AP STATISTICS Test Booklet
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1. 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.
A team of psychologists studied the concept of visualization in basketball, where players visualize making a
basket before shooting the ball. They conducted an experiment in which 20 basketball players with similar
abilities were randomly assigned to two groups. The 10 players in group 1 received visualization training, and the
10 players in group 2 did not.
Each player stood 22 feet from the basket at the same location on the basketball court. Each player was then
instructed to attempt to make the basket until two consecutive baskets were made. The players who received
visualization training were instructed to use visualization techniques before attempting to make the basket. The
total number of attempts, including the last two attempts, were recorded for each player.
The total number of attempts for each of the 20 players are summarized in the following boxplots.
(a) Based on the boxplots, did basketball players who received visualization training tend to need fewer attempts
to make two consecutive baskets from a distance of 22 feet than players who did not receive the training? Explain
your reasoning.
(b) State and check conditions for conducting a two-sample -test for a difference in means.
Because both distributions, visualization and no visualization , are skewed, the psychologists
conducted a simulation to test for a difference in medians rather than means. For each trial of the simulation, the
20 values of the total number of attempts observed in the experiment were combined into one group and then
randomly split into two groups of 10. The difference in the medians of the groups was calculated for
each trial. The following dotplot shows the difference in the medians for 100 trials of the simulation.
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(c) Using the observed difference in medians and the results of the simulation, estimate a -value for
a test for the difference in medians. Show the work needed to calculate this -value.
(d) Based on the -value in part (c), is there convincing statistical evidence that basketball players similar to the
ones in this study who receive visualization training need fewer attempts to make two consecutive baskets from a
distance of 22 feet than those who do not receive such training? Justify your answer.
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2. 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.
A consumer group compared the longevity of two types of razor blades: an inexpensive generic blade and a more
expensive name-brand blade. Two independent samples were randomly selected—one consisted of ten generic
blades and the other consisted of ten name-brand blades. The number of shaves obtained before a blade became
too dull to provide a close shave was recorded for each blade in the two independent samples.
a. Create a back-to-back stemplot to display the distributions of the data.
b. In a few sentences, compare the distributions in part (a).
c. To test whether the name-brand blades tend to last longer than the generic blades on average, a two-
sample t-test was performed. The conditions for inference were checked and verified. The test statistic
was t=0.87 with a p-value of 0.20. What should be concluded about whether the name-brand blades last
significantly longer than the generic blades? Explain.
Also of interest to the consumer group is whether the generic (G) blades have more variability in the
number of close shaves than the name-brand (B) blades. The consumer group intends to use the ratio of
the sample standard deviations , as a test statistic to test the hypothesis verses
.
The group conducted a simulation study to investigate the sampling distribution of the test statistic when
the population standard deviations were actually the same. For each repetition of the simulation, two
independent samples of size 10 were selected from the same normally distributed population, and the
ratio of their sample standard deviations was computed.
d. Explain why the median of the distribution of a sample of 1,000 of these simulated ratios will be about 1.
e. A histogram for the 1,000 simulated values of the ratio is shown below.
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Summary statistics for the original data (number of close shaves) are provided in the table below.
Is there convincing evidence that the generic blades have more variability in the number of close shaves than the
name-brand blades? Explain.
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3. 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.
A city council is considering funding a proposal to create a new city park. The council members will fund the
proposal if they conclude that more than 60 percent of the city residents support the proposal. A survey of 2,000
randomly selected city residents will be conducted to investigate the level of support for the proposal. Let X
represent the number of city residents in the sample who support the proposal. Assume that X is a binomial
random variable.
a. Determine the mean and the standard deviation of the random variable X, assuming that 60 percent of city
residents support the proposal.
The figure below shows a normal distribution that can be used to approximate the binomial probability
distribution of X. The shaded region is bounded by k1 and k2 on the figure and represents the middle 0.997 of the
area under the curve.
b. Assume that 60 percent of the city residents support the proposal. Use a normal approximation and the mean
and standard deviation from part (a) to determine the values of k1 and k2 .
k1 =
k2 =
c. The survey was conducted, and 1,293 of the 2,000 city residents surveyed supported the proposal. Do your
answers in part (b) and the survey results support the funding of the proposal? Justify your answer.
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4. 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.
The graph below shows the monthly average temperatures, in degrees Fahrenheit, for two
cities–Madison,Wisconsin, and Juneau, Alaska–in the United States.
a. Based on the graph, compare the two cities with respect to the monthly average temperatures over the
year. Address both similarities and differences in the overall patterns.
Similarities:
Differences:
b. For which of the two cities is the standard deviation of the 12 monthly average temperatures greater?
Justify your answer without performing any calculations.
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5. 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.
A university researcher is interested in comparing the percents of high school seniors who took a college entrance
exam for two different regions of the country, Region I and Region II. The researcher recorded the percent of
seniors taking the exam for each high school within the two regions. The boxplots of the distributions of the
percents of seniors who took the college entrance exam are shown below.
a. Compare the distributions of percents of seniors who took the college entrance exam for the two regions.
b. In writing a report, the researcher produced a single histogram of the combined data for Region I and
Region II. Describe the shape of the histogram for the combined data.
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6. 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.
The following histogram shows the relative frequencies of the heights, recorded to the nearest inch, of a
population of women. The mean of the population is inches, and the standard deviation is inches.
One woman from the population will be selected at random.
(a) Based on the histogram, what is the probability that the selected woman will have a height of at least
inches? Show your work.
(b) What is the area of the bar corresponding to a height of inches in the graph, and what does the area
represent in terms of probability?
(c) The histogram displays a discrete probability model for height. However, height is often considered a
continuous variable that follows a normal model. Consider a normal model that uses the mean and standard
deviation of the population of women as its parameters.
(i) Use the normal model and the relationship between area and relative frequency to find the probability that the
randomly selected woman will have a height of at least inches. Show your work.
(ii) Does your answer in part (c-i) match your answer in part (a) ? If not, give a reason for why the answers might
be different.
(d) Let the random variable represent the height of a woman in the population. represents the
probability of randomly selecting a woman with height less than inches. Based on the information given, the
probability can be found using either the discrete model or the normal model.
(i) Give an example of a probability of that can be found using the discrete model but not the normal model.
Explain why.
(ii) Give an example of a probability of that can be found using the normal model but not the discrete model.
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Explain why.
7. River Run campground has sites for people to use for camping. The sites can be reserved for a certain number of
days. To help with cleaning and maintenance, the campground requests an exit time (the time at which campers
leave the site) of 9 A.M. on the last day of the reservation.
To estimate the typical exit time, the manager of River Run selected a random sample of 60 sites. Of the selected
sites, 40 were reserved by people without young children, and 20 were reserved by people with young children.
The following histograms summarize the exit times, recorded as minutes relative to 9 A.M. For example, an exit
time of 9:30 A.M. is 30 minutes relative to an exit time of 9 A.M. Each interval contains possible values from the
left endpoint up to but not including the right endpoint.
(a) Consider the two histograms.
(i) How many of the 60 sites had an exit time before 8:30 A.M.?
(ii) How many of the 60 sites had an exit time of 11:00 A.M. or later?
(b) Compare the distributions of the exit times for those without young children and those with young children.
(c) Based on the histograms, what is a reasonable estimate of the median exit time for the random sample of 60
sites? Explain your reasoning.
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8. Population pyramids are a type of bar chart that show the distribution of ages of a country’s population. The
distributions of ages of men and women for two countries, A and B, for the year 2015 are shown in the population
pyramids below. The age-groups, in years, are listed in the center columns, and the proportions are shown on the
horizontal axes. Each bar represents the proportion of the age-group in the population for that sex.
a. Is the proportion of the female population age 60 or older in Country A greater than the proportion of the
female population age 60 or older in Country B? Justify your answer.
b. One of the two countries experienced an increase in the birth rate in the years 1946 to 1955 and another
increase in the birth rate about 20 years later. Based on the graphs, which country experienced the
described increases in birth rate? Justify your answer using information from the graphs.
c. For Country A, in which age-group is the median age of the male population? Justify your answer.
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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.
The students enrolled in honors biology at a high school were given the task of using a spreadsheet program to investigate
a topic in genetics. All students in the class had similar background knowledge of the topic. Some students in the class had
no spreadsheet experience, Group R, and needed time to learn the program to complete the task. The rest of the students,
Group S, had previous spreadsheet experience and typically took less time to complete the task. Each of the histograms
below show the distribution of completion times, in minutes, for one of the two groups.
9. (a) Of the two histograms shown, I and II, which is more likely to be the distribution of completion times for the
students in Group R? Justify your answer.
(b) Describe the shape of a histogram created from the data of the two groups of students combined.
(c) Consider the population of all students in honors biology classes in the high school’s state who are given the
task of using the spreadsheet program to investigate the topic in genetics. The distribution of the completion times
has a shape similar to the combined histogram of students at the high school, with mean 70 minutes and standard
deviation 26.5 minutes. For random samples of 50 students taken from the population, describe the sampling
distribution of the sample mean completion time.
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10. 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.
The following frequency table summarizes the total amounts, in dollars, for 91 orders from a food truck during a
certain day.
Amount Frequency
$0 up to but not including $5 8
$5 up to but not including $10 14
$10 up to but not including $15 25
$15 up to but not including $20 27
$20 up to but not including $25 12
$25 up to but not including $30 5
(a-i) Use the data in the table to create a histogram showing the distribution of the amounts of the orders.
(a-ii) Describe the shape of the distribution of amounts.
(b) Identify a possible amount for the median of the distribution. Justify your answer.
11. 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.
Kathy and her brother Clay recently ran in a local marathon. The distribution of finishing time for women was
approximately normal with mean 259 minutes and standard deviation 32 minutes. The distribution of finishing
time for men was approximately normal with mean 242 minutes and standard deviation 29 minutes.
(a) The finishing time for Clay was 289 minutes. Calculate and interpret the standardized score for Clay’s
marathon time. Show your work.
(b) The finishing time for Kathy was 272 minutes. What proportion of women who ran the marathon had a
finishing time less than Kathy’s? Show your work.
(c) The standard deviation of finishing time is greater for women than for men. What does this indicate about the
finishing times of the women who ran the marathon compared to the finishing times of the men who ran the
marathon?
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12. 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.
The distribution of levels for all community swimming pools in a large county is approximately normal with
mean 7.5 and standard deviation 0.2. According to swimming pool studies, the safest levels for water in
swimming pools are between 7.2 and 7.8.
(a) One community swimming pool in the county will be selected at random. What is the probability that the
selected pool has a level that is not considered safe?
The county health inspector will select a random sample of 4 community swimming pools in the county to
investigate the levels.
(b) Describe the sampling distribution of the sample mean for samples of size 4.
(c) Consider the situation in which the health inspector finds the sample mean of the 4 pools to be outside the safe
levels. As a result, the inspector declares that the population mean is not 7.5. However, if the population mean
really is 7.5, the inspector will have made an error. Such an error is called a Type error. Find the probability that
the inspector will make a Type error with the sample of 4 pools. Show your work.
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