married 6 Married Widowed Divorced Perform a goodness of fit test to determine if the marital status distribution of U.S. males 5 to 9 years old differs from that of the U.S. adult population. H : The marital-status distribution of 5- to 9-year-old U.S. males is the same as that of the population as a whole. H A : The marital-status distribution of 5- to 9-year-old U.S. males is different from that of the population as a whole. SRS, given Independent samples Expected counts are all 5 [ ] Chi-square goodness of fit (Ο Ε) ( 14.5) ( 46) =... = + + E = , withdf = 3 p-value = P( χ > ) Since the p-value is so small, we reject H. We have evidence to show that the marital-status distribution of 5- to 9-year-old U.S. males is different from that of the population as a whole. 16

17 13.3 GENETICS: CROSSING TOBACCO PLANTS Researchers want to cross two yellow-green tobacco plants with genetic makeup (Gg). Here is a Punnet square for this genetic experiment: G g G GG gg g Gg gg This shows that the expected ratio of green (GG) to yellow-green (Gg) to albino (gg) tobacco plants is 1::1. When the researchers perform the experiment, the resulting offspring are green, 5 yellowgreen, and 1 albino seedlings. Use a chi-square goodness of fit test to assess the validity of the researchers genetic model. H : The genetic model is valid (the different colors occur in the stated ratio of 1::1) H A : The genetic model is not valid SRS, given Independent samples Expected counts are all 5 [ 1 4 1] Chi-square goodness of fit (Ο Ε) ( 1) (5 4) (1 1) = = + + E = 5.43, withdf = p-value = P( χ > 5.43) =.66 Since the p-value is greater than 5%, we fail to reject H. We do not have evidence to show that the genetic model is not valid. 17

18 13.4 In recent years, a national effort has been made to enable more members of minority groups to have increased educational opportunities. You want to know if the policy of affirmative action and similar initiatives have had any effect in this regard. You obtain information on the ethnicity distribution of holders of the highest academic degree, the doctor of philosophy degree, for the year 1981: Race/ethnicity White, non-hispanic Black, non-hispanic Hispanic Asian of Pacific Islander American Indian/Alaskan Native Nonresident alien Percent A random sample of 3 doctoral degree recipients in 1994 showed the following frequency distribution: Race/ethnicity Percent White, non-hispanic 189 Black, non-hispanic 1 Hispanic 6 Asian of Pacific Islander 14 American Indian/Alaskan Native 1 Nonresident alien 8 (a) Perform a goodness of fit test to determine if the distribution of doctoral degrees in 1994 is significantly different from the distribution in Note: Although two expected counts are less than 5, the chi-square test still gives adequate results in this setting. H : The ethnicity distribution of the PhD degree in 1994 is the same as it was in H : The ethnicity distribution of the PhD degree in 1994 is not the same as it was in A SRS, given Independent samples Expected counts: [ ] Chi-square goodness of fit (Ο Ε) ( ) (8 38.4) = = E = 61.98, withdf = 5 p-value = P( χ > 61.98) Since the p-value is so small, we reject H. We have evidence to show that the ethnicity distribution of the PhD degree has changed from 1981 to

19 (b) In what categories have the greatest changes occurred and in what direction? The greatest change is that many more nonresident aliens than expected received the Ph.D. degree in 1994 over the 1981 figures. To a lesser extent, a smaller proportion of white, non- Hispanics received the Ph.D. degree in IS YOUR RANDOM NUMBER GENERATOR WORKING? In this exercise you will use your calculator to simulate sampling from the following uniform distribution: : P(): You will then perform a goodness of fit test to see if a randomly generated sample distribution comes form a population that is different from this distribution. (a) State the null and alternative hypotheses for this test. H : p = p = p =... = p = H : At least one of the p s is not equal to.1. A (b) Use the randint function to randomly generate digits from to 9, and store these values in L4. Using randint (, 9, ) L4, we obtained these counts for digits to 9: 19, 17, 3,, 19,, 5, 1, 7, 16. (c) Plot the data as a histogram with Window dimensions sets as follows: [-.5,9.5] 1 and Y[-5, 3] 5. (You may have to increase the vertical scale.) Then TRACE to see the frequencies of each digit. Record these frequencies (observed values) in L1. (d) Determine the expected counts for a sample size of, and store them in L. They should all = (e) Complete a goodness of fit test. Report your chi-square statistic, the P-value, and your conclusion with regard to the null and alternative hypotheses. SRS independent sample All expected counts 5 (Ο Ε) (19 ) (16 ) = = E = 8.9, withdf = 9 p-value =.447 p-value is large, so we fail to reject H. There is no evidence that the sample data were generated from a distribution that is different from the uniform distribution. 19

20 13.8 ROLL THE DICE Simulate rolling a fair, six-sided die 3 times on your calculator. Plot a histogram of the results, and then perform a goodness of fit test of the hypothesis that the die is fair. H : The die is fair, all the p s = 1/6. Use randint(1, 6, 3) L1 H A : The die is not fair To simulate rolling a fair die 3 times. SRS Independent samples In our simulation, we obtained the following frequency distribution: All expected counts = 5 5 side Chi-square goodness of fit freq (Ο Ε) (57 5) (43 4) = = E 5 5 = 3.6, withdf = 5 p-value = P( χ > 3.6) =.68 Since the p-value is so large, we fail to reject H. We have no evidence to show that the die is not fair CARNIVAL GAMES A wheel of fortune at a carnival is divided into four equal parts: Part I: Win a doll Part II: Win a candy bar Part III: Win a free ride Part IV: Win nothing You suspect that the wheel is unbalanced (i.e., not all parts of the wheel are equally likely to be landed upon when the wheel is spun.) The results of 5 spins of the wheel are as follows: Part: Frequency: I 95 II 15 III 135 IV 165 Perform a goodness of fit test. Is there evidence that the wheel is not in balance? H : The wheel is balanced. (All four outcomes are uniformly distributed) H A : The wheel is not balanced. SRS Independent samples All expected counts = 15 5 Chi-square goodness of fit (Ο Ε) (95 15) (165 15) = = E = 4, withdf = 3 p-value = Since the p-value is so small, we reject H. We have strong evidence to show that the wheel is not balanced. Since Part IV: Win nothing shows the greatest deviation from the expected result, there may be reason to suspect that the carnival game operator may have tampered with the wheel to make it harder to win.

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