Practice AP Statistics Exam Saturday May 2, 2015 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 1 2 3 4 5 6

Score

DO NOT BEGIN UNTIL INSTRUCTED TO DO SO.

STATISTICS SECTION II

Part A Questions 1-5

Spend about 65 minutes on this part of the exam. Percent of Section II grade ----75

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. A student interested in purchasing her first vehicle wanted to research the typical fuel efficiency of

some of the most popular vehicles. The following data show the city miles per gallon (mpg) of the twelve best-selling vehicles in the previous calendar year, as reported by one widely used car buying website that the student visited.

27 16 29 17 27 26 25 17 18 28 30 19 (a) Display these data in a dotplot.

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(b) Use your dotplot from part (a) to describe the main features of this city mpg distribution.

(c) Why would it be misleading for this student to use only a measure of center for this city mpg distribution as an indication of the typical fuel efficiency for popular vehicles?

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2. The remains of five concentric agricultural terraces set in circles of increasing depths can be found in Peru, at the ancient site of Moray in the Sacred Valley of the Incas. These terraces were used by the Incas to grow crops of varying species. The current local inhabitants wish to use them to compare the yields of four varieties of corn: Kulli black, Oaxacan green, Chullpi yellow, and Incan giant.

Due to their design, each terrace differs greatly in terms of soil type, irrigation level, and amount of sunlight. Each terrace has been divided into eight sections, resulting in 40 sections total. The diagram below is an overhead view of the five terraces.

To study yields, the inhabitants plan to assign the four corn varieties completely at random to one of the 40 sections while ensuring that each corn variety is represented the same number of times.

(a) A second way to design the experiment is to use blocking while still ensuring that each corn

variety is represented the same number of times within each block. Identify the factor to be used to create the blocks and justify your choice.

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(b) Describe a process by which to assign the corn varieties to the sections in the randomized complete block design.

(c) In the context of this situation, describe one statistical advantage of selecting a randomized complete block design as opposed to the completely randomized design.

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3. A triathlon is an athletic event that consists of swimming, cycling and running. At the Lake Tahoe Ironman Triathlon competitors swim for 2.4 miles, cycle for 112 miles, and run for 26.2 miles. Competitors are timed for each individual event and then receive an overall time which is the sum of the three different event times. The winner of the triathlon is the competitor with the lowest total time. The times for all competitors for each event at the Lake Tahoe Ironman Triathlon are approximately normally distributed. Their means and standard deviations, in minutes, are summarized in the table below.

Mean Standard Deviation

Swimming 80 23

Cycling 390 60

Running 300 46

(a) What is the probability that a randomly chosen competitor has a swim time less than 1 hour?

(b) How fast would a competitor need to run in order to be in the fastest 2.5% of runners?

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(c) On her last triathlon, Christine’s total time was 900 minutes. She would like to know how well she performed relative to the other competitors. She determines that the mean for the distribution of total times is equal to

𝜇𝜇T = 𝜇𝜇S + 𝜇𝜇C + 𝜇𝜇𝑅𝑅 = 770 minutes

and the standard deviation for the distribution of total times is equal to

𝜎𝜎T = �𝜎𝜎S2 + 𝜎𝜎C2 + 𝜎𝜎R2 = √6245 ≈ 79.03 minutes.

What assumption did Christine make in her calculations? Comment on the validity of this assumption.

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4. A psychologist conducted a study to investigate the theory that firstborn children are smarter than their younger siblings. The psychologist randomly selected 10 families, each family consisting of just two children who both attend a school within a local school district. An IQ test was administered to each of the two siblings in the 10 families. The results are presented in the table below. Family 1 2 3 4 5 6 7 8 9 10 IQ Score for Firstborn Sibling 125 112 130 113 115 88 83 103 107 93 IQ Score for Younger Sibling 123 99 125 99 106 82 85 96 107 94

Do the data provide convincing evidence that in this school district firstborn children have higher IQ scores, on average, than their younger siblings?

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5. The proficiency levels on a statewide End of Course Exam are shown in the table below for a random sample of high school students. The results are also classified by gender.

Proficiency Level Gender Below Basic Basic Proficient Advanced Total Male 2757 6552 9951 1734 20994 Female 5131 6878 8550 1083 21642 Total 7888 13430 18501 2817 42636

(a) If a student is to be selected at random, what is the probability that the student scored basic or above on this exam?

(b) If a female student is to be selected at random, what is the probability that she was at the advanced proficiency level?

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If a chi-square test of homogeneity were to be performed the hypotheses would be: H0: The proportions in each proficiency level category are the same for both genders. Ha: The proficiency level category proportions are not all the same for both genders.

The computer output below gives the results from performing this test. For each cell, the observed and expected counts are reported, as well as the contribution of each cell [(observed – expected)2/expected] to the chi-square statistic.

Below Basic Basic Proficient Advanced

Male Observed Count (Expected count) (Contributions to Chi-Square)

2757

(3884.06) (327.04)

6552

(6612.94) (0.56)

9951

(9109.91) (77.66)

1734

(1387.09) (86.76)

Female Observed Count (Expected count) (Contributions to Chi-Square)

5131

(4003.94) (317.25)

6878

(6817.06) (0.54)

8550

(9391.09) (75.33)

1083

(1429.91) (84.16)

Statistic DF Value P-value

Chi-square 3 969.31168 <0.0001

(c) Given the results of the test, is there statistically convincing evidence that the proficiency level proportions are not the same across genders? Justify your decision.

(d) The state's Department of Education would like to determine which one of these eight groups needs more tutoring assistance. Which group would you recommend? Use the contributions to chi-square results shown in the table above to justify your choice.

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STATISTICS SECTION II

Part B Question 6

Spend about 25 minutes on this part of the exam. Percent of Section II grade ----25

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. Each flu season, medical researchers estimate the effectiveness of the flu vaccine that was administered that season. At the end of the most recent flu season, 2321 adults were randomly selected to participate in the U.S. Influenza Vaccine Effectiveness Study. Each participant was classified by both whether or not they chose to receive the flu vaccine and whether or not they were diagnosed with the flu that flu season. The results are presented in the table below.

Flu No Flu Total Not Vaccinated 239 846 1085

Vaccinated 124 1112 1236 Total 363 1958 2321

(a) Is this study an experiment or an observational study? Explain your answer and discuss the implications this has for establishing a causal relationship between receiving the flu vaccine and contracting the flu.

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(b) The conditions for inference have been met. Construct and interpret a 95 percent confidence interval for the difference between the proportion of adults not receiving the vaccine who contract the flu and the proportion of adults receiving the vaccine who contract the flu this particular flu season.

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In many of these types of studies, researchers are interested in the ratio of the odds of contracting the flu for those not receiving the vaccine and those receiving the vaccine. This ratio is usually referred to as an odds ratio (OR), and is given by

OR =𝑝𝑝NV (1 − 𝑝𝑝NV)⁄𝑝𝑝V (1 − 𝑝𝑝V)⁄

where 𝑝𝑝NV represents the proportion of adults not receiving the vaccine who contract the flu and 𝑝𝑝V represents the proportion of adults receiving the vaccine who contract the flu. For example, an odds ratio of 1 indicates that the odds of contracting the flu are the same for adults who do not and who do receive the flu vaccine. Whereas, an odds ratio of 1.5 indicates that the odds of contracting the flu for adults not receiving the flu vaccine are 1.5 times the odds for adults receiving the vaccine. An estimator of the odds ratio is the sample odds ratio

OR� =�̂�𝑝NV (1 − �̂�𝑝NV)⁄�̂�𝑝V (1 − �̂�𝑝V)⁄ .

(c) Using the data from the U.S. Influenza Vaccine Effectiveness Study presented above, compute the estimate of the odds ratio.

The sampling distribution of OR� is skewed. However, when both sample sizes 𝑛𝑛NV and 𝑛𝑛V are relatively large, the distribution of 𝑙𝑙𝑛𝑛�OR��, the natural logarithm of the sample odds ratio, is approximately normal with a mean of 𝑙𝑙𝑛𝑛(OR) and an estimated standard error of

�1

𝑛𝑛NV,F+

1𝑛𝑛NV,NF

+1𝑛𝑛V,F

+1

𝑛𝑛V,NF

where 𝑛𝑛NV,F represents the number of adults in the sample that were not vaccinated and did contract the flu, 𝑛𝑛NV,NF represents the number of adults in the sample that were not vaccinated and did not contract the flu, 𝑛𝑛V,F represents the number of adults in the sample there were vaccinated and did contract the flu, and 𝑛𝑛V,NF represents the number of adults in the sample that were vaccinated and did not contract the flu. When a 95 percent confidence interval for 𝑙𝑙𝑛𝑛(OR) is known, an approximate 95 percent confidence interval for OR can be constructed by exponentiating (applying the inverse of the natural logarithm to) the endpoints of the confidence interval for 𝑙𝑙𝑛𝑛(OR).

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(d) The conditions for inference are met, and a 95 percent confidence interval for 𝑙𝑙𝑛𝑛(OR) based on the data from the study presented above is (0.69495, 1.16421). Construct and interpret a 95 percent confidence interval for the odds ratio of contracting the flu for those not receiving the vaccine to those receiving the vaccine.

(e) What is an advantage of using the interval in part (d) over using the interval in part (b)?

END OF EXAMINATION

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