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This quiz focuses on Representing Two Categorical Variables, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Statistics.
AP Statistics Quiz: Representing Two Categorical Variables
20 questions · exam conditions
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Representing Two Categorical VariablesQuestion 1 of 20
A school surveyed 200 students about whether they participated in a school sport this year and whether they reported sleeping at least 8 hours on most school nights. The two-way table shows the results. Which comparison is appropriate for assessing whether sleep amount is associated with sports participation using conditional distributions?
| $\ge 8$ hours | $<8$ hours | |
|---|---|---|
| Plays a sport | 54 | 46 |
| Does not play a sport | 38 | 62 |
ACompare the overall proportion who sleep ≥8 hours to the overall proportion who sleep <8 hours.
BCompare the proportion who play a sport among those who sleep ≥8 hours to the proportion who play a sport among those who sleep <8 hours.
CCompare the proportion who sleep ≥8 hours among those who play a sport to the proportion who sleep ≥8 hours among those who do not play a sport.
DCompare the counts of students who sleep ≥8 hours in the two sport categories.
ECompare the overall proportion who play a sport to the overall proportion who do not play a sport.
AP Statistics Quiz
AP Statistics Quiz: Representing Two Categorical Variables
Practice Representing Two Categorical Variables in AP Statistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.
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.
All questions
Question 1
A school surveyed 200 students about whether they participated in a school sport this year and whether they reported sleeping at least 8 hours on most school nights. The two-way table shows the results. Which comparison is appropriate for assessing whether sleep amount is associated with sports participation using conditional distributions?
| $\ge 8$ hours | $<8$ hours | |
|---|---|---|
| Plays a sport | 54 | 46 |
| Does not play a sport | 38 | 62 |
- Compare the overall proportion who sleep ≥8 hours to the overall proportion who sleep <8 hours.
- Compare the proportion who play a sport among those who sleep ≥8 hours to the proportion who play a sport among those who sleep <8 hours.
- Compare the proportion who sleep ≥8 hours among those who play a sport to the proportion who sleep ≥8 hours among those who do not play a sport. (correct answer)
- Compare the counts of students who sleep ≥8 hours in the two sport categories.
- Compare the overall proportion who play a sport to the overall proportion who do not play a sport.
Explanation: This question assesses the skill of representing two categorical variables, specifically using conditional distributions to evaluate the association between sports participation and sleep amount. The appropriate comparison is to condition on sports participation and compare the proportions of students sleeping at least 8 hours across the two sports categories, as this reveals whether the distribution of sleep differs by sports group. For example, the conditional proportion of >=8 hours given plays a sport is 54/100 = 0.54, while given does not play is 38/100 = 0.38, indicating an association. A common distractor is choice B, which conditions on sleep instead and compares sports participation across sleep groups, but the question's setup with sports as rows suggests conditioning on sports for the comparison. Another distractor is choice A, which uses overall proportions without conditioning, failing to assess association properly. In a mini-lesson on conditional distributions: these are found by dividing cell counts by the relevant row or column total, allowing us to examine how one variable's outcomes vary depending on the category of the other variable, which is key for detecting associations in two-way tables.
Question 2
A university sampled 220 students, recording whether each student lives on campus and whether they have a meal plan. The two-way table shows the results. Which comparison is appropriate for assessing whether having a meal plan is associated with living on campus using conditional distributions?
| Has meal plan | No meal plan | |
|---|---|---|
| On campus | 88 | 22 |
| Off campus | 44 | 66 |
- Compare the overall proportion with a meal plan to the overall proportion without a meal plan.
- Compare the proportion living on campus among those with a meal plan to the proportion living on campus among those without a meal plan.
- Compare the counts of students with a meal plan in the on-campus and off-campus groups.
- Compare the proportion with a meal plan among on-campus students to the proportion with a meal plan among off-campus students. (correct answer)
- Compare the overall proportion living on campus to the overall proportion living off campus.
Explanation: The skill involved is representing two categorical variables with conditional distributions to assess association between living situation and meal plan status. The appropriate comparison conditions on living situation and compares meal plan proportions: 88/110 = 0.80 for on-campus versus 44/110 = 0.40 for off-campus, showing association. This aligns with row conditioning. Distractor choice B reverses by conditioning on meal plan and comparing living situation. Choice A uses overall meal plan proportions without conditioning. Mini-lesson: Conditional distributions are row or column percentages (cell divided by marginal), used to compare how one variable's distribution differs across categories of the other, with differences evidencing association.
Question 3
A streaming service sampled users and recorded whether they have a premium subscription (Premium/Standard) and whether they watched at least 10 hours last week (10+ hours/Under 10). The two-way table summarizes the sample. Which comparison is appropriate for assessing whether weekly viewing level is associated with subscription type?
Categorical variables: Subscription type and Weekly viewing level.
- Compare the percent premium among 10+ hour viewers to the percent premium among under-10-hour viewers, because viewing level should be the explanatory variable.
- Compare the overall percent premium to the overall percent standard.
- Compare the percent 10+ hours among premium users to the percent 10+ hours among standard users. (correct answer)
- Compare the marginal percent 10+ hours to the marginal percent under 10 hours.
- Compare the counts of 10+ hour viewers in each subscription group, without using conditional percents.
Explanation: This question evaluates the skill of representing two categorical variables, subscription type and weekly viewing level, via conditional distributions to check for association. The suitable comparison is the percent of 10+ hours viewers among premium users versus among standard users, conditioning viewing level on subscription type as the explanatory variable. This fits the question's inquiry into whether viewing level is associated with subscription type, suggesting we examine viewing distributions given subscription. Choice A is a typical distractor, reversing the conditioning to treat viewing as explanatory, which contradicts the question's phrasing. For a mini-lesson, conditional distributions compute the proportions of one variable's categories within levels of another; compare percent 10+ hours for premium and standard groups. Significant differences in these percentages signal an association. Proper identification of the explanatory variable is key to selecting the right conditional comparison.
Question 4
A company surveyed employees about work arrangement (Remote or In-office) and whether they report high job satisfaction (High/Not high). Use conditional distributions to describe the association between work arrangement and job satisfaction.
Which comparison is appropriate?
- Compare the overall percent of employees who work remotely to the overall percent who work in-office.
- Compare the overall percent with high satisfaction to the overall percent without high satisfaction.
- Compare the percent with high satisfaction among remote employees to the percent with high satisfaction among in-office employees. (correct answer)
- Compare the percent who are remote among those with high satisfaction to the percent who are remote among those without high satisfaction.
- Compare the number of remote employees with high satisfaction to the number of in-office employees with high satisfaction.
Explanation: This question requires identifying the appropriate conditional distribution to examine the association between work arrangement and job satisfaction. To determine if work arrangement affects satisfaction, we need to condition on work arrangement and compare satisfaction rates. Option C correctly compares the percent with high satisfaction among remote employees to the percent with high satisfaction among in-office employees - this conditional comparison directly reveals whether satisfaction differs by work arrangement. Options A and B examine marginal distributions that don't address the relationship, while option D reverses the conditioning by looking at work arrangement given satisfaction level. Option E uses counts rather than percentages, failing to account for potentially different group sizes. The principle remains consistent: condition on the explanatory variable (work arrangement) and compare the response variable (satisfaction) across its categories.
Question 5
A wildlife center recorded whether rescued birds were treated with a new rehabilitation protocol (New/Standard) and whether they were successfully released (Released/Not Released). The table summarizes outcomes. Which comparison is appropriate for assessing whether release success is associated with protocol type?
Categorical variables: Protocol type and Release outcome.
- Compare the percent released among birds treated with the new protocol to the percent released among birds treated with the standard protocol. (correct answer)
- Compare the overall percent released to the overall percent not released.
- Compare the percent treated with the new protocol among released birds to the percent treated with the new protocol among not released birds, because release outcome should be the explanatory variable.
- Compare the marginal totals for New and Standard protocols.
- Compare the counts released for the new protocol to the counts released for the standard protocol.
Explanation: This question assesses representing two categorical variables, protocol type and release outcome, using conditional distributions to check association. The fitting comparison is the percent released among new protocol birds versus standard, conditioning outcome on protocol as explanatory. This suits assessing if release success is associated with protocol type, suggesting protocol affects release. Choice C distracts by flipping variables, conditioning protocol on outcome and assuming outcome is explanatory. As a mini-lesson, conditional distributions find response percentages within explanatory levels; release rates for new and standard groups. Varied rates suggest association. Question intent directs the conditioning choice.
Question 6
A city collected data from 180 residents on whether they commute primarily by car or not and whether they support building more bike lanes. The results are shown. Which comparison is appropriate for assessing whether support for bike lanes is associated with commuting method using conditional distributions?
| Support | Do not support | |
|---|---|---|
| Commute by car | 45 | 75 |
| Do not commute by car | 42 | 18 |
- Compare the overall proportion who support to the overall proportion who do not support.
- Compare the proportion who commute by car among supporters to the proportion who commute by car among non-supporters.
- Compare the counts who support in the car and non-car groups.
- Compare the overall proportion who commute by car to the overall proportion who do not commute by car.
- Compare the proportion who support among car commuters to the proportion who support among non-car commuters. (correct answer)
Explanation: The skill here involves representing two categorical variables through conditional distributions to assess the association between commuting method and support for bike lanes. The appropriate comparison conditions on commuting method and examines the proportions supporting bike lanes, like 45/120 = 0.375 for car commuters versus 42/60 = 0.70 for non-car, suggesting an association. This aligns with conditioning on the row variable, commuting method. Choice B is a distractor that conditions on support instead, comparing car commuting across support groups, which is the reverse perspective. Choice A distracts by using overall support proportions without conditioning, missing the relational aspect. Mini-lesson: Conditional distributions normalize data within rows or columns (e.g., row percentages) to compare how the distribution of one variable shifts across levels of the other, providing evidence of association if the proportions differ notably.
Question 7
A retailer tracked whether customers used a coupon (Coupon/No Coupon) and whether they made a purchase (Purchase/No Purchase). The table summarizes the data from one weekend. Which comparison is appropriate for assessing whether making a purchase is associated with coupon use?
Categorical variables: Coupon use and Purchase outcome.
- Compare the percent who used a coupon among purchasers to the percent who used a coupon among non-purchasers, because purchase outcome must be the explanatory variable.
- Compare the overall percent purchase to the overall percent no purchase.
- Compare the marginal counts in the Coupon and No Coupon rows.
- Compare the percent who made a purchase among coupon users to the percent who made a purchase among non-coupon users. (correct answer)
- Compare the counts of purchasers in the coupon and no-coupon groups, without using conditional percents.
Explanation: This question probes the skill of representing two categorical variables, coupon use and purchase outcome, using conditional distributions for association. The proper comparison is the percent who made a purchase among coupon users versus non-users, conditioning purchase on coupon as explanatory. This aligns with assessing if making a purchase is associated with coupon use, implying coupon influences purchase. Choice A is a distractor that reverses, conditioning coupon on purchase and assuming purchase is explanatory. As a mini-lesson, conditional distributions compute response percentages within explanatory levels; purchase rates for coupon and no-coupon. Differences suggest association. Intent from wording determines conditioning.
Question 8
A company surveyed 250 employees about whether they work remotely at least 3 days per week and whether they report being satisfied with work-life balance. The results are shown. Which comparison is appropriate for assessing whether satisfaction is associated with remote-work status using conditional distributions?
| Satisfied | Not satisfied | |
|---|---|---|
| Remote $\ge 3$ days | 90 | 35 |
| Remote $<3$ days | 72 | 53 |
- Compare the overall proportion satisfied to the overall proportion not satisfied.
- Compare the counts of satisfied employees in the two remote-work categories.
- Compare the proportion satisfied among employees remote ≥3 days to the proportion satisfied among employees remote <3 days. (correct answer)
- Compare the proportion remote ≥3 days among satisfied employees to the proportion remote ≥3 days among not-satisfied employees.
- Compare the overall proportion remote ≥3 days to the overall proportion remote <3 days.
Explanation: This question tests the skill of using conditional distributions for two categorical variables to assess association between remote-work status and satisfaction. The proper comparison conditions on remote status and compares satisfaction proportions: 90/125 = 0.72 for >=3 days versus 72/125 = 0.576 for <3 days, indicating association. This conditions on the row variable. Choice D distracts by conditioning on satisfaction and comparing remote status across those groups. Choice A uses overall satisfaction without conditioning, a common mistake. Mini-lesson: Calculate conditional distributions as proportions within each row or column to examine the conditional behavior of one variable given the other, where varying distributions suggest the variables are associated.
Question 9
A health clinic asked 210 patients whether they received a flu shot this season and whether they reported having the flu in the past year. The table summarizes responses. Which comparison is appropriate for evaluating whether flu incidence is associated with receiving a flu shot using conditional distributions?
| Had flu | Did not have flu | |
|---|---|---|
| Flu shot | 24 | 96 |
| No flu shot | 36 | 54 |
- Compare the proportion who had the flu among those who got a flu shot to the proportion who had the flu among those who did not get a flu shot. (correct answer)
- Compare the overall proportion who had the flu to the overall proportion who did not have the flu.
- Compare the counts who had the flu in the flu-shot and no-flu-shot groups.
- Compare the proportion who got a flu shot among those who had the flu to the proportion who got a flu shot among those who did not have the flu.
- Compare the overall proportion who got a flu shot to the overall proportion who did not get a flu shot.
Explanation: The skill here is representing two categorical variables via conditional distributions to evaluate association between flu shot receipt and flu incidence. The correct comparison conditions on flu shot status and contrasts flu proportions: 24/120 = 0.20 for shot versus 36/90 = 0.40 for no shot, revealing an association. This follows the row-based conditioning in the table. Distractor choice D reverses it by conditioning on flu status and comparing shot receipt. Choice B uses overall flu proportions without conditioning, missing the association check. Mini-lesson: Conditional distributions are obtained by normalizing within rows or columns (cell/row total), allowing comparison of how one variable's proportions change across the other's categories, with discrepancies pointing to association.
Question 10
A streaming service sampled 300 users and recorded whether each user has a premium subscription and whether they usually watch on a mobile device. The table summarizes the data. Which comparison is appropriate for evaluating the relationship between subscription type and device preference using conditional distributions?
| Usually mobile | Usually not mobile | |
|---|---|---|
| Premium | 72 | 48 |
| Free | 96 | 84 |
- Compare the proportion who are premium among mobile users to the proportion who are premium among non-mobile users.
- Compare the overall proportion of premium users to the overall proportion of free users.
- Compare the counts of mobile users in the premium and free groups.
- Compare the proportion who usually watch on mobile among premium users to the proportion who usually watch on mobile among free users. (correct answer)
- Compare the overall proportion who usually watch on mobile to the overall proportion who usually do not watch on mobile.
Explanation: This question targets the skill of representing two categorical variables by using conditional distributions to investigate the relationship between subscription type and device preference. The correct approach compares the conditional proportions of usually watching on mobile across premium and free subscribers, such as 72/120 = 0.60 for premium versus 96/180 = 0.53 for free, highlighting potential association. This conditions on subscription type, aligning with the table's row structure. A frequent distractor is choice A, which reverses the conditioning by comparing premium proportions across device groups, but the setup emphasizes conditioning on subscription. Choice B is another distractor, relying on overall proportions without any conditioning, which does not reveal associations. Mini-lesson on conditional distributions: compute them by dividing each cell by its row total (or column, depending on the focus) to see the percentage distribution of one variable within categories of the other, enabling assessment of whether the variables are independent or associated.
Question 11
A university compared course format (Online or In-person) with whether students earned an A or B (A/B or Below B). Use conditional distributions to describe the association between course format and earning an A/B.
Which comparison is appropriate?
- Compare the overall percent of students in online courses to the overall percent in in-person courses.
- Compare the percent earning an A/B among online students to the percent earning an A/B among in-person students. (correct answer)
- Compare the percent who are online among those earning an A/B to the percent who are online among those below B.
- Compare the overall percent earning an A/B to the overall percent below B.
- Compare the number of students earning an A/B in online courses to the number earning an A/B in in-person courses.
Explanation: This question requires understanding conditional distributions to examine the association between course format and academic performance. To determine if course format affects grades, we need to condition on course format and compare grade distributions. Option B correctly compares the percent earning an A/B among online students to the percent earning an A/B among in-person students - this conditional comparison directly reveals whether success rates differ by course format. Option A examines marginal distributions of course formats, option C reverses the conditioning by looking at format given grades, and option D looks at marginal grade distributions. Option E uses counts rather than percentages, which doesn't control for different enrollment sizes. When studying educational outcomes, condition on the instructional variable (course format) and compare the performance variable (grades) across its categories.
Question 12
A teacher recorded homework completion (Completed or Not completed) and whether the student passed the unit test (Pass/Fail). To interpret the relationship using conditional distributions, identify the correct conditional comparison.
Which comparison is appropriate?
- Compare the number of students who passed to the number who failed.
- Compare the percent who passed among students who completed the homework to the percent who passed among students who did not complete the homework. (correct answer)
- Compare the overall percent who completed the homework to the overall percent who did not.
- Compare the percent who completed the homework among students who passed to the percent who completed the homework among students who failed.
- Compare the number who completed the homework and passed to the number who did not complete the homework and passed.
Explanation: This question tests identifying the correct conditional distribution to analyze the relationship between homework completion and test performance. To determine if homework completion is associated with passing rates, we should condition on homework completion status and compare pass rates. Option B correctly compares the percent who passed among students who completed homework to the percent who passed among those who didn't - this directly shows whether completion affects passing rates. Option A uses counts without considering group sizes, option C examines marginal distributions of homework completion, and option D reverses the natural conditioning by looking at homework rates given test results. Option E compares counts within just the passing group. When studying cause-and-effect relationships, condition on the potential cause (homework completion) and examine how the effect (passing) varies.
Question 13
A health clinic recorded whether patients received a flu shot this season (Shot/No Shot) and whether they reported getting the flu (Flu/No Flu). The two-way table shows the results. Which comparison is appropriate for assessing whether getting the flu is associated with receiving a flu shot?
Categorical variables: Flu shot status and Flu occurrence.
- Compare the overall percent who got the flu to the overall percent who did not get the flu.
- Compare the marginal totals for Shot and No Shot.
- Compare the percent who got the flu among those who received a shot to the percent who got the flu among those who did not receive a shot. (correct answer)
- Compare the percent who received a shot among those who got the flu to the percent who received a shot among those who did not get the flu, because flu status should be the explanatory variable.
- Compare the counts who got the flu between shot and no-shot groups.
Explanation: This question tests representing two categorical variables, flu shot status and flu occurrence, with conditional distributions to assess association. The suitable comparison is the percent who got the flu among shot recipients versus non-recipients, conditioning flu on shot status as explanatory. This fits evaluating if getting the flu is associated with receiving a shot, suggesting shot affects flu risk. Choice D distracts by inverting, conditioning shot on flu and deeming flu explanatory. For a mini-lesson, conditional distributions are response proportions within explanatory groups; flu rates in shot and no-shot. Disparities indicate association. Question structure guides explanatory selection.
Question 14
A teacher recorded whether 150 students attended after-school tutoring at least once this month and whether they passed the most recent quiz. The two-way table is shown. Which comparison is appropriate for determining whether quiz outcome is associated with tutoring attendance using conditional distributions?
| Passed | Did not pass | |
|---|---|---|
| Tutoring | 48 | 12 |
| No tutoring | 60 | 30 |
- Compare the proportion who attended tutoring among those who passed to the proportion who attended tutoring among those who did not pass.
- Compare the overall proportion who passed to the overall proportion who did not pass.
- Compare the proportion who passed among tutoring students to the proportion who passed among non-tutoring students. (correct answer)
- Compare the counts of students who passed in the tutoring and no-tutoring groups.
- Compare the overall proportion who attended tutoring to the overall proportion who did not attend tutoring.
Explanation: This question evaluates the skill of using conditional distributions for two categorical variables to determine association between tutoring attendance and quiz outcome. The proper comparison conditions on tutoring and contrasts passing proportions: 48/60 = 0.80 for tutoring versus 60/90 = 0.667 for no tutoring, indicating a possible link. This follows the table's structure by conditioning on rows. Choice A distracts by conditioning on outcome and comparing tutoring across pass/fail groups, reversing the typical explanatory-response dynamic. Choice B uses overall passing proportions without conditioning, a common error that overlooks associations. Mini-lesson on conditional distributions: They are percentages within a row or column, calculated as cell value divided by marginal total, allowing us to observe if one variable's outcomes depend on the other's categories, with differences suggesting association.
Question 15
A school surveyed students about grade level (9th or 12th) and whether they participate in an after-school club (Yes/No). Use the two-way table to decide which comparison is appropriate for describing the relationship between grade level and club participation using conditional distributions.
Which comparison is appropriate?
- Compare the overall percent of students who are 9th grade to the overall percent who are 12th grade.
- Compare the percent who participate (Yes) among 9th graders to the percent who participate (Yes) among 12th graders. (correct answer)
- Compare the percent who are 9th grade among club participants (Yes) to the percent who are 9th grade among nonparticipants (No).
- Compare the overall percent who participate (Yes) to the overall percent who do not (No).
- Compare the number of club participants (Yes) in 9th grade to the number of club participants (Yes) in 12th grade.
Explanation: This question tests understanding of conditional distributions when analyzing the relationship between two categorical variables: grade level and club participation. To examine if there's an association between these variables, we need to compare the participation rates across different grade levels. The correct approach (B) compares the percent who participate among 9th graders to the percent who participate among 12th graders - this is a conditional distribution that shows how participation varies by grade level. Option A incorrectly compares marginal distributions of grades, while options C and D examine different conditioning variables or marginal distributions that don't directly address the relationship. When analyzing associations between categorical variables, always condition on one variable (here, grade level) and compare the distribution of the other variable (participation) across those conditions.
Question 16
An environmental group recorded region (Coastal or Inland) and whether households recycle regularly (Yes/No). To interpret the relationship using conditional distributions, focus on comparing conditional percentages across regions.
Which comparison is appropriate?
- Compare the overall percent of households that are coastal to the overall percent that are inland.
- Compare the overall percent that recycle regularly (Yes) to the overall percent that do not (No).
- Compare the percent that recycle regularly (Yes) among coastal households to the percent that recycle regularly (Yes) among inland households. (correct answer)
- Compare the percent that are coastal among households that recycle regularly (Yes) to the percent that are coastal among households that do not (No).
- Compare the number of coastal households that recycle regularly (Yes) to the number of inland households that recycle regularly (Yes).
Explanation: This question tests identifying the correct conditional distribution to analyze the relationship between region and recycling behavior. To assess whether region is associated with recycling habits, we should condition on region and compare recycling rates. Option C correctly compares the percent that recycle regularly among coastal households to the percent among inland households - this conditional comparison directly shows if recycling behavior differs by region. Options A and B examine marginal distributions that don't reveal the relationship, while option D reverses the conditioning by looking at region given recycling behavior. Option E uses counts instead of percentages, failing to account for potentially different population sizes in each region. The key insight: condition on the explanatory variable (region) and compare the response variable (recycling) across its categories.
Question 17
A city collected data on commute method (Car or Public transit) and whether the commuter arrived late at least once last month (Yes/No). To describe the association using conditional distributions, focus on comparing conditional percentages.
Which comparison is appropriate?
- Compare the overall percent who commute by car to the overall percent who use public transit.
- Compare the percent who commute by car among those who were late (Yes) to the percent who commute by car among those who were not late (No).
- Compare the percent who arrived late (Yes) among car commuters to the percent who arrived late (Yes) among public-transit commuters. (correct answer)
- Compare the overall percent who arrived late (Yes) to the overall percent who did not (No).
- Compare the number of late commuters (Yes) who use public transit to the number of on-time commuters (No) who use public transit.
Explanation: This question tests understanding of conditional distributions for examining the association between commute method and punctuality. To determine if commute method affects lateness, we should condition on commute method and compare lateness rates. Option C correctly compares the percent who arrived late among car commuters to the percent who arrived late among public-transit commuters - this directly shows whether lateness rates differ by transportation mode. Option A compares marginal distributions of commute methods, while option B reverses the conditioning by looking at commute methods given lateness status. Options D and E examine marginal distributions or counts that don't reveal the relationship. When investigating associations, identify which variable is explanatory (commute method) and which is the response (lateness), then condition on the explanatory variable.
Question 18
A hospital reviewed patients by smoking status (Smoker or Nonsmoker) and whether they had high blood pressure (Yes/No). Use conditional distributions to evaluate the relationship between smoking status and high blood pressure.
Which comparison is appropriate?
- Compare the overall percent of patients who are smokers to the overall percent who are nonsmokers.
- Compare the percent who are smokers among those with high blood pressure (Yes) to the percent who are smokers among those without (No).
- Compare the overall percent with high blood pressure (Yes) to the overall percent without (No).
- Compare the percent with high blood pressure (Yes) among smokers to the percent with high blood pressure (Yes) among nonsmokers. (correct answer)
- Compare the number of patients with high blood pressure (Yes) who are smokers to the number with high blood pressure (Yes) who are nonsmokers.
Explanation: This question examines the relationship between smoking status and high blood pressure using conditional distributions. To assess whether smoking is associated with blood pressure, we need to condition on smoking status and compare blood pressure rates. Option D correctly compares the percent with high blood pressure among smokers to the percent with high blood pressure among nonsmokers - this conditional comparison directly reveals if blood pressure rates differ by smoking status. Options A and C examine marginal distributions that don't address the relationship, while option B reverses the conditioning by looking at smoking rates given blood pressure status. Option E uses counts instead of percentages, which doesn't properly control for different group sizes. The key principle: condition on the potential explanatory variable (smoking) and compare the distribution of the outcome (blood pressure) across its categories.
Question 19
A streaming service sampled users and recorded subscription type (Basic or Premium) and whether the user watched at least 10 hours last week (Yes/No). Use conditional distributions to assess the relationship.
Which comparison is appropriate?
- Compare the percent who watched at least 10 hours (Yes) among Basic users to the percent who watched at least 10 hours (Yes) among Premium users. (correct answer)
- Compare the overall percent of users who are Basic to the overall percent who are Premium.
- Compare the percent who are Premium among those who watched at least 10 hours (Yes) to the percent who are Premium among those who did not (No).
- Compare the overall percent who watched at least 10 hours (Yes) to the overall percent who did not (No).
- Compare the number of Premium users who watched at least 10 hours (Yes) to the number of Basic users who watched at least 10 hours (Yes).
Explanation: This question requires identifying the appropriate conditional distribution to analyze the relationship between subscription type and viewing habits. The goal is to determine if subscription type is associated with watching behavior, so we should condition on subscription type and compare viewing rates. Option A correctly compares the percent who watched at least 10 hours among Basic users to the percent among Premium users - this conditional comparison directly reveals whether viewing behavior differs by subscription type. Option B compares marginal distributions of subscription types, which doesn't address the relationship. Option C conditions on viewing behavior instead, while options D and E use marginal distributions or counts rather than conditional percentages. Remember: to study how one variable relates to another, condition on the explanatory variable and compare the response variable's distribution across categories.
Question 20
A gym tracked members by membership type (Monthly or Annual) and whether they attended at least 8 times in the last month (Yes/No). To interpret the relationship using conditional distributions, choose the correct comparison.
Which comparison is appropriate?
- Compare the percent who attended at least 8 times (Yes) among Monthly members to the percent who attended at least 8 times (Yes) among Annual members. (correct answer)
- Compare the overall percent who are Monthly members to the overall percent who are Annual members.
- Compare the overall percent who attended at least 8 times (Yes) to the overall percent who did not (No).
- Compare the percent who are Annual members among those who attended at least 8 times (Yes) to the percent who are Annual members among those who did not (No).
- Compare the number of Monthly members who attended at least 8 times (Yes) to the number of Annual members who attended at least 8 times (Yes).
Explanation: This question tests understanding of conditional distributions when analyzing the relationship between membership type and gym attendance. To assess whether membership type is associated with attendance frequency, we should condition on membership type and compare attendance rates. Option A correctly compares the percent who attended at least 8 times among Monthly members to the percent among Annual members - this conditional comparison directly shows if attendance patterns differ by membership type. Options B and C examine marginal distributions that don't reveal the relationship, while option D reverses the conditioning by looking at membership type given attendance. Option E uses counts instead of percentages, which doesn't properly control for different group sizes. Remember: to study associations between categorical variables, condition on one variable and compare the distribution of the other across those conditions.