AP Statistics Quiz: Representing A Categorical Variable With Tables

What this quiz covers

This quiz focuses on Representing A Categorical Variable With Tables, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. The total number of moviegoers in the sample.
2. The average rating moviegoers gave for their preferred movie type. (correct answer)
3. The number of moviegoers who prefer Comedy.
4. 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

1. 20 (correct answer)
2. 5
3. 95
4. 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

1. Low: 0.42, Medium: 0.30, High: 0.28 (correct answer)
2. Low: 63, Medium: 45, High: 42
3. Low: 0.28, Medium: 0.30, High: 0.42
4. 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

1. The claim is correct because the frequency for Car (165) is the highest among all categories.
2. The claim is correct because the relative frequency for Car is 0.55, which is greater than 0.50. (correct answer)
3. The claim is incorrect because 135 commuters use a mode of transport other than a car.
4. 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

1. Strength training is preferred by about 44% of surveyed members.
2. Cycling is the most preferred new class type in the sample.
3. Yoga is preferred by the largest proportion of surveyed members, about 32.5%. (correct answer)
4. Dance is preferred by 28% of surveyed members.
5. 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

1. Public transit is the most common commute mode in the sample.
2. Fewer than one-fourth of sampled adults reported public transit as their primary mode.
3. About 49% of sampled adults reported driving alone, which is the largest category. (correct answer)
4. Carpool and bike/walk together account for more than half of the sample.
5. 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

1. More students preferred small-group sessions than online resources.
2. About $35\%$ of surveyed students preferred one-on-one tutoring, the highest proportion. (correct answer)
3. Teacher office hours were preferred by about 18% of surveyed students.
4. Peer mentoring was preferred by about 6% of surveyed students.
5. 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

1. Email is preferred by more patients than phone calls in the sample. (correct answer)
2. Text messages are preferred by about 55% of the sampled patients.
3. No reminder is preferred by about 6% of the sampled patients.
4. Phone calls are preferred by about 22.0% of the sampled patients.
5. A majority of the sampled patients preferred either email or phone calls.

Explanation: The skill here involves representing a categorical variable with tables by analyzing frequency and relative frequency for preferred reminder methods in 110 patients. Choice A is supported because email has a frequency of 27, exceeding phone calls at 22, indicating more patients prefer email. Evidence from the table confirms 27 > 22, directly comparing counts. A common distractor is choice B, claiming text messages at about 55%, but it's 0.500 or 50%, an overestimation. Choice D says phone calls at about 22.0%, mistaking frequency for relative frequency (20%). A mini-lesson on categorical tables is to compare frequencies for count-based claims and relative frequencies for proportions, calculating totals or sums to validate statements about majorities or preferences.

Question 9

1. Phones are used by the largest proportion of students in the sample.
2. Tablets are used by about 18% of students in the sample.
3. Laptops are the most common primary device in the sample, at about 43.3%. (correct answer)
4. Desktops and tablets together account for about 63.3% of the sample.
5. The frequency for phone is 0.267 students.

Explanation: This problem focuses on the skill of representing a categorical variable with tables, interpreting frequency and relative frequency for primary homework devices in 90 students. Choice C is supported, with laptops at 0.433 or about 43.3%, the most common, surpassing phones at 26.7%. This is backed by 39/90 ≈ 0.433, the highest. A typical distractor is choice E, stating the frequency for phone is 0.267 students, confusing relative frequency with the actual count of 24. Choice D wrongly adds desktops and tablets to 63.3%, but it's 30%. For interpreting categorical tables, use relative frequencies to assess prevalence and add them for combinations, ensuring you reference the correct column to avoid mixing counts and proportions.

Question 10

1. TikTok is the least common platform in the sample.
2. Instagram is the most common platform in the sample, with 56 respondents (40%). (correct answer)
3. YouTube is associated with the brand by about 28% of respondents.
4. Facebook and YouTube together account for about 10% of respondents.
5. More respondents selected Facebook than YouTube.

Explanation: The skill evaluated is representing a categorical variable with tables, interpreting frequency and relative frequency for social media platforms associated with a brand in 140 customers. Choice B is supported, with Instagram at 56 respondents and 0.400 or 40%, the most common. Evidence includes 56/140 = 0.4, outperforming others. A key distractor is choice C, claiming YouTube at about 28%, but 28 is frequency, not the 20% relative. Choice D errs by adding Facebook and YouTube to 10%, when it's 30%. A transferable mini-lesson for categorical tables is to use frequencies for counts and relative frequencies for percentages, verifying combinations by summing proportions to check aggregate claims accurately.

Question 11

1. Always is the most common response in the sample.
2. Often is the most common response in the sample, at about 34%. (correct answer)
3. Sometimes and never together account for about 88% of the sample.
4. The frequency for never is 0.120 students.
5. More students responded always than often.

Explanation: This question targets the skill of representing a categorical variable with tables by examining frequency and relative frequency for recycling behaviors in 100 students. The statement in choice B is supported, as often has 0.340 or about 34%, the highest relative frequency, exceeding always at 28%. This is evidenced by 34/100 = 0.34, making it the mode. A key distractor is choice D, which states the frequency for never is 0.120 students, but 0.120 is the relative frequency, while frequency is 12. Choice A incorrectly identifies always as most common, ignoring the higher count for often. In working with categorical tables, always calculate and compare relative frequencies to spot the dominant category, and combine them for aggregate claims to avoid errors in interpretation.

Question 12

1. Juice is chosen by about 21% of the sampled students.
2. Soda is chosen by fewer students than coffee/tea.
3. Water is the most common beverage in the sample, at about 42%. (correct answer)
4. More than half of sampled students chose soda or juice.
5. The relative frequency for coffee/tea is 30.

Explanation: This problem evaluates the ability to represent a categorical variable with tables by analyzing frequency and relative frequency for lunch beverage choices among 150 students. The data supports choice C since water has a relative frequency of 0.420, or 42%, and it's the highest, surpassing soda at 24%. Supporting evidence includes the frequency of 63, where 63/150 = 0.42, clearly the mode. A typical distractor is choice A, stating juice is about 21%, but 0.140 is 14%, possibly mistaking the frequency 21 for a percentage. Choice E errs by saying the relative frequency for coffee/tea is 30, confusing it with the frequency count. For interpreting categorical tables, remember to differentiate counts from proportions and use relative frequencies to determine popularity, calculating sums for combined categories to check claims like majorities.

Question 13

1. Clubs account for about 31% of the students in the sample.
2. Sports are the most common primary activity in the sample, at about 40%. (correct answer)
3. Homework/study is the least common primary activity in the sample.
4. More students reported part‑time job than homework/study.
5. Sports and clubs together account for fewer than half of the sample.

Explanation: This question tests representing a categorical variable with tables, using frequency and relative frequency for after-school activities in 125 students. Choice B is supported as sports has 0.400 or 40%, the most common, higher than clubs at 24.8%. This is evidenced by 50/125 = 0.4, the mode. A frequent distractor is choice A, stating clubs at about 31%, but 31 is the frequency, not the 24.8% relative. Choice C incorrectly calls homework the least, despite it exceeding part-time jobs. In interpreting categorical tables, identify the highest relative frequency for the most common category and compare counts carefully for rankings, avoiding confusion between absolute and proportional values.

Question 14

1. E-books are preferred by a majority of the sampled patrons.
2. Audiobooks are preferred by about 12% of the sampled patrons.
3. Print is the least common preferred format in the sample.
4. Print is the most common preferred format in the sample, at about 55%. (correct answer)
5. Other formats account for 4% of the sampled patrons.

Explanation: The skill here is representing a categorical variable with tables, interpreting frequency and relative frequency for preferred book formats in a sample of 80 patrons. Choice D is backed by the data, with print at 0.550 or 55%, the highest proportion, outranking e-books at 25%. Evidence comes from 44/80 = 0.55, establishing it as the most common. A common distractor is choice B, claiming audiobooks at about 12%, but it's 0.150 or 15%, likely a misreading of the table. Choice E says other at 4%, but it's 0.050 or 5%, a slight but incorrect approximation. A useful mini-lesson on categorical tables is to rely on relative frequencies for percentage claims and compare them to identify the least or most preferred options, ensuring approximations align closely with exact values.

Question 15

1. The sum of the frequencies will be equal to the number of distinct eye colors.
2. Each frequency must be less than the corresponding relative frequency.
3. The sum of the frequencies will be equal to 80. (correct answer)
4. The category with the highest frequency will have a relative frequency of 1.0.

Explanation: The sum of the frequencies (counts) in a frequency table must equal the total number of observations in the dataset, which is 80 in this case. The other statements are incorrect definitions or properties of frequency tables.

Question 16

1. The number of categories.
2. The mode of the distribution.
3. The mean of the distribution.
4. The total number of observations. (correct answer)

Explanation: A relative frequency is a proportion. To find the original count (frequency), you must multiply the proportion by the total number of observations (the sample size). Without the total number of observations, the original frequencies cannot be recovered.

Question 17

1. 1.20
2. 0.30
3. 0.48 (correct answer)
4. 120

Explanation: The relative frequency is the count for the category divided by the total number of observations. In this case, it is $\frac{120}{250} = 0.48$. Choice B is the relative frequency for entering the workforce ($\frac{30}{250} = 0.12$, a calculation error could lead to 0.30). Choice D is the frequency, not the relative frequency.

Question 18

1. 0.20 (correct answer)
2. 0.80
3. 0.15
4. 0.25

Explanation: The sum of all relative frequencies in a distribution must be 1. The sum of the given relative frequencies is $0.35 + 0.25 + 0.20 = 0.80$. Therefore, the relative frequency for Platform D must be $1 - 0.80 = 0.20$.

Question 19

1. 15 (correct answer)
2. 0.25
3. 0.45
4. 45

Explanation: The total number of dogs is 60. The number of specified breeds is $18 + 15 + 12 = 45$. The number of mixed-breed dogs is the total minus the specified breeds: $60 - 45 = 15$. The other choices confuse frequency with relative frequency or represent incorrect calculations.

Question 20

1. 45% (correct answer)
2. 50%
3. 30%
4. 25%

Explanation: First, find the total number of votes: $450 + 300 + 250 = 1000$. The relative frequency for Candidate A is $450/1000 = 0.45$. To express this as a percentage, multiply by 100, which gives 45%.