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AP Statistics Help: Representing A Categorical Variable With Tables

Review real example questions for Representing A Categorical Variable With Tables in AP Statistics.

Question 1 / 10

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A researcher collects data on the preferred type of movie from a sample of moviegoers. The categories are Action, Comedy, Drama, and Sci-Fi. Which of the following is NOT information that can be obtained from a frequency table of this data?

All questions

Question 1

The total number of moviegoers in the sample.
The average rating moviegoers gave for their preferred movie type. (correct answer)
The number of moviegoers who prefer Comedy.
The movie type that was most frequently preferred.

Explanation: A frequency table for a categorical variable like 'preferred movie type' can show counts for each category, the total count, and the most frequent category (the mode). It cannot provide information about a quantitative variable like an 'average rating,' as that data is not part of the categorical frequency count.

Question 2

A library tracks the genres of books checked out by patrons. A relative frequency table shows the following: Fiction: 0.52, Non-Fiction: 0.28, Young Adult: 0.15. The rest are Children's books. If 400 books were checked out in total, what is the frequency of Children's books?

20 (correct answer)
5
95
380

Explanation: First, find the relative frequency of Children's books. The sum of relative frequencies must be 1. $1 - (0.52 + 0.28 + 0.15) = 1 - 0.95 = 0.05$ . Then, find the frequency by multiplying this relative frequency by the total number of books: $0.05 \times 400 = 20$ .

Question 3

An environmental agency tested 150 rivers and classified their pollution level as Low, Medium, or High. They found 63 rivers had a Low pollution level and 45 had a Medium pollution level.

Which of the following represents the relative frequency table for the pollution levels of the 150 rivers?

Low: 0.42, Medium: 0.30, High: 0.28 (correct answer)
Low: 63, Medium: 45, High: 42
Low: 0.28, Medium: 0.30, High: 0.42
Low: 0.63, Medium: 0.45, High: 0.42

Explanation: First, find the frequency for High: $150 - 63 - 45 = 42$ . Then, calculate relative frequencies: Low is $63/150 = 0.42$ , Medium is $45/150 = 0.30$ , and High is $42/150 = 0.28$ . Choice A correctly lists these relative frequencies. Choice B lists frequencies. Choices C and D use incorrect calculations.

Question 4

A survey of 300 commuters in a large city asked for their primary mode of transport. A frequency table of the results is shown below: Car: 165 Bus: 75 Subway: 45 Walk/Bike: 15

A city planner claims that cars are the primary mode of transport for a majority of commuters. Which of the following statements correctly evaluates this claim using the provided data?

The claim is correct because the frequency for Car (165) is the highest among all categories.
The claim is correct because the relative frequency for Car is 0.55, which is greater than 0.50. (correct answer)
The claim is incorrect because 135 commuters use a mode of transport other than a car.
The claim is incorrect because the frequency for Car (165) is not more than twice the next highest frequency (75).

Explanation: The term 'majority' means more than 50%. To evaluate this, we need the relative frequency. The relative frequency for Car is $165/300 = 0.55$ . Since $0.55 > 0.50$ , the claim is supported by the data. While choice A is true (Car is the mode), it doesn't directly address the 'majority' claim. Choices C and D use flawed reasoning to evaluate the claim.

Question 5

A fitness center manager is researching which class type members most want added to the schedule. From the population of current members, a random sample of 160 members is surveyed and each member selects a preferred new class type (a categorical variable). The distribution is shown below.

Which statement is supported by the data?

Preferred new class type	Frequency	Relative frequency
Yoga	52	0.325
Strength training	44	0.275
Cycling	36	0.225
Dance	28	0.175

Strength training is preferred by about 44% of surveyed members.
Cycling is the most preferred new class type in the sample.
Yoga is preferred by the largest proportion of surveyed members, about 32.5%. (correct answer)
Dance is preferred by 28% of surveyed members.
More than half of surveyed members preferred cycling or dance.

Explanation: The skill assessed is representing a categorical variable with tables, using frequency and relative frequency to evaluate preferred new class types among 160 members. Choice C is supported by yoga's relative frequency of 0.325, or 32.5%, the largest proportion, greater than strength training at 27.5%. Evidence includes 52/160 = 0.325, confirming its lead. A common distractor is choice A, claiming strength training at about 44%, but that's the frequency, not the 27.5% relative. Choice D similarly mistakes dance's frequency 28 for 28%. A transferable lesson for categorical tables is to distinguish frequencies from relative frequencies when making proportion-based claims, and verify modes by comparing decimals or percentages directly.

Question 6

A city transportation office is studying commuting habits of adults who live within city limits. A random sample of 200 adults is asked to report their primary commute mode (a categorical variable). The distribution is shown below.

Which statement is supported by the data?

Primary commute mode	Frequency	Relative frequency
Drive alone	98	0.490
Carpool	34	0.170
Public transit	46	0.230
Bike/Walk	22	0.110

Public transit is the most common commute mode in the sample.
Fewer than one-fourth of sampled adults reported public transit as their primary mode.
About 49% of sampled adults reported driving alone, which is the largest category. (correct answer)
Carpool and bike/walk together account for more than half of the sample.
The frequency for bike/walk is 0.110 adults.

Explanation: The skill tested here involves representing a categorical variable with tables, focusing on interpreting frequencies and relative frequencies for commute modes in a sample of 200 adults. Choice C is supported as the relative frequency for driving alone is 0.490, or about 49%, and it is the largest category, exceeding public transit at 23% and others. This is evidenced by the frequency of 98, which is 98/200 = 0.49, confirming it's the mode. A frequent distractor is choice E, claiming the frequency for bike/walk is 0.110 adults, but 0.110 is the relative frequency, while the actual frequency is 22, mixing up the two columns. Choice D incorrectly adds carpool and bike/walk to over half (0.170 + 0.110 = 0.280), which is less than 0.5. A transferable mini-lesson for categorical tables is to always verify proportions by dividing frequencies by the total and compare them to assess dominance or combinations, avoiding assumptions about aggregates without calculation.

Question 7

A school counselor wants to understand how 9th-grade students at Central High prefer to receive academic support. The counselor surveys a random sample of 120 ninth graders and records each student's preferred support method (a categorical variable). Results are summarized below.

Which statement is supported by the data in the table?

Preferred support method	Frequency	Relative frequency
One-on-one tutoring	42	0.350
Small-group sessions	30	0.250
Online resources	24	0.200
Teacher office hours	18	0.150
Peer mentoring	6	0.050

More students preferred small-group sessions than online resources.
About $35\%$ of surveyed students preferred one-on-one tutoring, the highest proportion. (correct answer)
Teacher office hours were preferred by about 18% of surveyed students.
Peer mentoring was preferred by about 6% of surveyed students.
A majority of surveyed students preferred either online resources or teacher office hours.

Explanation: This question assesses the skill of representing a categorical variable with tables by interpreting frequency and relative frequency to identify supported statements about students' preferred academic support methods. The data supports choice B because the relative frequency for one-on-one tutoring is $0.350$ , or $35\%$ , which is indeed the highest proportion among the options, as confirmed by comparing it to $0.250$ , $0.200$ , $0.150$ , and $0.050$ . Evidence from the table shows 42 students chose this out of 120, yielding exactly $35\%$ , outperforming others like small-group sessions at $25\%$ . A common distractor is choice C, which claims about 18% preferred teacher office hours, but the actual relative frequency is $0.150$ or 15%, likely confusing the frequency count of 18 with a percentage. Another distractor, choice D, states about 6% for peer mentoring, but it's precisely 5%, highlighting the need for accurate reading. In interpreting categorical tables, a key lesson is to use relative frequencies for proportions and compare them directly to identify modes or rankings, ensuring calculations align with the total sample size for validity.

Question 8

A table reports athletes 80, musicians 50, neither 30, for 120 students. Counts sum to 160. The table is invalid because...

Counts should be percents
The categories overlap (correct answer)
Categories should be sorted
Relative tables fix this

Explanation: The categories are not mutually exclusive: a student can be both an athlete and a musician, so the counts double-count those students. That is why 80 + 50 + 30 exceeds 120. The tempting wrong answer is that counts should be percents, but percents would not fix double-counting from overlapping categories.

Question 9

Class X: 10 of 25 prefer A. Class Y: 10 of 100 prefer A. A frequency table shows 10 votes each. Why is it misleading?

Relative frequencies exceed 1
Counts are always too small
Same count, different rates (correct answer)
Totals cannot be computed here

Explanation: In Class X, 10 out of 25 means 40% prefer A, while in Class Y, 10 out of 100 means only 10%, so equal counts hide very different rates. The tempting mistake is treating the equal vote totals as comparable without considering each class size.

Question 10

A survey of 1,200 residents: 480 rent, 360 own, 240 other. A table says rent 0.40, own 0.30, other 0.30. What is wrong?

Other should be 0.20, not 0.30 (correct answer)
Own should be 0.45, not 0.30
Rent should be 0.48, not 0.40
Total is 1.00, so it is valid

Explanation: Among 1,200 residents, each proportion is count divided by 1,200: rent 480/1,200 = 0.40, own 360/1,200 = 0.30, other 240/1,200 = 0.20. So the table's 0.30 for other is wrong. The tempting mistake is thinking rent is 0.48, which comes from dividing 480 by 1,000 instead of 1,200.