Learn Extracted exam questions AP Statistics 2021 Free Response
2021 Free Response
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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 | 1 |
[Dotplot titled "Length of Stay (days)"; x-axis labeled Length of Stay (days), tick marks at 5, 7, 9, 11, 13, 15, 17, 19, 21. Stacked dots above each value show the counts from the table: 4 dots at 5, 13 dots at 6, 14 dots at 7, 11 dots at 8, 6 dots at 9, 1 dot at 12, 1 dot at 21 (no dots shown at 6, 10, 11, or 13-20 other than the isolated points at 12 and 21).]
Determine the five-number summary of the distribution of length of stay.
Consider two rules for identifying outliers, method A and method B. Let method A represent the $1.5 \times \text{IQR}$ rule, and let method B represent the 2 standard deviations rule.
Using method A, determine any data points that are potential outliers in the distribution of length of stay. Justify your answer.
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.
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.
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.
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.
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.
Selecting a random sample creates a reasonable representative sample of the target population. Explain the benefit of using a representative sample from the population.
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.
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.
Consider the probability that a particular employee receives at least one gift card in a 52-week year.
Define the random variable of interest and state how the random variable is distributed.
Determine the probability that a particular employee receives at least one gift card in a 52-week year. Show your work.
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.
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.
The manager of a large company that sells pet supplies online wants to increase sales by encouraging repeat purchases. The manager believes that if past customers are offered $10 off their next purchase, more than 40 percent of them will place an order. To investigate the belief, 90 customers who placed an order in the past year are selected at random. Each of the selected customers is sent an e-mail with a coupon for$10 off the next purchase if the order is placed within 30 days. Of those who receive the coupon, 38 place an order.
Is there convincing statistical evidence, at the significance level of $\alpha = 0.05$, that the manager's belief is correct? Complete the appropriate inference procedure to support your answer.
Based on your conclusion from part (a), which of the two errors, Type I or Type II, could have been made? Interpret the consequence of the error in context.
A research center conducted a national survey about teenage behavior. Teens were asked whether they had consumed a soft drink in the past week. The following table shows the counts for three independent random samples from major cities.
| Baltimore | Detroit | San Diego | Total | |
|---|---|---|---|---|
| Yes | 727 | 1,232 | 1,482 | 3,441 |
| No | 177 | 431 | 798 | 1,406 |
| Total | 904 | 1,663 | 2,280 | 4,847 |
Suppose one teen is randomly selected from each city's sample. A researcher claims that the likelihood of selecting a teen from Baltimore who consumed a soft drink in the past week is less than the likelihood of selecting a teen from either one of the other cities who consumed a soft drink in the past week because Baltimore has the least number of teens who consumed a soft drink. Is the researcher's claim correct? Explain your answer.
Consider the values in the table.
Construct a segmented bar chart of relative frequencies based on the information in the table.
[Blank segmented-bar-chart template with three horizontal bars labeled Baltimore, Detroit, and San Diego; x-axis labeled "Relative Frequency of Response" ranging from 0 to 1.0 in increments of 0.1. Each bar is drawn as a single unsegmented rectangle spanning the full 0 to 1.0 width, left for the student to divide into segments representing "Yes" and "No" relative frequencies.]
Which city had the smallest proportion of teens who consumed a soft drink in the previous week? Determine the value of the proportion.
Consider the inference procedure that is appropriate for investigating whether there is a difference among the three cities in the proportion of all teens who consumed a soft drink in the past week.
Identify the appropriate inference procedure.
Identify the hypotheses of the test.
Attendance at games for a certain baseball team is being investigated by the team owner. The following boxplots summarize the attendance, measured as average number of attendees per game, for 47 years of the team's existence. The boxplots include the 30 years of games played in the old stadium and the 17 years played in the new stadium.
[Two vertical boxplots side by side, y-axis "Average Attendance (thousands)" from 10 to 28 in increments of 2. "Old Stadium" boxplot: whisker from about 11.5 to 20, box (Q1 to Q3) from about 14 to 17.5, median line at about 16. "New Stadium" boxplot: whisker from about 16 to 27.5, box (Q1 to Q3) from about 22 to 26.5, median line at about 25.]
Compare the distributions of average attendance between the old and new stadiums.
The following scatterplot shows average attendance versus year.
[Scatterplot titled with y-axis "Average Attendance (thousands)" from 10 to 28 in increments of 2, x-axis "Year" from 1970 to 2020 in increments of 10 (gridlines at 1970, 1980, 1990, 2000, 2010, 2020). Legend distinguishes open squares = Old Stadium and filled circles = New Stadium. Old Stadium square points span roughly 1970-2000, scattered between about 11 and 20 with no strong trend (values fluctuate between roughly 13-20 throughout). New Stadium circle points span roughly 2000-2018, showing a clear upward trend from about 16-21 in the early 2000s rising steadily to about 26-27.5 by the late 2010s.]
Compare the trends in average attendance over time between the old and new stadium.
Consider the following scatterplots.
[Graph I: scatterplot titled "Graph I", y-axis "Average Attendance (thousands)" (label shown on Graph II) from 10 to 28 in increments of 2, x-axis "Number of Games Won" from 75 to 105 in increments of 5. All points plotted as plain dots (stadium not distinguished) showing a positive, fairly linear association: points cluster from about (75,11-15) at the low end rising to about (100-103,26-28) at the high end.]
[Graph II: scatterplot titled "Graph II", same axes as Graph I ("Average Attendance (thousands)" from 10 to 28, "Number of Games Won" from 75 to 105). Same data as Graph I but now marked with open squares = Old Stadium and filled circles = New Stadium (legend shown). Old Stadium (square) points cluster in the lower-left region, roughly games won 75-86 and attendance 11-20. New Stadium (circle) points cluster in the upper-right region, roughly games won 75-103 and attendance 16-27.5, appearing to lie along a steeper positive trend than the Old Stadium points.]
Graph I shows the average attendance versus number of games won for each year. Describe the relationship between the variables.
Graph II shows the same information as Graph I, but also indicates the old and new stadiums. Does Graph II suggest that the rate at which attendance changes as number of games won increases is different in the new stadium compared to the old stadium? Explain your reasoning.
Consider the three variables: number of games won, year, and stadium. Based on the graphs, explain how one of those variables could be a confounding variable in the relationship between average attendance and the other variables.