Practice AP Statistics Exam Saturday April 29, 2017 University of Delaware. Section II: Free Response

Transcription

1 Practice AP Statistics Exam Saturday April 29, 2017 University of Delaware Section II: Free Response Name: School: Instructions: 1. No electronic devices except an approved calculator are permitted, including cell phones and dictionaries. If you brought a cell phone or an electronic device other than an approved calculator, please turn it off and put it away. 2. Calculators may not be shared. For Scoring Use Only: Question Score DO NOT BEGIN UNTIL INSTRUCTED TO DO SO.

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9 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. Mercury exposure can present serious health risks. In particular, there is heightened concern about exposure in pregnant women. Partially due to bioaccumulation, where organisms have even higher levels than their surroundings, pregnant women are often told to avoid eating certain types of fish to avoid exposure to high levels of mercury. A study was conducted in Maine to measure the amount of mercury present in the state's lakes as part of a larger study on water quality. The graph below displays the distribution of mercury concentrations (in parts per million, ppm) in samples from 120 lakes in Maine. (a) Use the provided graphical display to describe the distribution of mercury levels in Maine's lakes. -1-

10 The researchers believed there could be a difference in lakes from the Southern and Northern parts of Maine. To investigate, they constructed parallel boxplots for the Northern (n = 18) and Southern (n = 102) lakes. (b) Use the boxplots shown above to comment on how the distribution of mercury levels in the Northern lakes is different from the distribution of mercury levels in the Southern lakes. (c) Based on the boxplots shown above, how would you expect the mean mercury level to compare for the two different regions? Explain. -2-

11 2. The South Carolina Department of Natural Resources (SCDNR) is interested in conducting a study to determine which type of nesting box will result in higher fledgling rates for Wood Ducks, the traditional wooden boxes they place around the state or a new plastic nesting box. The SCDNR has decided to use a river system where they have been placing nesting boxes for many years in this study. This river system, as can be seen in the figure below, has four sections: the upper lake and dam, the upper tailrace section of river, the lower lake and dam, and the lower tailrace section of river. A tailrace is a flowing section of river controlled by a dam. UPPER LAKE LOWER LAKE UPPER TAILRACE LOWER TAILRACE Each of the four sections of the river system has 10 nesting sites where ducks have successfully nested in wooden boxes in the past, with no significant difference in success between the upper and lower regions, for a total of 40 nesting box sites. To conduct their study, SCDNR has decided to randomly assign 20 wooden nesting boxes and 20 plastic boxes to the 40 sites along the river system. They are considering two blocking strategies, Plan 1 and Plan 2, pictured below. For both plans, SCDNR will randomly assign 10 wooden nesting boxes and 10 plastic nesting boxes to sites throughout Block A. They will follow the same process for Block B. -3-

12 (a) Which of the blocking schemes, 1 or 2, is better for this experiment? Explain your answer. (b) Once the blocking scheme is selected, SCDNR will have to randomly assign the wooden and plastic boxes to the nesting sites within each block. What is the purpose of randomization in the context of this experiment? -4-

13 3. In Hogwarts School of Witchcraft and Wizardry, the Slytherin dormitory has long been suspected of hosting illegal after-hours gambling games, even though the caretaker Mr. Filch and his cantankerous cat Mrs. Norris have failed to catch anyone in the act. One evening Slytherin students Malfoy and Crabbe decide to develop a new game of chance involving two dice. Die A has three 4's and three 7's on its faces. Die B has four 2's and two 12's on its faces. When either of these dice is rolled, each face has an equal chance of landing on top. Malfoy selects one of the dice and rolls it. Crabbe then rolls the remaining die. The winner is the player whose die has the highest number on top. (a) List the four possible outcomes of playing one round of this game along with the letter of the winning die (A or B) and the associated probabilities of each of the four outcomes. Show your work in the table below. Outcome Winning Die Probability (b) Malfoy, who never likes to lose, needs to know which die he should select. Calculate the probability of winning if Malfoy chooses die A or die B first. Based on these calculations, which die should Malfoy select? (c) Suppose now that the player using die A receives 30 galleons (wizarding gold currency) each time he wins the game. How many galleons must the player using die B receive each time he wins in order for this to be a fair game? Slytherin students are short-tempered and quickly resort to fist fights if games are not fair! [Hint: A fair game is one in which the expected gains for all players are equal.] -5-

14 4. Chantix is a prescription pill used to help smokers who want to quit. However, some doctors are concerned about the potential negative side effects of the drug including nausea, abnormal dreams, insomnia, and headache. A study of the safety and effectiveness of Chantix included 1,493 smokers wanting to quit who volunteered to participate in a clinical trial of the drug. Of the 751 smokers randomly assigned to take Chantix over a 24 week period, 28 experienced serious side effects. Of the 742 smokers randomly assigned to take a placebo over the same period, 16 experienced serious side effects. At the 5% significance level, do the data provide convincing statistical evidence that taking Chantix would increase the risk of serious side effects among all smokers similar to the volunteers? -6-

15 5. Visual memory has been defined as a specific type of recollection based on what you have seen. Two students in an AP Statistics class decided to undertake a project to determine if one gender tended to do better on visual memory tasks than the other. To carry out their study, they took a random sample of sixteen females and sixteen males from their large high school and had each student play an online memory game in which they had to match pairs of pictures. The mean score for the female students was 48.7 points and the mean score for the male students was 38.4 points. A graph of the scores on this game for each gender is given below. (a) Was this an experiment or an observational study? Justify your response. (b) What condition for a two-sample t-test for a difference in means has not been met? -7-

16 (c) The students decided to simulate the distribution of xx female xx male. Under the assumption that the null hypothesis is true, 100 values of xx female xx male were simulated. The dotplot below shows the results of the simulation. Based on the results of the simulation, is there evidence of a significant difference between the mean memory score of male and female students at this high school at a 5% significance level? -8-

17 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. Monthly record snowfalls data for the months November, December, January, February and March are provided in the following table for the cities of Buffalo, Denver and St. Paul. MONTHLY RECORD SNOWFALLS IN INCHES Nov Dec Jan Feb Mar Buffalo Denver St. Paul (a) On the grid below, construct a graphical display that compares the monthly record snowfalls for the three cities and that also shows how the record snowfalls vary by months. (b) Describe similarities and differences among the cities for the record snowfalls over the 5 months. -9-

18 (c) Complete the table by replacing the monthly record snowfalls by the ranks 1, 2, 3 (the lowest record snowfall is assigned the rank 1, the middle record snowfall is assigned the rank 2, and the highest record snowfall is assigned the rank 3) for the three cities for the month of March and for each city calculate and record the mean rank of the five months in the last column of the table. RANKED MONTHLY RECORD SNOWFALLS FOR THE THREE CITIES Nov Dec Jan Feb Mar Mean Rank Buffalo Denver St. Paul If there are no significant differences in the distributions of monthly record snowfalls among the three cities then the expected mean rank for each of the three cities would be 2. To test the hypotheses H0: no differences in the distributions of monthly record snowfalls among the three cities Ha: there are differences in the distributions of monthly record snowfalls among the three cities, the following test statistic can be used: Q = 5[(RBuffalo. 2) 2 + (RDenver. 2) 2 + (RStPaul. 2) 2 ] where RBuffalo., RDenver., and RStPaul. are the mean ranks calculated in the last column of the above table. (d) Would you reject the null hypothesis of no differences in the distributions of monthly record snowfalls among the three cities for large or small values of the test statistic Q? (e) Calculate the test statistic Q for this data. -10-

19 One thousand simulations of the test statistic Q were calculated assuming no differences in the distributions of monthly record snowfalls among the three cities, and the frequencies of Q are given in the following table. Tally for Simulated Values of the Test Statistic Q Assuming No Differences in the Distribution of Record Snowfalls among the Three Cities Values of Q Frequency Cumulative Frequency Relative Frequency Cumulative Relative Frequency (f) Use this table and the calculated value of Q in part (d) to provide the conclusion for the test of the null hypothesis of no differences in the distributions of monthly record snowfalls among the three cities assuming that the conditions for inference have been met. END OF EXAMINATION -11-