Unit 1 test

AP Statistics Unit 1 Test

15 multiple-choice questions and 1 free-response question, every one from Unit 1: Exploring One-Variable Data and Collecting Data, written in the format of the AP Statistics Exam.

Take the multiple choice on its own, timed at the exam's pace or in practice mode with each answer checked as you give it. Or sit the whole test, free response included, with each section on its own clock; the free-response scoring guide opens when that section ends.

AP Statistics Unit 1 Test

15 multiple-choice and 1 free-response question, a shorter set laid out like the AP Statistics Exam. Each section runs on its own clock and ends when time runs out.

Section	Questions	Time
Section I: Multiple Choice	15 of the exam's 42	32 minutes 9 seconds
Section II: Free Response	1 of the exam's 4	22 minutes 30 seconds

A graphing calculator with statistical capabilities is expected on both sections.

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Multiple choice is scored when its section ends, with an explanation for every choice.
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At the end you get your score and the time you used for each section.
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Time for each section Standard time Time and a half Double time Untimed Study mode

Choose before Section I starts. Time and a half and double time match common AP testing accommodations; untimed shows the time you use without ending a section. Study mode is untimed and shows each answer as you go.

Section I: Multiple Choice

15 questions · 32 minutes 9 seconds

Before collecting any data, a county fair board wrote this investigative question: "What proportion of the county's households attended the county fair last summer?" The board then surveyed a random sample of 600 households in the county. In the results, households with children had attended at a much higher rate than households without children. A board member suggests replacing the investigative question with "Do households with children attend the fair at a higher rate than households without children?" and reporting only the answer to that question. Which response is best?
Answer: D.
1. A
  An investigative question has a defined purpose that is fixed before the data are collected. Rewriting it to fit a pattern the data happened to show turns a planned question into one chosen after the fact, and the study was not designed to answer it.
2. B
  A comparison is not a better question in itself. The board's purpose was to estimate the share of households that attended, and a question about two groups does not serve that purpose. Swapping questions after seeing the results is the problem either way.
3. C
  Sample size is not the issue. A random sample of 600 households will include plenty of households with children. The objection is to changing the question because of the results, and a bigger sample would not fix that.
4. D
  Correct. The board wrote its investigative question before collecting data, for a defined purpose: the proportion of the county's households that attended. An investigative question should not be changed based on the results. The difference between households with and without children can be reported as something the data showed, and it could become the question for a new study planned to answer it.
For each car it services, a tire shop records three variables: the tread depth of the front left tire, measured to the nearest 0.1 millimeter; the number of tires replaced, from 0 to 4; and the number of the service bay, 1 through 6, where the work was done. Which statement is correct?
Answer: C.
1. A
  Rounding to the nearest 0.1 millimeter is a limit of the gauge, not of the variable. Tread depth is a measured quantity that can take any value in an interval, so it is continuous.
2. B
  A bay number is a label that happens to be written with digits. An average bay number would mean nothing, so service bay is categorical.
3. C
  Correct. The number of tires replaced is a count. It takes whole-number values from 0 to 4 that can be listed one by one, which makes it a discrete quantitative variable.
4. D
  Having only a few possible values does not make a variable categorical. The values 0 to 4 are counts of tires, and the differences between them mean something (3 tires is one more than 2), so the variable is quantitative.
A dental office asked each of the 160 patients who joined the practice last year how they first heard about it. The answers were friend or family 56, insurance directory 48, online search 40 and mailer 16. Which list gives the relative frequencies of these four answers, in the order listed?
Answer: A.
1. A
  Correct. Each relative frequency is the count divided by the total of 160: $56 / 160 = 0.35$ , $48 / 160 = 0.30$ , $40 / 160 = 0.25$ and $16 / 160 = 0.10$ . They add to 1, as the relative frequencies of all the categories must.
2. B
  These are cumulative proportions: each one adds in the categories before it. A relative frequency is the proportion in one category alone, such as $48 / 160 = 0.30$ for the insurance directory.
3. C
  These treat each count as a percent, which works only when the total is 100. Here the total is 160, so each count is divided by 160. These four values add to 1.60, which no set of relative frequencies can.
4. D
  These compare each count with the largest count, 56. A relative frequency divides by the total number of patients, 160, not by the count in the most common category.
An airline recorded the main cause of every delayed departure last month at two airports it serves. The counts are shown in the table, and a bar chart of the counts for each airport is drawn from it.
$\begin{matrix} Weather & Crew & Maintenance & Air traffic control & Total \\ North & 80 & 120 & 100 & 100 & 400 \\ South & 56 & 40 & 24 & 40 & 160 \end{matrix}$
Which claim do the data support?
Answer: B.
1. A
  North's air traffic control bar is taller because North had far more delays in all, 400 against 160. As shares, $100 / 400 = 0.25$ and $40 / 160 = 0.25$ , so air traffic control caused the same share of delays at both airports. Comparing counts between groups of different sizes is the mistake.
2. B
  Correct. Weather caused $56 / 160 = 0.35$ of South's delays and $80 / 400 = 0.20$ of North's. South's weather bar is shorter only because South had fewer delays in all. A claim comparing two groups of different sizes has to compare shares, not counts.
3. C
  56 is a count, not a percent. South had 160 delays in all, so weather caused $56 / 160 = 0.35$ of them, or 35%. Reading a count off a bar chart as a percent works only when the total is 100.
4. D
  Crew problems were the most common cause at North, 120 of 400. At South, weather was the most common cause, 56 of 160, ahead of crew problems and air traffic control at 40 each. The pattern at one airport does not carry over to the other.
A food pantry recorded the ages of its 22 volunteers:

16, 18, 19, 20, 20, 23, 27, 30, 31, 34, 38, 40, 40, 41, 45, 52, 55, 60, 63, 67, 68, 71

A histogram of the ages uses bins 10 years wide starting at 10, and each bin includes its left endpoint. Which list gives the heights of the histogram's bars, from left to right?
Answer: B.
1. A
  This counts each different age once, so the two volunteers aged 20 and the two aged 40 are counted as one each. A bar's height is the number of volunteers in its bin, repeated values included.
2. B
  Correct. With each bin including its left endpoint, the bins hold ages 10 to 19 (16, 18, 19), 20 to 29 (20, 20, 23, 27), 30 to 39 (30, 31, 34, 38), 40 to 49 (40, 40, 41, 45), 50 to 59 (52, 55), 60 to 69 (60, 63, 67, 68) and 70 to 79 (71). The heights add to 22, one for each volunteer.
3. C
  These are cumulative counts, each one adding all the bins before it. A histogram's bar shows only the volunteers inside its own bin, so its last bar is 1, not 22.
4. D
  This puts each age that falls on a boundary, such as 20, 30, 40 and 60, in the bin to its left. Each bin includes its left endpoint, so a volunteer aged 20 belongs in the bin that starts at 20.
A retired teachers' association asked 50 of its members the age at which they retired. A histogram of the ages has bins 5 years wide, with these counts.
$\begin{matrix} Age at retirement & Members \\ 45 to 49 & 1 \\ 50 to 54 & 2 \\ 55 to 59 & 6 \\ 60 to 64 & 15 \\ 65 to 69 & 20 \\ 70 to 74 & 6 \end{matrix}$
Which describes the shape of the distribution?
Answer: D.
1. A
  Skew is named for the longer tail, not for where most of the values are. The bars trail off toward the younger ages, so the tail is on the left and the distribution is skewed left.
2. B
  There is one peak, but the two sides are not mirror images. Left of the tallest bar the counts fall over four bins (15, 6, 2, 1); to its right they drop to 6 in one bin and stop. One peak alone does not make a distribution symmetric.
3. C
  Two neighboring tall bars are one peak spread over two bins, not two peaks. A bimodal distribution has two separate humps with lower bars between them, and here the counts rise steadily to 20 and then fall.
4. D
  Correct. The counts rise slowly from 1 member at 45 to 49 to a peak of 20 at 65 to 69, then drop to 6 at 70 to 74. The long tail stretches toward the younger retirement ages, so the distribution is skewed left.
A zoo has 25 reptile enclosures. A keeper recorded the temperature of each enclosure at noon in degrees Fahrenheit. The temperatures have a mean of 86.0, a standard deviation of 2.7 and an interquartile range (IQR) of 3.6. To share the data with a partner zoo, she converts every temperature to degrees Celsius using $C = (F - 32) / 1.8$ . What are the mean, standard deviation and IQR of the temperatures in degrees Celsius?
Answer: A.
1. A
  Correct. Every temperature has 32 subtracted and is then divided by 1.8, and the mean follows both steps: $(86.0 - 32) / 1.8 = 30.0$ . Subtracting 32 moves every value by the same amount and leaves the spread alone, so the standard deviation and IQR are only divided by 1.8: $2.7 / 1.8 = 1.5$ and $3.6 / 1.8 = 2.0$ .
2. B
  The IQR changes with the units just as the standard deviation does. Being resistant means the IQR is not pulled around by outliers; it still shrinks when every value is divided by 1.8, to $3.6 / 1.8 = 2.0$ .
3. C
  Subtracting 32 does not change the spread, but dividing by 1.8 does. Every distance between two temperatures shrinks by a factor of 1.8, so the standard deviation becomes 1.5 and the IQR becomes 2.0.
4. D
  47.8 is $86.0 / 1.8$ , which leaves out subtracting 32. The mean goes through the whole conversion: $(86.0 - 32) / 1.8 = 30.0$ .
A bakery recorded the number of loaves of rye bread left unsold at closing on each of 11 days:

4, 6, 7, 7, 8, 9, 10, 11, 12, 13, 15

When the data were typed into a spreadsheet, the 15 was entered as 51 by mistake. Which summary statistics for the data as entered differ from those for the correct data?
Answer: C.
1. A
  The median and IQR do not change. The median is still the 6th of the 11 ordered values, 9, and the quartiles are still 7 and 12, because 51 sits at the top of the ordered list just as 15 did. Only the statistics that use the size of every value change.
2. B
  The standard deviation measures how far the values fall from the mean, and 51 is much farther from the mean than 15 is. It jumps from about 3.29 to about 13.03, so it differs too.
3. C
  Correct. The mean rises from about 9.27 to about 12.55 and the standard deviation from about 3.29 to about 13.03, because both use the size of every value. The median stays 9 and the quartiles stay 7 and 12, so the IQR stays 5: 51 still sits at the top of the ordered list, just as 15 did.
4. D
  This has resistance backwards. The median and IQR depend only on where values fall in order, and changing the largest value to an even larger one does not change the order. The mean and standard deviation use every value's size, so they are the ones that change.
A teacher recorded the number of pages in each of the 60 novels on her school's summer reading list. The five-number summary is shown, and no novel is an outlier by the $1.5 \times IQR$ rule.
$\begin{matrix} Min & Q_{1} & Median & Q_{3} & Max \\ 150 & 220 & 265 & 300 & 410 \end{matrix}$
Which statement does a boxplot of these data support?
Answer: D.
1. A
  $Q_{3}$ is the value with about 75% of the data at or below it, so only about 25% of the novels have more than 300 pages.
2. B
  Each whisker covers about a quarter of the novels, whatever its length, so the boxplot gives no reason to think the top quarter holds more novels than the bottom quarter. The right whisker is longer because the top quarter is spread over a wider span of page counts, from 300 to 410.
3. C
  80 pages is the interquartile range, $300 - 220$ . The range is the maximum minus the minimum, $410 - 150 = 260$ pages.
4. D
  Correct. $Q_{1} = 220$ and $Q_{3} = 300$ mark off the middle half of the data, so about half of the 60 novels have from 220 to 300 pages. That middle half is what the box shows.
Ten customers at each of two restaurants rated the same new dessert on a scale from 1 to 9. Dotplots of the ratings are drawn from the counts in the table. No customer gave a rating of 1 or 9.
$\begin{matrix} Rating & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ Restaurant P & 5 & 0 & 0 & 0 & 0 & 0 & 5 \\ Restaurant Q & 1 & 0 & 2 & 4 & 2 & 0 & 1 \end{matrix}$
Both sets of ratings have a mean of 5. Which restaurant's ratings have the larger standard deviation, and why?
Answer: C.
1. A
  The number of different values is not what the standard deviation measures. Restaurant Q's ratings take five different values, but most sit at or next to 5. Restaurant P's take only two values, and every one of them is 3 points from the mean.
2. B
  The same mean and the same range do not make the same spread. The range uses only the two most extreme values. Every one of Restaurant P's ratings is at an extreme, while only two of Restaurant Q's are, so P's ratings vary more about the mean.
3. C
  Correct. Every one of Restaurant P's ratings is 3 points from the mean of 5. Eight of Restaurant Q's ten are within 1 point of 5, and only two are 3 points away. So P's typical distance from the mean is larger: its standard deviation is about 3.16, against about 1.56 for Q.
4. D
  Uneven stack heights describe how the counts vary, not how the ratings vary. The standard deviation measures how far the ratings fall from their mean, and Restaurant P's ratings are all far from 5 even though its two stacks are the same height.
A public health researcher enrolls 400 adults who work downtown in a large city. When they enroll, each adult reports whether they usually bike to work. Every six months for the next three years, the researcher measures each adult's blood pressure, and then she compares the adults who bike to work with those who do not. Which describes this study?
Answer: A.
1. A
  Correct. The researcher does not assign a way of getting to work; each adult already chose one, so this is an observational study. It selects the adults now and gathers data from that point into the future, which makes it prospective.
2. B
  A retrospective study selects its units now and gathers data from the past. Here the blood pressure readings are collected over the next three years, so the study is prospective, even though the adults were biking or not before it began.
3. C
  Careful, scheduled measurement does not make a study an experiment. An experiment imposes the treatments, and here the adults chose for themselves whether to bike to work.
4. D
  A census records data on every member of a population. These 400 adults are a sample of the people who work downtown, not all of them.
An airline wants to survey the passengers who flew with it on one Monday. It numbers the 220 flights it operated that day, uses a random number generator to select 12 of the flights, and surveys every passenger on those 12 flights. Which sampling method did the airline use?
Answer: B.
1. A
  A stratified sample selects units at random from within every group. Here 12 whole flights were chosen, and the other 208 flights contribute no passengers at all.
2. B
  Correct. The flights are the clusters. The airline took a simple random sample of 12 flights and then surveyed every passenger on each of them, which is a cluster random sample.
3. C
  In a simple random sample of passengers, every set of passengers of the same size has the same chance of being chosen. Here passengers come into the sample a whole flight at a time, so a sample with passengers from 200 different flights could never be chosen.
4. D
  A systematic sample starts at a random point and takes units at a fixed interval, such as every 20th passenger on a list. The airline chose its flights with a random number generator, with no fixed interval.
A city council is considering closing four blocks of Main Street to cars on weekends. It posted a link to an online form on the city's website and social media pages, and 2,400 residents filled it out; 71% of them supported the closure. The council also phoned a random sample of 300 adult residents, and 291 of them answered; 46% of those supported the closure. Which is the best explanation for the difference between the two results?
Answer: A.
1. A
  Correct. The online form is a voluntary response sample: the residents chose to take part. People who care most about the closure, here mostly those who want it, are the ones who take the time to respond, so 71% very likely overstates support among all adult residents. The phone sample was chosen at random, and 97% of it answered.
2. B
  A large sample does not fix a biased method. The 2,400 forms came from volunteers, and more of them repeat the same bias. The smaller random sample is the more trustworthy of the two.
3. C
  Nine nonrespondents out of 300 cannot explain a 25-point gap. Even if all 9 had supported the closure, the phone result would be only about 48%. Nonresponse is worth checking in any survey, but it is small here.
4. D
  Chance variation between samples of these sizes is a few percentage points. A 25-point gap points to a systematic difference in how the two samples were collected, which is bias, not chance.
A dog trainer wants to compare three methods for teaching puppies to come when called. She has 36 puppies: 12 of small breeds, 12 of medium breeds and 12 of large breeds. Within each size group, she randomly assigns 4 puppies to each method. After four weeks, she records how many of 10 calls each puppy obeys. Which describes this design?
Answer: D.
1. A
  In a completely randomized design, the methods are assigned to all 36 puppies at once, with no grouping first. Here the trainer grouped the puppies by size and randomized separately within each group, so size is a blocking variable.
2. B
  A matched pairs design compares two treatments within pairs of similar units. Here three methods are compared within groups of 12 puppies, which is a randomized block design with more than two treatments.
3. C
  A stratified random sample is a way of selecting units from a population. The trainer did not select puppies from a larger population; she assigned methods to the 36 she had, which makes this an experiment, and the size groups are blocks.
4. D
  Correct. The puppies were first grouped by a variable that could affect how quickly they learn, their size, and then the three methods were assigned at random within each group. That is a randomized block design with size as the blocking variable.
A software company offers a 30-day free trial of its photo-editing app. For the next 2,000 people who start a trial, the app will randomly assign each person to see either a new video tutorial or the current written guide when they first open it. The company will compare the percent of each group who buy the app at the end of the trial. A manager asks what the random assignment accomplishes in this study. Which is the best answer?
Answer: B.
1. A
  That is what random selection does, not random assignment. These 2,000 people are simply the next ones to start a trial, not a random sample of everyone who might download the app, so the results apply to people like them.
2. B
  Correct. Random assignment spreads other differences, such as editing experience and how serious each person is about buying, roughly evenly across the two groups. Then if one group buys at a clearly higher rate, the tutorial is the explanation that is left, which is what allows a cause-and-effect conclusion.
3. C
  Random assignment makes the groups similar, not identical. Even if the tutorial had no effect at all, chance alone would give the two groups somewhat different buying rates.
4. D
  That describes blocking. Random assignment does not make each group more alike inside; it makes the two groups alike as a whole, so that they differ mainly in the treatment.

AP Statistics Unit 1 Test

AP Statistics Unit 1 Test

Section I: Multiple Choice

Section II: Free Response

Question 1: Multi-Focus on Practices 1 and 2 10 points

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