- A school surveyed students about grade level (9th or 10th) and whether they participate in a school club (Yes or No). The results are shown in a table. The principal wants to compare club participation between grade levels. Which comparison is appropriate for assessing the relationship between grade level and club participation using conditional distributions?
Answer: Compare the percent who participate in a club within 9th grade to the percent who participate in a club within 10th grade.
This question tests understanding of conditional distributions when analyzing relationships between two categorical variables. To assess whether grade level is associated with club participation, we need to compare the proportion of students who participate in clubs within each grade level - that is, the percent who participate in clubs among 9th graders versus the percent who participate in clubs among 10th graders. This is a conditional distribution because we're looking at club participation conditioned on grade level. Choice C correctly identifies this comparison. The other options either compare marginal distributions (A, E), raw counts (D), or condition on the wrong variable (B). When examining relationships between categorical variables, we compare conditional distributions by holding one variable constant and examining how the other variable's distribution changes.
- 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?
Answer: Compare the percent who participate (Yes) among 9th graders to the percent who participate (Yes) among 12th graders.
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.
- A researcher studied pet ownership (Owns a pet or Does not own a pet) and whether a person reports daily stress as high (High/Not high). Use the conditional distribution approach to describe the relationship.
Which comparison is appropriate?
Answer: Compare the percent with high stress among pet owners to the percent with high stress among non-owners.
This question examines the relationship between pet ownership and stress levels using conditional distributions. To determine if pet ownership is associated with stress levels, we need to condition on pet ownership and compare stress rates. Option D correctly compares the percent with high stress among pet owners to the percent with high stress among non-owners - this conditional comparison directly reveals whether stress levels differ by pet ownership status. Options A and E examine marginal distributions that don't address the relationship, option B uses counts rather than percentages, and option C reverses the conditioning by looking at pet ownership given stress level. When investigating potential benefits or associations, condition on the characteristic of interest (pet ownership) and examine how the outcome (stress) varies across its categories.
- 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?
<table>
<tr><th></th><th>Passed</th><th>Did not pass</th></tr>
<tr><td>Tutoring</td><td>48</td><td>12</td></tr>
<tr><td>No tutoring</td><td>60</td><td>30</td></tr>
</table>
Answer: Compare the proportion who passed among tutoring students to the proportion who passed among non-tutoring students.
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.
- 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?
Answer: Compare the percent earning an A/B among online students to the percent earning an A/B among in-person students.
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.
- A bookstore tracked 160 customers, recording whether each customer used a coupon and whether they bought at least one hardcover book. The two-way table is shown. Which comparison is appropriate for assessing whether hardcover purchase is associated with coupon use using conditional distributions?
<table>
<tr><th></th><th>Bought hardcover</th><th>Did not buy hardcover</th></tr>
<tr><td>Used coupon</td><td>28</td><td>52</td></tr>
<tr><td>No coupon</td><td>40</td><td>40</td></tr>
</table>
Answer: Compare the proportion who bought a hardcover among coupon users to the proportion who bought a hardcover among non-coupon users.
This question focuses on the skill of using conditional distributions for two categorical variables to assess association between coupon use and hardcover purchase. The appropriate method compares hardcover purchase proportions conditioned on coupon use: 28/80 = 0.35 for users versus 40/80 = 0.50 for non-users, suggesting an association. This aligns with conditioning on the row variable. Choice E is a distractor that conditions on purchase type and compares coupon use across those groups instead. Choice A distracts with overall coupon proportions, ignoring conditional aspects. Mini-lesson: To find conditional distributions, divide cell counts by the row or column subtotal to get percentages, then compare these across categories to see if one variable influences the other's distribution, indicating association if they differ.
- A streaming service recorded whether users have a Premium subscription and whether they primarily watch on a phone or a TV. The two categorical variables are Subscription type (Premium/Basic) and Primary device (Phone/TV). Using the two-way table, which comparison is appropriate for assessing whether subscription type is related to primary device?
Which comparison is appropriate?
Answer: Compare the percent who watch on a phone among Premium users to the percent who watch on a phone among Basic users.
This question requires identifying the correct conditional comparison to assess association between subscription type and primary device. To determine if subscription type is related to primary device, we should compare the conditional distribution of primary device given subscription type - specifically, the percent who watch on phones among Premium users versus the percent who watch on phones among Basic users. Option C correctly identifies this comparison. Option A reverses the conditioning by looking at subscription rates given device type, while option B examines marginal distributions that don't reveal associations. Options D and E compare counts rather than percentages, which fails to account for different group sizes. The key principle is to condition on the explanatory variable (subscription type) and compare the distribution of the response variable (primary device) across those conditions.
- 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?
Answer: Compare the percent that recycle regularly (Yes) among coastal households to the percent that recycle regularly (Yes) among inland households.
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.
- A fitness center recorded whether members attend morning classes or evening classes and whether they renewed their membership. The two categorical variables are Class time (Morning/Evening) and Renewal (Renewed/Did not renew). Using the two-way table, which comparison is appropriate for assessing whether class time is associated with renewal?
Which comparison is appropriate?
Answer: Compare the percent who renewed among morning-class members to the percent who renewed among evening-class members.
This question requires identifying the correct conditional comparison to assess association between class time preference and membership renewal. To determine if class time is associated with renewal, we should compare the conditional distribution of renewal given class time - specifically, the percent who renewed among morning-class members versus the percent who renewed among evening-class members. Option B correctly identifies this comparison. Option C reverses the conditioning by looking at class time preferences given renewal status, while option A examines marginal distributions that don't reveal associations. Options D and E compare counts rather than conditional percentages. The fundamental principle in analyzing associations is to condition on the explanatory variable (class time) and compare the response variable (renewal) across those conditions.
- A gym surveyed members about membership plan (Monthly or Annual) and whether they attend at least 3 times per week (Yes or No). The results are shown in a table. Which comparison is appropriate for assessing the relationship between membership plan and frequent attendance using conditional distributions?
Answer: Compare the percent who attend at least 3 times per week among Monthly-plan members to the percent who attend at least 3 times per week among Annual-plan members.
This question asks about comparing conditional distributions to assess the relationship between membership plan and gym attendance frequency. To determine if these variables are associated, we should compare the proportion of members who attend frequently within each membership plan - that is, the percent who attend at least 3 times per week among Monthly-plan members versus the percent who attend at least 3 times per week among Annual-plan members. Choice D correctly identifies this comparison of conditional distributions. Options A and E examine marginal distributions, C compares raw counts rather than proportions, and B conditions on attendance frequency instead of membership plan. The key is to condition on the explanatory variable (membership plan) and examine how the response variable (attendance frequency) differs across those conditions.
- 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?
Answer: Compare the percent who passed among students who completed the homework to the percent who passed among students who did not complete the homework.
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.
- 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?
Answer: Compare the percent with high satisfaction among remote employees to the percent with high satisfaction among in-office employees.
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.
- 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?
Answer: Compare the percent who arrived late (Yes) among car commuters to the percent who arrived late (Yes) among public-transit commuters.
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.
- A teacher recorded whether students attended after-school tutoring at least once and whether they passed the final exam. The two categorical variables are Tutoring attendance (Attended/Did not attend) and Final exam result (Passed/Did not pass). Using the two-way table, which comparison is appropriate for assessing whether tutoring is associated with passing?
Which comparison is appropriate?
Answer: Compare the percent who passed among those who attended tutoring to the percent who passed among those who did not attend tutoring.
This question requires identifying the appropriate comparison to assess whether tutoring attendance is associated with passing the exam. To evaluate this association, we should compare the conditional distribution of exam results given tutoring attendance - specifically, the percent who passed among those who attended tutoring versus the percent who passed among those who didn't attend. Option C correctly identifies this comparison. Option B reverses the conditioning by examining tutoring rates given exam results, while options A and D look at marginal distributions. Option E compares specific cell counts rather than conditional percentages, which doesn't properly assess association. The key principle is to condition on the explanatory variable (tutoring attendance) and compare the response variable (exam result) across those conditions.
- 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?
Answer: 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.
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.
- 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?
<table>
<tr><th></th><th>Support</th><th>Do not support</th></tr>
<tr><td>Commute by car</td><td>45</td><td>75</td></tr>
<tr><td>Do not commute by car</td><td>42</td><td>18</td></tr>
</table>
Answer: Compare the proportion who support among car commuters to the proportion who support among non-car commuters.
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.
- 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.
Answer: 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.
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.
- 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?
Answer: Compare the percent with high blood pressure (Yes) among smokers to the percent with high blood pressure (Yes) among nonsmokers.
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.
- A city tracked whether households have access to curbside recycling pickup and whether they report recycling at least weekly. The two categorical variables are Curbside access (Yes/No) and Recycles weekly (Yes/No). Based on the two-way table, which comparison is appropriate for assessing whether curbside access is associated with weekly recycling?
Which comparison is appropriate?
Answer: Compare the percent who recycle weekly among households with curbside access to the percent who recycle weekly among households without curbside access.
This question tests understanding of conditional distributions when assessing associations between curbside access and recycling behavior. To determine if curbside access is associated with weekly recycling, we need to compare the conditional distribution of recycling behavior given curbside access - that is, the percent who recycle weekly among households with curbside access versus the percent who recycle weekly among households without access. Option D correctly identifies this comparison. Option A reverses the conditioning (looking at access rates given recycling behavior), while option B examines marginal distributions. Options C and E compare counts rather than percentages, failing to account for different group sizes. When examining associations in two-way tables, always condition on the explanatory variable and compare the response variable's distribution.
- A clinic compared whether patients received a text-message reminder and whether they showed up for their appointment. The two categorical variables are Reminder (Text/No text) and Appointment outcome (Showed/No-show). Using the two-way table, which comparison is appropriate for assessing whether receiving a text reminder is associated with showing up?
Which comparison is appropriate?
Answer: Compare the percent who showed up among patients who received a text to the percent who showed up among patients who did not receive a text.
This question requires identifying the appropriate comparison to assess whether text reminders are associated with appointment attendance. To evaluate this association, we should compare the conditional distribution of appointment outcomes given reminder status - specifically, the percent who showed up among patients who received texts versus the percent who showed up among patients who didn't receive texts. Option D correctly identifies this comparison. Option B reverses the conditioning by examining reminder rates given appointment outcome, while option C looks at marginal distributions that don't reveal associations. Options A and E compare counts rather than percentages, failing to account for potentially different group sizes. The fundamental principle is to condition on the explanatory variable (reminder status) and compare the response variable (appointment outcome) across those conditions.
