1. You cannot be serious ?!? Question #5

2. Intent of Question The primary goals of this question were to assess a student’s ability to (1)calculate proportions from a two-way table; and (2) interpret a confidence interval for the difference between two proportions.

3. Question #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’sWomen’s Correct 15372 Incorrect 394231

4. Part (a) Calculate the proportion of all challenges that were determined to be correct. Part (a) solution

5. Part (a) Scoring Part (a) is scored as follows: Essentially correct (E) if the response correctly calculates the proportion with work shown. Partially correct (P) if the response provides the correct answer with no work shown. Incorrect (I) if the response does not meet the criteria for E or P. Note: Answers reported as fractions rather than decimals are acceptable and can be considered to have work shown.

6. Question #5 continued (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 ± 0.06. 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.

7. Part (b) solution

8. Part (b) scoring Part (b) is scored as follows: Essentially correct (E) if the response states that because the interval contains 0, it does not provide convincing statistical evidence that there is a difference in the effectiveness with which men and women use the challenge system. Partially correct (P) if the response indicates that it is necessary to check whether the value of 0 is in the computed interval, but there are errors in implementation. Examples of errors include: The response notes that 0 is within the interval but does not draw a conclusion. The response has an arithmetic error in the computation of the endpoints of the interval but provides a correct conclusion with justification that is consistent with the computed interval. Incorrect (I) if the response does not recognize how to use the confidence interval to determine whether there is a difference in men and women; OR if the response states that the interval shows that the difference in proportions is equal to 0; OR if the response notes that 0 is within the interval and concludes that there is a difference in proportions; OR if the response otherwise does not meet the criteria for E or P.

9. Question #5 continued 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 6111 Incorrect 111120

10. Part (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: Part (c) solution

11. Part (c) scoring Part (c) is scored as follows: Essentially correct (E) if the response correctly performs both calculations with work shown. Partially correct (P) if the response correctly performs one of the two calculations with work shown; OR if the response provides both correct answers with no work shown. Incorrect (I) if the response does not meet the criteria for E or P.

12. Question #5 continued Part (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 ± 0.09. All conditions for inference were met. Does this confidence interval provide statistical evidence to support the scientific findings detailed above? Justify your answer. Part (d) Solution Yes. The confidence interval is (0.18, 0.36). Because this interval is entirely above zero, this suggests that the proportion of correct challenges is higher when the umpire calls the ball ‘out’ than when the umpire calls the ball ‘in,’ and the confidence interval does provide statistical evidence to support the theory 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.’

13. Scoring part (d) Part (d) is scored as follows: Essentially correct (E) if the response concludes that there is statistical evidence to support the scientific findings because the interval is entirely above 0. Partially correct (P) if the response indicates that it is necessary to determine whether the interval is entirely above 0 but there are errors in implementation; OR if the response notes that 0 is not in the interval and concludes that there is statistical evidence of a difference in proportions, but does not address the direction of the relationship between the two proportions; OR if the response notes that the interval is entirely above 0 but confuses the order of subtraction and argues there is statistical evidence that the better time to challenge is when the ball is called 'in.' Incorrect (I) if no argument is made based on whether or not 0 is in the interval; OR if the response otherwise does not meet the criteria for E or P.

14. Scoring Each essentially correct (E) part counts as 1 point. Each partially correct (P) part counts as ½ point. 4Complete Response 3Substantial Response 2Developing Response 1Minimal Response If a response is between two scores (for example, 2½ points), score down.