Statistics for Two Categorical Variables
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AP Statistics › Statistics for Two Categorical Variables
A university compared two residence types (On-campus vs Off-campus) and surveyed whether students participated in at least one club (Yes/No). The purpose is to compare conditional proportions of club participation by residence type. Based on the two-way table, which comparison is supported?
Two-way table of Residence by Club Participation (proportions of all students):
- On-campus & Yes: 0.35, On-campus & No: 0.15
- Off-campus & Yes: 0.28, Off-campus & No: 0.22
Off-campus students have a higher participation rate because 0.28 is close to 0.35.
A larger proportion of on-campus students participate in a club than off-campus students.
More off-campus students do not participate because 0.22 is greater than 0.15, so off-campus has a higher non-participation rate.
The participation rates are the same because both residence types have total proportion 0.50.
Among students who participate, a larger proportion live off-campus than on-campus.
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.
A teacher compared two homework policies (Optional vs Required) and noted whether students turned in the assignment (Turned in/Did not turn in). The purpose is to compare conditional proportions of turning in homework by policy. Based on the two-way table, which comparison is supported?
Two-way table of Policy by Turn-in Status (proportions of all students):
- Optional & Turned in: 0.21, Optional & Did not: 0.29
- Required & Turned in: 0.36, Required & Did not: 0.14
The turn-in rate is the same for both policies because both totals add to 0.50.
Among students who turned in the homework, a larger proportion were under the Optional policy than the Required policy.
A larger proportion of students under the Required policy turned in homework than under the Optional policy.
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.
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.
A store compared two checkout options (Self-checkout vs Cashier) and recorded whether customers used a coupon (Used/Did not use). The purpose is to compare conditional proportions of coupon use by checkout option. Based on the two-way table, which comparison is supported?
Two-way table of Checkout Option by Coupon Use (proportions of all customers):
- Self-checkout & Used: 0.14, Self-checkout & Did not: 0.36
- Cashier & Used: 0.18, Cashier & Did not: 0.32
A larger proportion of customers at self-checkout used a coupon than customers with a cashier.
Coupon use is the same for both options because both options have total proportion 0.50.
Among customers who used a coupon, a larger proportion used self-checkout than a cashier.
More customers did not use a coupon at self-checkout (0.36) than at a cashier (0.32), so self-checkout increases coupon use.
A larger proportion of customers with a cashier used a coupon than customers at self-checkout.
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.
A school surveyed students about whether they participate in a school club and whether they prefer online or in-person homework help. The goal is to compare groups by looking at conditional proportions (preference within each participation group). Which comparison is supported by the data?
Counts: Club: Online 72, In-person 48 (Total 120). No club: Online 60, In-person 90 (Total 150).
More club participants prefer online help than nonparticipants prefer online help, so club participants are more likely to prefer online help.
The proportion who prefer in-person help is the same for club participants and nonparticipants.
A higher proportion of club participants prefer online help than nonparticipants prefer online help.
A higher proportion of students who prefer online help participate in a club than students who prefer in-person help participate in a club.
A higher proportion of nonparticipants prefer online help than club participants prefer online help.
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.
A streaming service compared two recommendation layouts (Layout 1 vs Layout 2) and recorded whether users clicked a recommended title (Click/No click). The purpose is to compare the conditional proportions of clicking between layouts. Based on the two-way table, which comparison is supported?
Two-way table of Layout by Click (proportions of all users):
- Layout 1 & Click: 0.16, Layout 1 & No click: 0.24
- Layout 2 & Click: 0.18, Layout 2 & No click: 0.42
Layout 2 has a higher click rate because it has a larger total proportion (0.60) than Layout 1 (0.40).
More users clicked with Layout 2 because 0.18 > 0.16, so Layout 2 must have higher click probability.
A larger proportion of users shown Layout 1 clicked than users shown Layout 2.
The click rates are equal because both layouts have more “no click” than “click.”
Among users who clicked, a larger proportion used Layout 1 than Layout 2.
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.
A researcher compared two study strategies (Flashcards vs Practice problems) and recorded whether students passed an assessment (Pass/Fail). The purpose is to compare the conditional proportions of passing between strategies. Based on the two-way table, which comparison is supported?
Two-way table of Strategy by Result (proportions of all students):
- Flashcards & Pass: 0.26, Flashcards & Fail: 0.24
- Practice problems & Pass: 0.33, Practice problems & Fail: 0.17
A larger proportion of students using practice problems passed than students using flashcards.
The pass rates are the same because the overall pass proportion is $0.59$.
A larger proportion of students using flashcards passed than students using practice problems.
Among students who passed, a larger proportion used flashcards than practice problems.
More students passed using practice problems because 0.33 is greater than 0.26, so practice problems must have the higher pass rate.
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.
A gym recorded whether members attended a free class after receiving either a Phone call or an App notification. The purpose is to compare the conditional proportions of attending between contact methods. Based on the two-way table, which comparison is supported?
Two-way table of Contact Method by Class Attendance (proportions of all members):
- Phone & Attended: 0.12, Phone & Did not attend: 0.28
- App & Attended: 0.24, App & Did not attend: 0.36
Among members who attended, a larger proportion were contacted by Phone than by App.
A larger proportion of members contacted by App attended than members contacted by Phone.
Among members contacted by Phone, a smaller proportion did not attend than among members contacted by App.
More members attended after an App notification because 0.24 is twice 0.12, so the App group must be twice as large.
The attendance rate is the same because both methods have more “did not attend” than “attended.”
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).
Researchers surveyed commuters who either took the Bus or Drove a car and asked whether they were satisfied with their commute (Satisfied/Not satisfied). The purpose is to compare conditional proportions of satisfaction by commute type. Based on the two-way table, which comparison is supported?
Two-way table of Commute Type by Satisfaction (proportions of all commuters):
- Bus & Satisfied: 0.24, Bus & Not satisfied: 0.16
- Drive & Satisfied: 0.30, Drive & Not satisfied: 0.30
A larger proportion of drivers are satisfied than bus riders because 0.30 > 0.24.
A larger proportion of bus riders are satisfied than drivers.
More commuters are satisfied than not satisfied, so commute type does not matter.
Among commuters who are not satisfied, a smaller proportion drove than took the bus.
Among satisfied commuters, a larger proportion took the bus than drove.
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).
A school compared two ways to invite students to a tutoring session (Email vs Text) and recorded whether each student attended. The purpose is to compare the conditional proportions of attending between invitation methods. Based on the two-way table, which comparison is supported by the data?
Two-way table of Invitation Method by Attendance (proportions of all students):
- Email & Attended: 0.18, Email & Did not attend: 0.32
- Text & Attended: 0.22, Text & Did not attend: 0.28
Among students who attended, a larger proportion were invited by Email than by Text.
A larger proportion of Email-invited students did not attend than Text-invited students did not attend, so Email increases attendance.
More students attended from the Text group than from the Email group, so Text is more effective.
A larger proportion of students invited by Text attended than students invited by Email.
The proportion who attended is the same for Email and Text because 0.18 and 0.22 are close.
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.
A community center recorded whether adults enrolled in a fitness program (Enrolled/Not) and whether they met the weekly activity guideline (Met/Not). The purpose is to compare the guideline-meeting rates for enrolled vs not enrolled adults using conditional proportions. Which comparison is supported by the data?
Counts: Enrolled: Met 110, Not 90 (Total 200). Not enrolled: Met 66, Not 34 (Total 100).
A higher proportion of adults who met the guideline were enrolled than adults who did not meet the guideline were enrolled.
Enrolled and not enrolled adults have the same guideline-meeting rate.
Because more enrolled adults met the guideline (110) than not enrolled adults met the guideline (66), enrolled adults are more likely to meet it.
A higher proportion of not enrolled adults met the guideline than enrolled adults met the guideline.
A higher proportion of enrolled adults met the guideline than not enrolled adults met the guideline.
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.