AP Statistics Quiz: Statistics For Two Categorical Variables

What this quiz covers

This quiz focuses on Statistics For Two Categorical Variables, 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. A larger proportion of on-campus students participate in a club than off-campus students. (correct answer)
2. More off-campus students do not participate because 0.22 is greater than 0.15, so off-campus has a higher non-participation rate.
3. Among students who participate, a larger proportion live off-campus than on-campus.
4. The participation rates are the same because both residence types have total proportion 0.50.
5. Off-campus students have a higher participation rate because 0.28 is close to 0.35.

Explanation: This problem requires comparing club participation rates between residence types using conditional proportions. For on-campus students: 0.35/(0.35 + 0.15) = 0.35/0.50 = 0.70 or 70% participate in clubs. For off-campus students: 0.28/(0.28 + 0.22) = 0.28/0.50 = 0.56 or 56% participate in clubs. Since 70% > 56%, a larger proportion of on-campus students participate in clubs than off-campus students. Choice B incorrectly focuses on non-participation values without calculating proportions. Choice D wrongly assumes equal rates because both groups have 0.50 total proportion. Remember that equal group sizes in the sample doesn't mean equal participation rates within those groups.

Question 2

1. Among students who turned in the homework, a larger proportion were under the Optional policy than the Required policy.
2. A larger proportion of students under the Required policy turned in homework than under the Optional policy. (correct answer)
3. More students did not turn in homework under Optional because 0.29 is greater than 0.14, so Optional has a higher non-turn-in rate.
4. The turn-in rate is the same for both policies because both totals add to 0.50.
5. Among students who did not turn in homework, a smaller proportion were under Optional than Required.

Explanation: To compare homework turn-in rates between policies, calculate conditional proportions within each policy group. For Optional policy: 0.21/(0.21 + 0.29) = 0.21/0.50 = 0.42 or 42% turned in homework. For Required policy: 0.36/(0.36 + 0.14) = 0.36/0.50 = 0.72 or 72% turned in homework. Since 72% > 42%, a larger proportion of students under the Required policy turned in homework. Choice C mistakenly compares the "did not turn in" values directly (0.29 vs 0.14) without calculating proportions. Choice D incorrectly assumes equal rates because both policies have 0.50 total proportion. The key insight is that conditional proportions show the Required policy is more effective at getting students to turn in homework.

Question 3

1. Among customers who used a coupon, a larger proportion used self-checkout than a cashier.
2. A larger proportion of customers at self-checkout used a coupon than customers with a cashier.
3. A larger proportion of customers with a cashier used a coupon than customers at self-checkout. (correct answer)
4. More customers did not use a coupon at self-checkout (0.36) than at a cashier (0.32), so self-checkout increases coupon use.
5. Coupon use is the same for both options because both options have total proportion 0.50.

Explanation: To compare coupon usage rates between checkout options, calculate the conditional proportion using coupons within each group. For self-checkout: 0.14/(0.14 + 0.36) = 0.14/0.50 = 0.28 or 28% used coupons. For cashier checkout: 0.18/(0.18 + 0.32) = 0.18/0.50 = 0.36 or 36% used coupons. Since 36% > 28%, a larger proportion of customers with a cashier used coupons than customers at self-checkout. Choice D incorrectly interprets larger non-use at self-checkout (0.36 > 0.32) as evidence that self-checkout increases coupon use, which is backwards logic. Choice E wrongly claims equal usage because both options represent 50% of customers. The data shows cashiers may facilitate or encourage more coupon redemption.

Question 4

1. A higher proportion of club participants prefer online help than nonparticipants prefer online help. (correct answer)
2. More club participants prefer online help than nonparticipants prefer online help, so club participants are more likely to prefer online help.
3. A higher proportion of students who prefer online help participate in a club than students who prefer in-person help participate in a club.
4. A higher proportion of nonparticipants prefer online help than club participants prefer online help.
5. The proportion who prefer in-person help is the same for club participants and nonparticipants.

Explanation: This question tests understanding of conditional proportions when comparing preferences within participation groups. To find the proportion of club participants who prefer online help, we calculate 72/120 = 0.60 or 60%. For non-participants, the proportion preferring online help is 60/150 = 0.40 or 40%. Since 60% > 40%, a higher proportion of club participants prefer online help than non-participants. Choice B incorrectly focuses on raw counts (72 vs 60) rather than proportions, which is a common mistake when comparing groups of different sizes. When comparing categorical variables across groups, always use proportions within each group, not raw counts.

Question 5

1. Layout 2 has a higher click rate because it has a larger total proportion (0.60) than Layout 1 (0.40).
2. Among users who clicked, a larger proportion used Layout 1 than Layout 2.
3. A larger proportion of users shown Layout 1 clicked than users shown Layout 2. (correct answer)
4. More users clicked with Layout 2 because 0.18 > 0.16, so Layout 2 must have higher click probability.
5. The click rates are equal because both layouts have more "no click" than "click."

Explanation: This question asks you to compare click rates between website layouts using conditional proportions. For Layout 1: 0.16/(0.16 + 0.24) = 0.16/0.40 = 0.40 or 40% clicked. For Layout 2: 0.18/(0.18 + 0.42) = 0.18/0.60 = 0.30 or 30% clicked. Since 40% > 30%, a larger proportion of users shown Layout 1 clicked than users shown Layout 2. Choice A incorrectly uses total proportions (0.60 vs 0.40) which represent the distribution of users between layouts, not click rates. Choice D wrongly assumes 0.18 > 0.16 means Layout 2 has a higher click rate, ignoring that these must be divided by their respective totals. When comparing effectiveness, always calculate the success rate within each group.

Question 6

1. Among students who passed, a larger proportion used flashcards than practice problems.
2. More students passed using practice problems because 0.33 is greater than 0.26, so practice problems must have the higher pass rate.
3. A larger proportion of students using flashcards passed than students using practice problems.
4. A larger proportion of students using practice problems passed than students using flashcards. (correct answer)
5. The pass rates are the same because the overall pass proportion is $0.59$.

Explanation: This question tests comparing pass rates between study strategies using conditional proportions. For flashcards: 0.26/(0.26 + 0.24) = 0.26/0.50 = 0.52 or 52% passed. For practice problems: 0.33/(0.33 + 0.17) = 0.33/0.50 = 0.66 or 66% passed. Since 66% > 52%, a larger proportion of students using practice problems passed than students using flashcards. Choice B incorrectly assumes 0.33 > 0.26 directly indicates higher pass rate without calculating proportions. Choice C reverses the correct comparison. The key lesson is that when comparing success rates between strategies, you must calculate the proportion successful within each strategy group, not just compare the raw values in the table.

Question 7

1. A larger proportion of members contacted by App attended than members contacted by Phone. (correct answer)
2. More members attended after an App notification because 0.24 is twice 0.12, so the App group must be twice as large.
3. Among members who attended, a larger proportion were contacted by Phone than by App.
4. The attendance rate is the same because both methods have more "did not attend" than "attended."
5. Among members contacted by Phone, a smaller proportion did not attend than among members contacted by App.

Explanation: This problem tests calculating conditional proportions for attendance rates by contact method. For Phone contacts: 0.12/(0.12 + 0.28) = 0.12/0.40 = 0.30 or 30% attended. For App contacts: 0.24/(0.24 + 0.36) = 0.24/0.60 = 0.40 or 40% attended. Since 40% > 30%, a larger proportion of App-contacted members attended than Phone-contacted members. Choice B incorrectly assumes the 2:1 ratio in table values means the App group is twice as large, but we can't determine actual counts from proportions. Choice C reverses the comparison direction by looking at contact method given attendance. Always identify which variable is the explanatory variable (contact method) and which is the response (attendance).

Question 8

1. Among satisfied commuters, a larger proportion took the bus than drove.
2. A larger proportion of drivers are satisfied than bus riders because 0.30 > 0.24.
3. A larger proportion of bus riders are satisfied than drivers. (correct answer)
4. More commuters are satisfied than not satisfied, so commute type does not matter.
5. Among commuters who are not satisfied, a smaller proportion drove than took the bus.

Explanation: To compare satisfaction rates between commute types, calculate the conditional proportion satisfied within each group. For bus riders: 0.24/(0.24 + 0.16) = 0.24/0.40 = 0.60 or 60% are satisfied. For drivers: 0.30/(0.30 + 0.30) = 0.30/0.60 = 0.50 or 50% are satisfied. Since 60% > 50%, a larger proportion of bus riders are satisfied than drivers. Choice B incorrectly compares the raw table values 0.30 and 0.24 instead of calculating conditional proportions. Choice A examines the wrong conditional proportion (commute type given satisfaction). When the question asks about satisfaction by commute type, calculate P(satisfied|commute type), not P(commute type|satisfied).

Question 9

1. A larger proportion of students invited by Text attended than students invited by Email. (correct answer)
2. More students attended from the Text group than from the Email group, so Text is more effective.
3. Among students who attended, a larger proportion were invited by Email than by Text.
4. The proportion who attended is the same for Email and Text because 0.18 and 0.22 are close.
5. A larger proportion of Email-invited students did not attend than Text-invited students did not attend, so Email increases attendance.

Explanation: This question tests your ability to calculate and compare conditional proportions from a two-way table. To find the proportion of Text-invited students who attended, divide 0.22 by the total Text proportion (0.22 + 0.28 = 0.50), giving 0.22/0.50 = 0.44 or 44%. For Email-invited students who attended, divide 0.18 by the total Email proportion (0.18 + 0.32 = 0.50), giving 0.18/0.50 = 0.36 or 36%. Since 44% > 36%, a larger proportion of Text-invited students attended. Choice B incorrectly compares counts instead of proportions, while Choice C reverses the comparison direction. When comparing effectiveness between groups, always calculate the conditional proportion within each group rather than comparing raw values from the table.

Question 10

1. A higher proportion of enrolled adults met the guideline than not enrolled adults met the guideline.
2. Because more enrolled adults met the guideline (110) than not enrolled adults met the guideline (66), enrolled adults are more likely to meet it.
3. A higher proportion of not enrolled adults met the guideline than enrolled adults met the guideline. (correct answer)
4. A higher proportion of adults who met the guideline were enrolled than adults who did not meet the guideline were enrolled.
5. Enrolled and not enrolled adults have the same guideline-meeting rate.

Explanation: This question requires comparing guideline-meeting rates between enrolled and not enrolled adults. For enrolled adults, the rate is 110/200 = 0.55 or 55%. For not enrolled adults, the rate is 66/100 = 0.66 or 66%. Since 66% > 55%, a higher proportion of not enrolled adults met the guideline than enrolled adults. Choice B incorrectly uses raw count comparison (110 > 66) to justify an incorrect conclusion. This counterintuitive result might suggest that adults who already meet activity guidelines feel less need to enroll in formal fitness programs.

Question 11

1. A higher proportion of residents who support bike lanes bike to work than residents who oppose bike lanes bike to work.
2. Because more nonbikers support bike lanes (160) than bikers support bike lanes (135), nonbikers are more likely to support bike lanes.
3. A higher proportion of bikers support bike lanes than nonbikers support bike lanes. (correct answer)
4. A higher proportion of nonbikers support bike lanes than bikers support bike lanes.
5. Bikers and nonbikers have the same proportion who oppose bike lanes.

Explanation: This question tests comparing support rates for bike lanes between bikers and non-bikers. Among bikers, the support rate is 135/150 = 0.90 or 90%. Among non-bikers, the support rate is 160/200 = 0.80 or 80%. Since 90% > 80%, a higher proportion of bikers support bike lanes than non-bikers. Choice B incorrectly reasons from raw counts (160 > 135) without calculating proportions, which is misleading when group sizes differ. Remember that conditional proportions are calculated as (count in category)/(total in group), not by comparing raw counts across groups.

Question 12

1. A higher proportion of coupon users buy dessert than nonusers buy dessert.
2. A higher proportion of nonusers buy dessert than coupon users buy dessert. (correct answer)
3. Because more nonusers buy dessert (45) than coupon users buy dessert (40), nonusers have a higher dessert rate.
4. A higher proportion of dessert buyers used a coupon than no-dessert buyers used a coupon.
5. Coupon users and nonusers have the same dessert-purchase rate.

Explanation: This problem asks us to compare dessert purchase rates between coupon users and non-users. For coupon users, the dessert rate is 40/200 = 0.20 or 20%. For non-users, the dessert rate is 45/150 = 0.30 or 30%. Since 30% > 20%, a higher proportion of non-users buy dessert than coupon users. Choice C reaches the correct conclusion but uses flawed reasoning based on raw counts (45 > 40) rather than proportions. The data suggests that coupon users may be more price-conscious and less likely to add dessert to their order.

Question 13

1. A higher proportion of members who renewed attended classes than members who did not renew attended classes.
2. A higher proportion of class attendees renewed than nonattendees renewed. (correct answer)
3. More nonattendees renewed (140) than attendees renewed (96), so nonattendees are more likely to renew.
4. A higher proportion of nonattendees renewed than attendees renewed.
5. The renewal proportion is the same for attendees and nonattendees.

Explanation: This problem requires comparing renewal rates between gym members who attend classes and those who don't. For class attendees, the renewal rate is 96/120 = 0.80 or 80%. For non-attendees, the renewal rate is 140/200 = 0.70 or 70%. Since 80% > 70%, a higher proportion of class attendees renewed than non-attendees renewed. Choice C makes the error of comparing raw counts (140 > 96) without considering the different group sizes, leading to an incorrect conclusion. The key insight is that proportions, not counts, reveal the true likelihood of renewal within each group.

Question 14

1. A higher proportion of premium users watch documentaries weekly than standard users watch documentaries weekly. (correct answer)
2. Because more premium users watch weekly (90) than standard users watch weekly (80), premium users have a higher weekly-documentary rate.
3. A higher proportion of standard users watch documentaries weekly than premium users watch documentaries weekly.
4. A higher proportion of weekly documentary watchers are premium than non-weekly watchers are premium.
5. Premium and standard users have the same weekly-documentary rate.

Explanation: This question asks us to compare weekly documentary viewing rates between premium and standard users. For premium users, the rate is 90/150 = 0.60 or 60%. For standard users, the rate is 80/200 = 0.40 or 40%. Since 60% > 40%, a higher proportion of premium users watch documentaries weekly than standard users. Choice B incorrectly justifies the conclusion using raw counts (90 > 80) rather than proportions, which doesn't account for the larger number of standard users overall. The correct approach always involves calculating proportions within each group before making comparisons.

Question 15

1. Among customers who said "Yes," a larger proportion saw Design B than Design A.
2. A larger proportion of customers shown Design A said "Yes" than customers shown Design B. (correct answer)
3. More customers said "Yes" for Design A because 0.30 is greater than 0.25, so counts must be larger.
4. The designs have the same "Yes" rate because each design has total proportion 0.50.
5. Among customers shown Design A, a larger proportion said "No" than among customers shown Design B.

Explanation: This question requires calculating conditional proportions to compare customer responses between package designs. For Design A, the proportion saying "Yes" is 0.30/(0.30 + 0.20) = 0.30/0.50 = 0.60 or 60%. For Design B, the proportion saying "Yes" is 0.25/(0.25 + 0.25) = 0.25/0.50 = 0.50 or 50%. Since 60% > 50%, a larger proportion of customers shown Design A said "Yes" compared to Design B. Choice C mistakenly assumes that larger table values mean larger counts without knowing sample sizes. Choice D incorrectly claims equal rates just because row totals are equal. Remember that conditional proportions require dividing by the row total, not comparing individual cell values.

Question 16

1. Because more not-remote employees are satisfied (150) than remote employees are satisfied (84), not-remote employees are more likely to be satisfied.
2. The proportion satisfied among remote employees is higher than the proportion satisfied among not-remote employees.
3. The proportion satisfied among remote employees is lower than the proportion satisfied among not-remote employees. (correct answer)
4. The proportion remote is higher among satisfied employees than among not satisfied employees.
5. Remote and not-remote employees have the same satisfaction rate.

Explanation: This question asks us to compare satisfaction rates between remote and non-remote employees using conditional proportions. For remote employees, the satisfaction rate is 84/120 = 0.70 or 70%. For non-remote employees, the satisfaction rate is 150/200 = 0.75 or 75%. Since 70% < 75%, the proportion satisfied among remote employees is lower than among non-remote employees. Choice A incorrectly compares raw counts (150 vs 84) instead of proportions, which doesn't account for the different group sizes. When analyzing two-way tables, calculate the proportion within each category of the explanatory variable to make valid comparisons.

Question 17

1. A higher proportion of on-campus students attend office hours monthly than off-campus students attend office hours monthly.
2. A higher proportion of off-campus students attend office hours monthly than on-campus students attend office hours monthly. (correct answer)
3. Because more on-campus students attend office hours (75) than off-campus students attend office hours (60), on-campus students are more likely to attend.
4. A higher proportion of students who attend office hours live on campus than students who do not attend live on campus.
5. On-campus and off-campus students have the same office-hour attendance rate.

Explanation: This question tests understanding of office hour attendance rates for on-campus versus off-campus students. For on-campus students, the attendance rate is 75/200 = 0.375 or 37.5%. For off-campus students, the attendance rate is 60/100 = 0.60 or 60%. Since 60% > 37.5%, a higher proportion of off-campus students attend office hours monthly than on-campus students. Choice C incorrectly compares raw counts (75 > 60) without considering that there are twice as many on-campus students. When groups have different sizes, proportions provide the only valid basis for comparison.