2021 AP® Statistics Free-Response Questions © 2021 College Board. College Board, Advanced Placement, AP, AP Central, and the acorn logo are registered trademarks of College Board. Visit College Board on the web: collegeboard.org. AP Central is the official online home for the AP Program: apcentral.collegeboard.org. Formulas for AP Statistics I. Descriptive Statistics 2 1 ∑ x 1 2 ∑( x − x ) x = ∑ x = i s = ∑( x − x ) = i ni n x n − 1 i n − 1 yˆ = a + bx y = a + bx 1 xi − x yi − y sy r = ∑ b = r n − 1 sx sy sx II. Probability and Distributions PA( ∩ B) PA( ∪ B )()()(= PA+ PB − PA ∩ B ) PAB()| = PB() Probability Distribution Mean Standard Deviation µ = E( X) = ∑ xP x 2 Discrete random variable, X X i ( i ) σ = ∑ − µ X ( xi X ) Px( i ) If has a binomial distribution µ = np σ = − with parameters n and p, then: X X np(1 p) n x nx− PX( = x) = p (1 − p) x where x = 0, 1, 2, 3, , n If has a geometric distribution 1 1 − p with parameter p, then: µ = σ = X p X PX( = x) = (1 − p) x−1 p p where x = 1, 2, 3, III. Sampling Distributions and Inferential Statistics statistic − parameter Standardized test statistic: standard error of the statistic Confidence interval: statistic ± (critical value )( standard error of statistic ) (observed − expected)2 Chi-square statistic: χ 2 = ∑ expected III. Sampling Distributions and Inferential Statistics (continued) Sampling distributions for proportions: Random Parameters of Standard Error* Variable Sampling Distribution of Sample Statistic For one population: µ = p p(1 − p) pˆ (1 − pˆ ) pˆ σ pˆ = s pˆ = pˆ n n For two pˆ (1 − pˆ ) pˆ (1 − pˆ ) = 1 1 + 2 2 populations: s ˆ − ˆ µ = − p (1 − p ) p (1 − p ) p1 p2 ˆ − ˆ p p 1 1 2 2 n1 n2 ˆ − ˆ p1 p2 1 2 σ = + p1 p2 pˆ − pˆ 1 2 n1 n2 = When p1 p2 is assumed: = ˆ − ˆ 1 + 1 s pˆ − pˆ pc (1 pc ) 1 2 n1 n2 X + X where pˆ = 1 2 c + n1 n2 Sampling distributions for means: Random Standard Error* Parameters of Sampling Distribution Variable of Sample Statistic For one σ s population: µ = µ σ = s = X X n X n X For two µ = µ − µ 2 2 2 2 populations: X − X 1 2 σ σ s s 1 2 σ = 1 + 2 s = 1 + 2 X − X X − X X − X 1 2 1 2 n1 n2 1 2 n1 n2 Sampling distributions for simple linear regression: Random Standard Error* Parameters of Sampling Distribution Variable of Sample Statistic = s sb , σ sn− 1 For slope: σ = x b , b σ n 2 x ∑ − ˆ = βµ ( yi yi ) b where s = ∑ − 2 n − 2 ( xi x ) where σ = 2 x n ∑( x − x ) and s = i x n − 1 *Standard deviation is a measurement of variability from the theoretical population. Standard error is the estimate of the standard deviation. If the standard deviation of the statistic is assumed to be known, then the standard deviation should be used instead of the standard error. AP® Statistics 2021 Free-Response Questions Begin your response to QUESTION 1 on this page. STATISTICS SECTION II Total T ime—1 hour and 30 minutes 6 Questions Part A Questions 1-5 Spend about 1 hour and 5 minutes on this part of the exam. 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. The length of stay in a hospital after receiving a particular treatment is of interest to the patient, the hospital, and insurance providers. Of particular interest are unusually short or long lengths of stay. A random sample of 50 patients who received the treatment was selected, and the length of stay, in number of days, was recorded for each patient. The results are summarized in the following table and are shown in the dotplot. Length of stay (days) 5 6 7 8 9 12 21 Number of patients 4 13 14 11 6 1 (a) Determine the five-number summary of the distribution of length of stay. GO ON TO THE NEXT PAGE. Use a pencil or pen with black or dark blue ink only. Do NOT write your name. Do NOT write outside the box. © 2021 College Board. Visit College Board on the web: collegeboard.org. 4 166 V OFFSET Rear Bklt 8.625x11 AP® Statistics 2021 Free-Response Questions Continue your response to QUESTION 1 on this page. (b) Consider two rules for identifying outliers, method A and method B. Let method A represent the 1.5 ¥ IQR rule, and let method B represent the 2 standard deviations rule. (i) Using method A, determine any data points that are potential outliers in the distribution of length of stay. Justify your answer. (ii) The mean length of stay for the sample is 7.42 days with a standard deviation of 2.37 days. Using method B, determine any data points that are potential outliers in the distribution of length of stay. Justify your answer. (c) Explain why method A might identify more data points as potential outliers than method B for a distribution that is strongly skewed to the right. GO ON TO THE NEXT PAGE. Use a pencil or pen with black or dark blue ink only. Do NOT write your name. Do NOT write outside the box. © 2021 College Board. Visit College Board on the web: collegeboard.org. 5 166 V Front Bklt 8.625x11 AP® Statistics 2021 Free-Response Questions Begin your response to QUESTION 2 on this page. 2. Researchers will conduct a year-long investigation of walking and cholesterol levels in adults. They will select a random sample of 100 adults from the target population to participate as subjects in the study. (a) One aspect of the study is to record the number of miles each subject walks per day. The researchers are deciding whether to have subjects wear an activity tracker to record the data or to have subjects keep a daily journal of the miles they walk each day. Describe what bias could be introduced by keeping the daily journal instead of wearing the activity tracker. GO ON TO THE NEXT PAGE. Use a pencil or pen with black or dark blue ink only. Do NOT write your name. Do NOT write outside the box. © 2021 College Board. Visit College Board on the web: collegeboard.org. 6 166 V OFFSET Rear Bklt 8.625x11 AP® Statistics 2021 Free-Response Questions Continue your response to QUESTION 2 on this page. During the course of the study, the subjects will have their cholesterol levels measured each month by a doctor. The researchers will perform a significance test at the end of the study to determine whether the average cholesterol level for subjects who walk fewer miles each day is greater than for those who walk more miles each day. (b) Selecting a random sample creates a reasonable representative sample of the target population. Explain the benefit of using a representative sample from the population. (c) Suppose the researchers conduct the test and find a statistically significant result. Would it be valid to claim that increased walking causes a decrease in average cholesterol levels for adults in the target population? Explain your reasoning. GO ON TO THE NEXT PAGE. Use a pencil or pen with black or dark blue ink only. Do NOT write your name. Do NOT write outside the box. © 2021 College Board. Visit College Board on the web: collegeboard.org. 7 166 V Front Bklt 8.625x11 AP® Statistics 2021 Free-Response Questions Begin your response to QUESTION 3 on this page. 3. To increase morale among employees, a company began a program in which one employee is randomly selected each week to receive a gift card. Each of the company’s 200 employees is equally likely to be selected each week, and the same employee could be selected more than once. Each week’s selection is independent from every other week. (a) Consider the probability that a particular employee receives at least one gift card in a 52-week year. (i) Define the random variable of interest and state how the random variable is distributed. (ii) Determine the probability that a particular employee receives at least one gift card in a 52-week year. Show your work. GO ON TO THE NEXT PAGE. Use a pencil or pen with black or dark blue ink only. Do NOT write your name. Do NOT write outside the box. © 2021 College Board. Visit College Board on the web: collegeboard.org. 8 166 V OFFSET Rear Bklt 8.625x11 AP® Statistics 2021 Free-Response Questions Continue your response to QUESTION 3 on this page. (b) Calculate and interpret the expected value for the number of gift cards a particular employee will receive in a 52-week year. Show your work. (c) Suppose that Agatha, an employee at the company, never receives a gift card for an entire 52-week year. Based on her experience, does Agatha have a strong argument that the selection process was not truly random? Explain your answer. GO ON TO THE NEXT PAGE. Use a pencil or pen with black or dark blue ink only. Do NOT write your name. Do NOT write outside the box. © 2021 College Board. Visit College Board on the web: collegeboard.org. 9 166 V Front Bklt 8.625x11 AP® Statistics 2021 Free-Response Questions Begin your response to QUESTION 4 on this page.