AP Statistics Unit 1: Exploring One-Variable Data and Collecting Data

How to turn a question into data, how to display and describe one variable (categorical or quantitative), how to summarize it with resistant and non-resistant statistics, and how the way data are collected (sampling, observational study or experiment) decides what you can conclude.

Topics: 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 1.10, 1.11, 1.12, 1.13 · free response

Key formulas and facts

• Relative frequency of a category $=\dfrac{\text{count in the category}}{\text{total count}}$
• Sample mean $\bar x=\dfrac{\sum x_i}{n}$ and sample standard deviation $s_x=\sqrt{\dfrac{\sum (x_i-\bar x)^2}{n-1}}$
• Five-number summary: minimum, $Q_1$, median, $Q_3$, maximum, with $IQR=Q_3-Q_1$ (quartiles are the medians of the lower and upper halves)
• Outlier rule: a value below $Q_1-1.5\,IQR$ or above $Q_3+1.5\,IQR$
• z-score: $z=\dfrac{\text{value}-\text{mean}}{\text{standard deviation}}$
• Linear change $y=a+bx$: the mean, median and quartiles become $a+b(\cdot)$; the standard deviation, IQR and range are multiplied by $|b|$ and are not changed by $a$
• Scope of inference: random selection lets you generalize to the population; random assignment lets you conclude cause and effect

Worked examples

• Relative frequency: 18 of 120 surveyed students walk to school, so the relative frequency of walking is $\frac{18}{120}=0.15$.
• Mean and SD: for the data 3, 5, 7, 9, 11, $\bar x=7$, the squared deviations add to $40$, so $s_x=\sqrt{40/4}=\sqrt{10}\approx3.162$.
• Quartiles and outliers: for 4, 6, 7, 8, 9, 10, 12, 25, the median is $8.5$, $Q_1=6.5$ and $Q_3=11$, so $IQR=4.5$. The upper fence is $11+1.5(4.5)=17.75$, so 25 is an outlier.
• z-score: a 1,350 on a test with mean 1,050 and standard deviation 200 has $z=\frac{1350-1050}{200}=1.5$, so it is 1.5 standard deviations above the mean.
• Linear change: lengths with mean 12 inches and standard deviation 2 inches are converted to centimeters ($\times 2.54$). The new mean is $30.48$ cm and the new standard deviation is $5.08$ cm.
• Treatments: an experiment with 2 levels of light and 3 levels of water has $2\times3=6$ treatments; with 48 plants, 8 plants per treatment.
• Proportional stratified sample: a school of 1,200 has 300 seniors. In a stratified sample of 80 with proportional allocation, $80\cdot\frac{300}{1200}=20$ are seniors.

Common traps

• Describing a distribution without context or without all four parts: shape, center, variability and unusual features (outliers, gaps, clusters).
• Saying "the data are skewed toward the high values" for a left-skewed distribution: skew is named for the side of the long tail.
• Using the mean and standard deviation for a strongly skewed distribution or one with outliers: the median and IQR are resistant and describe it better.
• Calling any random sample a simple random sample: stratified, cluster and systematic samples are random but are not SRSs.
• Thinking a larger sample fixes bias: more data from a biased method only gives a more precise wrong answer.
• Claiming cause and effect from an observational study, or generalizing an experiment's results to a population that was not randomly sampled.
• Confusing blocking (grouping similar units before random assignment) with stratifying (grouping before random selection).

Practice questions

104 multiple-choice questions (4 options each, as on the new exam) and 2 free-response questions worth 10 points each. A calculator is allowed throughout. Try each one before opening the answer. Each line of a scoring guide is one point, scored 0 or 1 and earned independently of the other lines, the style of the new course's 10-point free-response scoring (no E/P/I part scores). A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Topic 1.1: What data can tell us: statistical questions and variability

8 multiple-choice questions.

Question 1 · Multiple choice

Which of the following is a statistical question?

1. How many hours are in a school week at Lakeview High?
2. What time does the first bell ring at Lakeview High?
3. How many hours of sleep do students at Lakeview High typically get on a school night?
4. How many hours of sleep did Maya get last night?

Question 2 · Multiple choice

Which of the following is NOT a statistical question?

1. What is the posted speed limit on Route 9 near the county fairgrounds?
2. Do weekend drivers on Route 9 tend to travel faster than weekday drivers?
3. How much does the daily traffic count on Route 9 vary over a month?
4. How fast do cars usually travel on Route 9 near the county fairgrounds?

Question 3 · Multiple choice

A city parks department wants to answer the question "How long do visitors typically stay at Cedar Point Park on a Saturday?" Which data would best answer it?

1. The length of the main walking trail at the park
2. The arrival and departure times of a random sample of Saturday visitors
3. The park's opening and closing times on Saturdays
4. The number of parking spaces at Cedar Point Park

Question 4 · Multiple choice

A teacher records the number of minutes each of her 28 students spent on a homework assignment. Why do statisticians say that data like these show variability?

1. There are 28 students, which is a small sample.
2. The values differ from one student to another.
3. The average time is larger than the shortest time.
4. The teacher may have recorded some times incorrectly.

Question 5 · Multiple choice

A random sample of 400 registered voters in Harlan County found that 58% support a new library tax. Which conclusion is best supported by these data?

1. Exactly 58% of all registered voters in Harlan County support the tax.
2. About 58% of all voters in the state support the tax.
3. Supporting the tax causes voters to use the library more.
4. Somewhere near 58% of all registered voters in Harlan County support the tax.

Question 6 · Multiple choice

A coach wants to know whether players on her soccer team who stretch before practice have fewer injuries. Which variable must she record for each player to answer this question?

1. The average length of practice
2. Only the total number of injuries on the team
3. Whether the player stretches before practice and the number of injuries the player has
4. Only the number of players who stretch

Question 7 · Multiple choice

Researchers asked 150 volunteers who answered an online ad how many cups of coffee they drink per day. Which statement about what these data can tell us is correct?

1. They show that coffee affects health.
2. They give the exact mean coffee consumption for all adults.
3. They describe the 150 volunteers but may not represent any larger population.
4. They cannot be summarized because the values vary.

Question 8 · Multiple choice

Which question could be answered by a statistical investigation that compares two groups?

1. How many miles is it from Elm Street to the office?
2. What is the maximum score on the stress questionnaire?
3. Do commuters who bike to work report lower stress scores than commuters who drive?
4. What was the stress score of the first commuter surveyed?

Topic 1.2: Variables: categorical and quantitative

8 multiple-choice questions.

Question 9 · Multiple choice

A school nurse records each student's grade level (9, 10, 11 or 12), height in centimeters, blood type and number of absences this year. Which of these variables are categorical?

1. Height and number of absences
2. Grade level and blood type
3. Blood type only
4. Grade level, blood type and number of absences

Question 10 · Multiple choice

Which of the following variables is quantitative and discrete?

1. The number of text messages a student sends in a day
2. The time it takes a student to run a mile
3. A student's favorite music genre
4. The weight of a student's backpack

Question 11 · Multiple choice

A data set has one row for each of 60 used cars for sale and the columns Make, Model year, Mileage and Color. What are the individuals in this data set?

1. The 60 used cars
2. Make, Model year, Mileage and Color
3. The people who will buy the cars
4. The four columns

Question 12 · Multiple choice

Which variable is continuous?

1. The number of rainy days in a city during April
2. Whether it rained on April 1
3. The amount of rainfall in a city during April, in millimeters
4. The month with the most rain

Question 13 · Multiple choice

A survey asks: "How many pets live in your home?" and "What kind of pet is your favorite?" Which statement is correct?

1. Both variables are categorical.
2. The first variable is quantitative and the second is categorical.
3. The first variable is categorical and the second is quantitative.
4. Both variables are quantitative.

Question 14 · Multiple choice

Zip codes such as 30301 and 60614 are numbers. How should a statistician treat zip code as a variable?

1. As quantitative, because it is recorded with digits
2. As continuous, because zip codes can be any number
3. As discrete quantitative, because zip codes are whole numbers
4. As categorical, because arithmetic on zip codes has no meaning

Question 15 · Multiple choice

A researcher records each tomato plant's variety, the number of tomatoes it produces and the total weight of its tomatoes in grams. How many of these variables are quantitative?

1. None
2. Three
3. Two
4. One

Question 16 · Multiple choice

A gym member rates each class as "poor," "fair," "good" or "excellent." What type of variable is the rating?

1. Neither categorical nor quantitative
2. Quantitative and continuous
3. Categorical
4. Quantitative and discrete

Topic 1.3: Frequency tables and relative frequencies for one categorical variable

8 multiple-choice questions.

Question 17 · Multiple choice

A random sample of students at Riverside High was asked how they usually get to school. The results are in the table. What proportion of the students in the sample usually walk?

1. 0.187
2. 0.230
3. 0.431
4. 0.813

Question 18 · Multiple choice

Use the same table of how Riverside High students get to school. What proportion of the students do NOT usually travel by car?

1. 0.433
2. 0.567
3. 0.750
4. 1.308

Question 19 · Multiple choice

The relative frequency table shows the favorite season of a sample of adults. What relative frequency belongs in place of the question mark?

1. 0.25
2. 0.39
3. 0.49
4. 0.61

Question 20 · Multiple choice

In a survey of 400 households, the relative frequency of households whose main heating fuel is propane is 0.235. How many of the surveyed households heat mainly with propane?

1. 17
2. 24
3. 85
4. 94

Question 21 · Multiple choice

The table shows the genres of books checked out of a middle school library on one Monday. Which genre has a relative frequency closest to 0.18?

1. Science
2. Other
3. Biography
4. History

Question 22 · Multiple choice

Using the same library table, what proportion of the books checked out were Fiction or Science?

1. 0.227
2. 0.387
3. 0.613
4. 0.657

Question 23 · Multiple choice

A random sample of 260 college students was asked which brand of phone they use. The relative frequencies are shown. How many more students in the sample use Brand A than Brand B?

1. 15
2. 39
3. 81
4. 120

Question 24 · Multiple choice

Twenty shoppers were asked whether they would use a self-checkout lane (Y = yes, N = no, U = unsure). The responses were: Y, N, N, Y, U, Y, N, Y, Y, U, N, Y, N, Y, Y, N, Y, U, Y, N. What proportion of the shoppers answered yes?

1. 0.15
2. 0.35
3. 0.50
4. 0.65

Topic 1.4: Bar charts and pie charts for one categorical variable

8 multiple-choice questions.

Question 25 · Multiple choice

The bar chart shows the primary language spoken at home by 200 randomly selected residents of a city. What proportion of the residents speak Spanish at home?

1. 0.23
2. 0.30
3. 0.39
4. 0.46

Question 26 · Multiple choice

Using the same bar chart of primary languages, how many more residents speak English at home than Spanish?

1. 46
2. 72
3. 82
4. 96

Question 27 · Multiple choice

The relative frequency bar chart shows the blood types of the 300 donors at a blood drive. How many donors had type B blood?

1. 13
2. 15
3. 39
4. 114

Question 28 · Multiple choice

Using the same blood-type chart, what proportion of donors had a blood type other than O or A?

1. 0.05
2. 0.13
3. 0.18
4. 0.82

Question 29 · Multiple choice

A news site showed a bar chart of the number of complaints about two airlines. The vertical axis started at 900, so Airline X's bar (980 complaints) looked four times as tall as Airline Y's bar (920 complaints). Why is the chart misleading?

1. Bar charts should never be used for counts.
2. The truncated vertical axis exaggerates a small difference in counts.
3. The categories should have been listed in alphabetical order.
4. The bars should have been drawn touching each other.

Question 30 · Multiple choice

A survey asked students to choose all the after-school activities they take part in. The percents for the activities add to 143%. Which display is NOT appropriate for these data?

1. A pie chart
2. A dotplot of the number of activities each student chose
3. A table of the number choosing each activity
4. A bar chart of the percent choosing each activity

Question 31 · Multiple choice

Which feature is the main difference between a bar chart and a histogram?

1. A bar chart shows counts, and a histogram shows only percents.
2. A histogram can only be used with fewer than 20 values.
3. A bar chart has a vertical axis, and a histogram does not.
4. A bar chart displays a categorical variable, and a histogram displays a quantitative variable.

Question 32 · Multiple choice

The bar chart shows the pizza topping chosen by each of 80 customers at a pizzeria. What proportion of the customers chose pepperoni?

1. 0.3100
2. 0.3875
3. 0.4900
4. 0.6200

Topic 1.5: Dotplots, stemplots and histograms for one quantitative variable

8 multiple-choice questions.

Question 33 · Multiple choice

The dotplot shows the number of books read over the summer by each of 20 members of a library reading club. How many members read more than 7 books?

1. 2
2. 5
3. 7
4. 15

Question 34 · Multiple choice

Using the same dotplot of books read, what proportion of the 20 members read 4 or fewer books?

1. 0.15
2. 0.25
3. 0.30
4. 0.50

Question 35 · Multiple choice

The histogram shows the one-way commute times of 50 employees at a company. What proportion of the employees have a commute of at least 50 minutes?

1. 0.08
2. 0.14
3. 0.22
4. 0.44

Question 36 · Multiple choice

Using the same histogram of commute times, which interval contains the median commute time?

1. 20 to 30 minutes
2. 30 to 40 minutes
3. 40 to 50 minutes
4. 10 to 20 minutes

Question 37 · Multiple choice

How many employees are represented in the commute-time histogram altogether? (Use the bar heights.)

1. 6
2. 15
3. 50
4. 60

Question 38 · Multiple choice

The stemplot shows the scores of 18 students on a 75-point quiz (key: 4 | 1 means 41 points). What is the range of the scores?

Key: 4 | 1 means 41 points

1. 4
2. 18
3. 38
4. 70

Question 39 · Multiple choice

Using the same stemplot of quiz scores, how many students scored in the 50s?

Key: 4 | 1 means 41 points

1. 3
2. 5
3. 6
4. 9

Question 40 · Multiple choice

A researcher has the resting heart rates of 2,400 adults. Which display is most practical for showing the shape of the distribution?

1. A histogram
2. A dotplot with one dot per adult
3. A stemplot with one leaf per adult
4. A pie chart

Topic 1.6: Describing a distribution: shape, center, variability, unusual features

8 multiple-choice questions.

Question 41 · Multiple choice

The dotplot shows the number of streaming services used by each of 16 households. Which best describes the shape of the distribution?

1. Skewed to the right
2. Uniform
3. Skewed to the left
4. Roughly symmetric

Question 42 · Multiple choice

The histogram shows the scores of 47 students on an easy final exam. Which best describes the shape?

1. Skewed to the left
2. Bimodal
3. Roughly symmetric
4. Skewed to the right

Question 43 · Multiple choice

The numbers of minutes 12 customers waited for a table at a restaurant were: 12, 14, 15, 15, 16, 17, 17, 18, 19, 20, 21, 34. Using the 1.5 × IQR rule, which of these values is an outlier?

1. 34 only
2. 12 only
3. There are no outliers
4. Both 12 and 34

Question 44 · Multiple choice

The prices (in dollars) of 13 board games at a store are: 21, 22, 22, 23, 24, 24, 25, 26, 28, 31, 35, 42, 58. How do the mean and median compare?

1. The mean and median are equal.
2. They cannot be compared without a graph.
3. The mean is less than the median.
4. The mean is greater than the median.

Question 45 · Multiple choice

The dotplot shows the heights of 20 adults. Which best describes the shape of the distribution?

1. Roughly symmetric and single-peaked
2. Skewed to the left
3. Skewed to the right
4. Bimodal

Question 46 · Multiple choice

A histogram of the weights of the dogs at an animal shelter has one cluster near 10 pounds and another near 60 pounds, with few dogs in between. Which is the best description of the shape?

1. Skewed to the right, because the mean is above 10 pounds
2. Roughly symmetric, because there are two clusters
3. Bimodal, possibly because small and large breeds are mixed together
4. Uniform, because the weights cover a wide range

Question 47 · Multiple choice

The numbers of minutes 14 students spent on a reading assignment were: 5, 31, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 66, 70. How many of these values are outliers by the 1.5 × IQR rule?

1. 0
2. 1
3. 2
4. 3

Question 48 · Multiple choice

Which description of a distribution of daily high temperatures in July is complete for an AP Statistics answer?

1. The data are spread out and the graph is normal.
2. The mean is 88 and the standard deviation is 4.
3. It is skewed with a high center and some variability.
4. The distribution of July high temperatures is roughly symmetric, centered near 88°F, with temperatures from about 79°F to 97°F and no obvious outliers.

Topic 1.7: Summary statistics for one quantitative variable

8 multiple-choice questions.

Question 49 · Multiple choice

The numbers of push-ups done by 8 students in one minute were 12, 15, 9, 18, 14, 11, 16 and 13. What is the mean number of push-ups?

1. 12.0
2. 13.5
3. 14.0
4. 15.0

Question 50 · Multiple choice

The numbers of emails 9 office workers received before noon were 7, 22, 13, 9, 30, 15, 11, 18 and 26. What is the median?

1. 13.0
2. 15.0
3. 16.8
4. 30.0

Question 51 · Multiple choice

A small bakery sold 4, 8, 6, 5 and 7 loaves of rye bread on five days. What is the sample standard deviation $s_x$ of these daily sales?

1. 1.000
2. 1.414
3. 1.581
4. 2.500

Question 52 · Multiple choice

The numbers of hours 10 teenagers spent gaming last week were 3, 5, 6, 8, 9, 10, 12, 15, 17 and 20. What is the interquartile range?

1. 6
2. 9
3. 10
4. 17

Question 53 · Multiple choice

The temperatures recorded at a weather station one week have a mean of 20°C and a standard deviation of 5°C. The temperatures are converted to Fahrenheit using $F=1.8C+32$. What is the mean of the temperatures in °F?

1. 36
2. 41
3. 52
4. 68

Question 54 · Multiple choice

Using the same weather-station temperatures (mean 20°C, standard deviation 5°C), what is the standard deviation in °F after converting with $F=1.8C+32$?

1. 5.0
2. 6.8
3. 9.0
4. 41.0

Question 55 · Multiple choice

A data set of house prices in a town includes one mansion priced far above every other house. Which pair of statistics is resistant to that extreme value?

1. The mean and the range
2. The median and the standard deviation
3. The median and the interquartile range
4. The mean and the standard deviation

Question 56 · Multiple choice

A student's score on a placement test is at the 80th percentile. In a group of 350 students who took the test, about how many scored at or below this student?

1. 70
2. 80
3. 270
4. 280

Topic 1.8: Boxplots, outliers and z-scores

8 multiple-choice questions.

Question 57 · Multiple choice

A sample of 11 used bicycles listed online had ages (in years) of 2, 5, 7, 8, 9, 11, 12, 14, 15, 18 and 31. What is the upper fence for outliers using the 1.5 × IQR rule?

1. 21.8
2. 25.5
3. 27.0
4. 31.0

Question 58 · Multiple choice

On a statewide exam, the mean score was 72 with standard deviation 8. Jalen scored 86. What is Jalen's z-score?

1. 0.57
2. 1.75
3. 10.75
4. 14.00

Question 59 · Multiple choice

The heights of the players in a youth league have mean 152 cm and standard deviation 6 cm. A player's height has a z-score of −1.5. What is the player's height in centimeters?

1. 143
2. 146
3. 150
4. 161

Question 60 · Multiple choice

The boxplot summarizes the battery life of 40 phones in a lab test. What is the interquartile range of battery life?

1. 7
2. 12
3. 17
4. 25

Question 61 · Multiple choice

Using the same boxplot of phone battery life, about what proportion of the 40 phones lasted longer than 21 hours?

1. 0.21
2. 0.25
3. 0.50
4. 0.75

Question 62 · Multiple choice

Using the same boxplot, what is the smallest battery life (in hours) that would be an outlier on the high side by the 1.5 × IQR rule? Give the value of the upper fence.

1. 26.0
2. 28.0
3. 31.5
4. 34.0

Question 63 · Multiple choice

What is the five-number summary of these 9 delivery times (minutes): 8, 10, 11, 13, 14, 16, 17, 19, 25?

1. Min 8, $Q_1$ 12, median 14.8, $Q_3$ 18, max 25
2. Min 8, $Q_1$ 10, median 13.5, $Q_3$ 19, max 25
3. Min 8, $Q_1$ 10.5, median 14, $Q_3$ 18, max 25
4. Min 8, $Q_1$ 11, median 14, $Q_3$ 17, max 25

Question 64 · Multiple choice

Ana scored 41 on a history test (class mean 35, standard deviation 4) and 78 on a math test (class mean 70, standard deviation 6). On which test did Ana do better relative to her class?

1. History, because her z-score is higher
2. Math, because 78 is greater than 41
3. They are equally good
4. Math, because her z-score is higher

Topic 1.9: Comparing distributions of a quantitative variable

8 multiple-choice questions.

Question 65 · Multiple choice

The boxplots compare the times for students to solve a puzzle while listening to music or in silence. By how many minutes does the median time for the music group exceed the median time for the silence group?

1. 1
2. 2
3. 3
4. 5

Question 66 · Multiple choice

Using the same boxplots, which group's times have the larger interquartile range?

1. Silence, because it has an outlier
2. They have the same IQR
3. Silence, with IQR 7 minutes
4. Music, with IQR 7 minutes

Question 67 · Multiple choice

Which statement correctly compares the two distributions in the puzzle boxplots?

1. The music group took longer because music causes slower thinking.
2. The silence times have a larger range than the music times.
3. Every student in the music group took longer than every student in the silence group.
4. The median time for the music group (13 minutes) is greater than the median for the silence group (10 minutes), and the music times are more variable.

Question 68 · Multiple choice

Quiz scores for two sections of a course are shown. Which section has the larger standard deviation of scores?

1. Section A
2. They are equal
3. It cannot be determined from the data
4. Section B

Question 69 · Multiple choice

Using the same quiz scores for Sections A and B, what is the difference in mean scores, $\bar x_B-\bar x_A$?

1. -1.500
2. 2.250
3. 2.500
4. 4.000

Question 70 · Multiple choice

Two histograms of weekly exercise hours are drawn for samples from Town P and Town Q. The Town Q histogram has the same shape as the Town P histogram shown, but every bar is shifted 2 hours to the right. Which comparison is correct?

1. Town Q's center is about 2 hours higher, with about the same variability.
2. Town Q's distribution is more variable.
3. Town Q's center is about 2 hours lower.
4. Town Q's distribution is skewed to the right.

Question 71 · Multiple choice

The table shows the miles per gallon recorded for 8 cars of each of two brands. What is the difference in medians, Brand X minus Brand Y?

1. 0.00
2. 1.00
3. 3.50
4. 4.75

Question 72 · Multiple choice

In the brand data above, Brand X has one car at 50 mpg. Which measure of center gives the fairest comparison of typical mileage for the two brands?

1. The mean, because it uses every value
2. The maximum, because it shows the best car
3. The range, because it shows the spread
4. The median, because it is resistant to the 50 mpg value

Topic 1.10: From investigative question to data collection: observational studies and experiments

8 multiple-choice questions.

Question 73 · Multiple choice

Researchers recorded the daily screen time and the sleep quality score of 500 randomly selected teenagers. Which type of study is this?

1. An observational study, because no treatment was imposed
2. An experiment, because the teenagers were randomly selected
3. A census, because many teenagers were included
4. An experiment, because two variables were measured

Question 74 · Multiple choice

A farmer randomly assigns 24 plots to receive either a new fertilizer or the old one and then measures the yield of each plot. Which type of study is this?

1. A sample survey
2. An experiment
3. An observational study
4. A census

Question 75 · Multiple choice

A school principal surveys every one of the 312 teachers in the district about a new schedule. What is this collection of data called?

1. A simple random sample
2. A census
3. A stratified sample
4. An experiment

Question 76 · Multiple choice

A poll selected 1,200 adults in Ohio at random and asked whether they own an electric vehicle. What is the population of interest?

1. Adults in Ohio who own an electric vehicle
2. All electric vehicles in Ohio
3. The 1,200 adults who were selected
4. All adults in Ohio

Question 77 · Multiple choice

An observational study found that people who eat breakfast daily have lower average blood pressure. Why can't we conclude that eating breakfast lowers blood pressure?

1. Other variables, such as exercise habits, may differ between the groups and be confounded with eating breakfast.
2. Observational studies can never show any association.
3. The sample was too large.
4. Blood pressure is a categorical variable.

Question 78 · Multiple choice

Which investigation requires an experiment rather than an observational study to answer its question?

1. Does a new flu-shot reminder text cause more patients to get the shot?
2. How long do patients typically wait at the clinic?
3. What proportion of patients got a flu shot last year?
4. Is age associated with getting a flu shot?

Question 79 · Multiple choice

A sample survey and an experiment both use randomness. What is the main purpose of the randomness in each?

1. Both use randomness only to make the samples larger.
2. Random selection in a survey lets results generalize to the population; random assignment in an experiment lets us conclude cause and effect.
3. Random selection allows causal conclusions; random assignment allows generalization.
4. Randomness guarantees the results are correct.

Question 80 · Multiple choice

A medical study follows 2,000 adults for 10 years, recording how many servings of fish each eats per week and whether each develops heart disease. What kind of study is this?

1. A prospective observational study
2. A randomized experiment
3. A retrospective observational study
4. A matched pairs experiment

Topic 1.11: Random sampling methods

8 multiple-choice questions.

Question 81 · Multiple choice

A college splits its students by class year (first-year, sophomore, junior, senior) and randomly selects 50 students from each class year. What sampling method is this?

1. Stratified random sampling
2. Cluster sampling
3. Simple random sampling
4. Systematic random sampling

Question 82 · Multiple choice

A city has 80 neighborhoods. An inspector randomly selects 6 neighborhoods and checks every restaurant in those neighborhoods. What sampling method is this?

1. Simple random sampling
2. Cluster sampling
3. Convenience sampling
4. Stratified random sampling

Question 83 · Multiple choice

A class has 32 students. The teacher puts their names in a hat and draws 4 names at random. What is the probability that a particular student, Leo, is chosen?

1. 0.031
2. 0.040
3. 0.125
4. 0.250

Question 84 · Multiple choice

A quality inspector takes a systematic sample of the 600 jars filled in an hour: she picks a random starting number from 1 to 20, then takes every 20th jar. The random start is 7. What is the number of the 5th jar sampled?

1. 67
2. 87
3. 100
4. 107

Question 85 · Multiple choice

A high school has 900 students: 360 in the lower grades and 540 in the upper grades. A stratified sample of 60 students is taken with each grade group represented in proportion to its size. How many upper-grade students are in the sample?

1. 24
2. 30
3. 36
4. 40

Question 86 · Multiple choice

A manager wants a simple random sample of 5 of the company's 250 employees using a table of random digits. Which method is correct?

1. Choose the 5 employees whose names come first in alphabetical order.
2. Label the employees 1 to 250, read single digits and take the first 5 digits.
3. Label the employees 001 to 250, read three-digit groups, skip 000, numbers above 250 and repeats, and take the first 5 labels found.
4. Label the employees 001 to 250 and take the first 5 three-digit groups even if some are above 250.

Question 87 · Multiple choice

Why might a researcher choose a stratified random sample instead of a simple random sample of the same size?

1. Stratified samples are always cheaper because only one group is visited.
2. Stratified samples allow cause-and-effect conclusions.
3. Stratified samples remove all bias from nonresponse.
4. If the strata differ on the response, stratifying gives a more precise estimate (less sampling variability).

Question 88 · Multiple choice

Which of the following is a simple random sample of 40 students from a school of 1,000?

1. Choose the first 40 students who arrive at school on Monday.
2. Assign each student a number from 1 to 1,000 and use a random number generator to choose 40 different numbers.
3. Choose one homeroom of 40 students at random.
4. Choose 10 students at random from each of the four grades.

Topic 1.12: Bias in sampling

8 multiple-choice questions.

Question 89 · Multiple choice

A radio host asks listeners to call in and say whether they support a new stadium. Of the 900 callers, 78% oppose it. What is the main problem with this sample?

1. Undercoverage: the host did not call every listener.
2. Voluntary response bias: people with strong opinions are more likely to call.
3. The sample is too small to be useful.
4. Response bias: the question was worded unfairly.

Question 90 · Multiple choice

A survey about internet access was done by email to a random sample of residents' email addresses. What is the most likely bias?

1. Voluntary response bias, which probably underestimates internet access
2. Response bias from the interviewer's tone
3. Undercoverage of residents without email, which probably overestimates internet access
4. There is no bias because the sample was random

Question 91 · Multiple choice

A random sample of 1,500 adults was mailed a survey and only 410 returned it. Which problem does this create?

1. Response bias, because answers were written
2. Undercoverage, because the sample was mailed
3. Voluntary response bias, because the sample was random
4. Nonresponse bias, if those who did not reply differ from those who did

Question 92 · Multiple choice

In the mailed survey above, 1,500 adults were selected and 410 returned the survey. What was the response rate?

1. 0.273
2. 0.365
3. 0.410
4. 0.727

Question 93 · Multiple choice

A survey asked, "Do you agree that the city's dangerous, crumbling roads need urgent repair funding?" What kind of bias does this wording create?

1. Nonresponse bias, which likely underestimates support
2. Undercoverage, which likely underestimates support
3. Response bias from a leading question, which likely overestimates support
4. Voluntary response bias, which has no predictable direction

Question 94 · Multiple choice

To estimate the average amount students spend on lunch, a student surveys the first 30 people in line at the school café. What is the problem?

1. It is a convenience sample; students who buy café lunches may spend more than those who bring lunch.
2. It is a census of café users, so there is no problem.
3. It is a stratified sample, so the estimate is too precise.
4. The sample size of 30 guarantees bias.

Question 95 · Multiple choice

A town has 12,000 adults. Its voter list, used as a sampling frame, includes 9,600 of them. What proportion of the town's adults is left out of the frame (undercoverage)?

1. 0.02
2. 0.20
3. 0.25
4. 0.80

Question 96 · Multiple choice

Which change would reduce sampling variability (the random sample-to-sample variation of an estimate) but NOT fix a bias problem?

1. Using a random method instead of a voluntary one
2. Following up with people who did not respond
3. Rewording a leading question
4. Increasing the sample size while using the same flawed method

Topic 1.13: Designing experiments

8 multiple-choice questions.

Question 97 · Multiple choice

An experiment tests the effect of watering frequency (daily, every other day, weekly) and fertilizer (none, organic, synthetic) on the growth of bean plants. How many treatments are there?

1. 2
2. 3
3. 6
4. 9

Question 98 · Multiple choice

In the bean experiment (9 treatments), 72 plants are available and each treatment gets the same number of plants. How many plants receive each treatment?

1. 6
2. 8
3. 12
4. 24

Question 99 · Multiple choice

Researchers test whether a new mouthwash reduces plaque. Volunteers are randomly assigned to use the new mouthwash or a look-alike rinse with no active ingredient, and neither the volunteers nor the dentists who measure plaque know who used which. What is this design feature called?

1. Stratified random sampling
2. A matched pairs design
3. Double-blinding with a placebo control
4. Blocking by mouthwash

Question 100 · Multiple choice

To compare two running shoe designs, each of 20 runners wears design A on one foot and design B on the other, with a coin flip deciding which foot gets A. What design is this?

1. A completely randomized design
2. An observational study
3. A randomized block design with two blocks
4. A matched pairs design

Question 101 · Multiple choice

An agricultural experiment compares 3 fertilizers on 30 plots. Half the plots are on a hillside and half are in a valley. The researcher randomly assigns fertilizers separately within the hillside plots and within the valley plots. Why?

1. To block on location, so that differences between hillside and valley do not add variability to the fertilizer comparison
2. To create a placebo group
3. To make sure every plot gets every fertilizer
4. So that the results can be generalized to all farms

Question 102 · Multiple choice

In an experiment on whether a study app improves test scores, 60 student volunteers were randomly assigned to use the app or not. Which conclusion is appropriate if the app group scored significantly higher?

1. The app causes higher scores for all students in the country.
2. The app is associated with higher scores, but no causal conclusion is possible.
3. No conclusion is possible because the subjects were volunteers.
4. The app caused higher scores for students like these volunteers, but the results may not generalize to all students.

Question 103 · Multiple choice

In an experiment about a new lighting system and worker productivity, what is the purpose of including a control group that keeps the old lighting?

1. To make sure the sample is random
2. To provide a baseline so the effect of the new lighting can be separated from other changes over time
3. To increase the number of treatments
4. To make the experiment double-blind

Question 104 · Multiple choice

A researcher lets volunteers choose whether to take a new vitamin or not, then compares their energy levels. Which problem prevents a cause-and-effect conclusion?

1. Energy levels vary from person to person.
2. The response variable is quantitative.
3. There are only two groups.
4. Confounding: people who choose the vitamin may differ in other ways that affect energy.

Free response

Write a complete answer, in context, as you would type it on the exam. The scoring guides are ours, written in the new course's style: 10 points per question, each point scored 0 or 1 and earned independently (no E/P/I part scores). They are not College Board scoring guidelines.

Question 105 · Free response (10 points)

A commuter recorded the time of her drive to work on 12 randomly chosen days using the highway and on 12 randomly chosen days using the back roads. The times are shown in the table.

1. (a) Find the five-number summary of the highway times. [2 points]
2. (b) Is the highway time of 52 minutes an outlier? Justify your answer with the 1.5 × IQR rule. [2 points]
3. (c) Find the median and IQR of the back-road times. [2 points]
4. (d) Compare the distributions of commute times for the two routes. [4 points]

Question 106 · Free response (10 points)

A sports scientist wants to know whether a new warm-up routine improves sprint times. She has 40 volunteer high school athletes: 20 are sprinters and 20 are distance runners. Each athlete will do either the new warm-up or the standard warm-up and then run a timed 100-meter sprint.

1. (a) Identify the experimental units, the treatments and the response variable. [2 points]
2. (b) Describe how to randomly assign the treatments in a completely randomized design so that 20 athletes get each warm-up. [2 points]
3. (c) Explain why blocking by event (sprinter or distance runner) would be a better design, and describe how to carry it out. [3 points]
4. (d) Suppose the new warm-up group has a significantly faster mean time. To whom can the conclusion be generalized, and what kind of conclusion is justified? [3 points]