STATISTICS SECTION II Part A Questions 1-5 Spend about 65 minutes on this part of the exam. Percent of Section II grade

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1 STATISTICS SECTION II Part A Questions 1-5 Spend about 65 minutes on this part of the exam. Percent of Section II grade Directions: Show all your work. Indicate clearly the methods you use, because you will be scored on the correctness of your methods as well as on the accuracy and completeness of your results and explanations. 1. Students in an AP Statistics class participated in an online memory game. All of the students first played the game at Level 1 (the lowest difficulty level), and then played the game again at Level 4 (a higher level of difficulty). The graphs below display the distribution of student scores for the two difficulty levels Level 1 and Level 4. Score (a) Use the graphical display above to compare the distribution of student scores for the two difficulty levels (Level 1 and Level 4) of the memory game. -1-

2 The difference in scores (Level 4 Level 1) on the memory game was calculated for each student. The graph below displays the distribution of the differences. (b) What added information does the graph above of the difference in scores (Level 4 Level 1) give you about students' scores on the two different levels of the game that was not apparent in the first graphical display? -2-

3 2. The National Park Service is interested in determining whether placing predator cages over the loggerhead turtle nests on Cape Lookout National Seashore will keep raccoons from stealing eggs from the nests. Due to budget constraints, funding for the predator cages will only be approved if the Park Service can provide convincing evidence that the predator cages increase the number of turtles that successfully fledge from the nests. The Park Service plans to collect data from a random sample of the loggerhead turtle nests which are laid at Cape Lookout one season. A test of significance will be conducted at a significance level of αα = 0.05 for the following hypotheses: H0 : μμ C = μμ NC Ha : μμ C > μμ NC, where μμ C is the mean number of eggs that successfully hatch per nest for all loggerhead turtle nests on Cape Lookout with a cage and μμ NC is the mean number of eggs that successfully hatch per nest for all nests on Cape Lookout with no cage. (a) Describe what a Type I error would be in the context of the study, and also describe a consequence of making this type of error. -3-

4 (b) Each season, the Park Service moves approximately half of the turtle nests at Cape Lookout very soon after they are laid because they are laid in locations that are vulnerable to extreme high tides. The Park Service decides to collect data for their study by randomly selecting 35 of the nests that were moved and placing a cage over them and comparing the fledgling rate to the rate for 35 randomly selected nests that were neither moved nor caged. This resulted in a p-value of for the hypotheses stated above. If it was reasonable to conduct a test of significance for the hypotheses stated above using the data collected, what would the p-value of lead you to conclude? (c) Describe the primary flaw in the study described in part (b), and explain why it is a concern. -4-

5 3. A smartphone manufacturer is concerned about the proportion of defectives produced at a certain plant which produces many smartphones every day. Historically, approximately 15% of phones produced at this plant have been defective. As part of their quality assurance testing, 4 smartphones are selected at random from a day's production (a) Let X represent the number of defective phones in the sample of 4 phones. Complete the table below for the probability distribution of X, assuming the historic defective rate holds. x P(x) (b) What is the expected number of defective phones in this sample? -5-

6 (c) What is the probability that all four phones were defective given that at least two defectives were found in the sample? (d) Suppose that each day the phones are produced independently of all other days' phones. If a sample of size 4 is taken every weekday (5 total samples), what is the probability that there are no defective phones found in any of the 5 samples taken? -6-

7 4. The National Sleep Foundation conducts an annual survey to track sleep related behaviors of U.S. adults. In their most recent survey, a random sample of 1,018 adults answered the question "About how much actual sleep would you estimate you typically get on work nights or weeknights?" The frequency table below summarizes the responses by whether they were less than, equal to, or more than the recommended 7 to 9 hours of sleep and by the age group of the respondent. Typical Weeknight Sleep Time Age Group Less Than Recommended More Than 7 Hours 7 to 9 Hours 9 Hours Total or older Total ,018 At the αα = 0.05 significance level, do the data provide convincing statistical evidence that there is an association between age group and typical weeknight sleep time for adults in the United States? -7-

8 5. In 2006, tennis introduced a challenge system in which a player can challenge a decision as to whether a tennis ball was correctly called in or out by an official. In the 2015 U.S. Open Tennis Championship there were a total of 850 player challenges between both the Men s and Women s Singles matches. These challenges were determined either to be correct (meaning the player was correct and the umpire was incorrect) or incorrect (meaning the player was incorrect and the umpire was correct). These 850 challenges are summarized in the table below. Player Challenge Men s Women s Correct Incorrect (a) Calculate the proportion of all challenges that were determined to be correct. (b) Using these data as a representative sample of all player challenges in professional tennis, a 95% confidence interval for the difference in the proportion of men s and women s challenges that are determined to be correct (men women) was found to be 0.04 ± All conditions for inference were met. Does this confidence interval provide convincing statistical evidence that there is a difference in the effectiveness with which men and women use the challenge system? Justify your answer. -8-

9 A scientific study has demonstrated that the best time for a player to use a challenge is when the tennis ball was called out but the player believes that the ball was actually in because when objects travel faster than the human eye the umpire is left to fill the gap with their own perception. The table below shows the Women s Singles challenges based on if the tennis ball was called out or in. Player Challenge Umpire Called Ball Out Umpire Called Ball In Correct Incorrect (c) Calculate the proportion of correct player challenges when the umpire called the ball out and the proportion of correct player challenges when the umpire called the ball in. Out proportion: In proportion: (d) Using these data as a representative sample of all player challenges in professional tennis, a 95% confidence interval for the difference in the proportion of correct challenges when the umpire calls the ball out and when the umpire calls the ball in (out in) was found to be 0.27 ± All conditions for inference were met. Does this confidence interval provide statistical evidence to support the scientific findings detailed above? Justify your answer. -9-

10 STATISTICS SECTION II Part B Question 6 Spend about 25 minutes on this part of the exam. Percent of Section II grade Directions: Show all your work. Indicate clearly the methods you use, because you will be scored on the correctness of your methods as well as on the accuracy and completeness of your results and explanations. 6. In a large manufacturing company every item produced is inspected for defects and will go through a repair process if there are serious defects. Management wanted to investigate whether items produced on Mondays are more likely to require repairing than items produced on the midweek day Wednesday. A random sample of 9 weeks from the past 5 years was taken and the number of items which required repairing for the 9 weeks are shown in the table below. Week Monday Wednesday Difference More Repairing on Monday Signed Rank of Difference A NO 1 B NO 2 C YES 6 D YES 3 E YES F YES G YES H YES I YES (a) A boxplot of the differences in number of items which required repairing on Monday and Wednesday for the 9 sampled weeks is shown below. -10-

11 Explain why management determined that the matched pair t-test of H0: μ difference = 0 Ha: μ difference > 0 (where μ difference is the mean of the differences in the number of produced items which required repairing on Monday and on Wednesday for all weeks in the past 5 years) was not appropriate after seeing the boxplot of the differences. A different possible set of hypotheses for this investigation could be H0: p = 0.5 Ha: p > 0.5 where p is the proportion of weeks where Monday had more produced items which required repairing than Wednesday. (b) Explain why the one-sample proportion z-test would not be appropriate for these data. -11-

12 A sign test of the hypotheses H0: p = 0.5 Ha: p > 0.5 can be used when the one-sample proportion z-test is not appropriate. The test statistic for the sign test is X = the number of weeks of the 9 sampled weeks where more items required repairing on Monday than Wednesday. (c) Assuming that the null hypothesis is true (that Mondays and Wednesdays are equally likely to have the most produced items which require repairing), calculate the p-value P(X 7) and use this p-value to provide the conclusion of the sign test for a significance level of αα = A signed rank test uses both the ranks of the absolute value of the differences and the signs of the differences to test the hypotheses H0: The distributions of the numbers of items which require repairing for Mondays and Wednesday are the same. Ha: The distribution of the numbers of items which require repairing for Mondays is shifted to the right of the distribution of the number of items which require repairing on Wednesdays. The test statistic for the signed rank test is the sum of the positive ranks. (d) Calculate the test statistic for the signed rank test by completing the signed rank of difference column in the table at the beginning of the problem and then adding up the positive ranks. -12-

13 Under the assumption that the null hypothesis of the distributions of the numbers of items requiring repairing for Mondays and Wednesday are the same, 1000 simulations were performed and the signed rank test statistic was calculated for each simulation. The frequency table below provide the frequencies for these 1000 simulated signed rank test statistics. Sign Rank Statistic Values Frequency (e) Based on the value of the signed rank test statistic calculated in part (d) and the distribution of the 1000 simulated signed rank test statistics above, what should be the conclusion for the manufacturing company for comparing the number of items requiring repairing on Mondays and Wednesdays? END OF EXAMINATION -13-