Categorical Association - College Statistics

1. Which of the following is the most likely explanation for this apparent contradiction?
   Answer: The College of Engineering enrolls a much higher proportion of students with moderate incoming GPAs, who have a lower graduation rate overall. This is an example of Simpson's Paradox. The reversal can occur when a lurking variable (incoming GPA) is associated with both the explanatory variable (college) and the response variable (graduation rate). If the College of Engineering has a disproportionately large number of students from the group with a lower overall success rate (Moderate GPA students), its overall average can be dragged down below that of the College of Arts & Sciences, even if Engineering's rate is higher within each GPA group. Choice A incorrectly dismisses the more detailed data. Choice B is false; Simpson's Paradox is a real statistical phenomenon. Choice D describes a general feature of the data but doesn't explain the reversal itself.
2. A research firm surveyed 500 randomly selected adults about their primary source of news (Television, Online, Print) and their level of trust in media (Low, Medium, High). The firm wants to determine if there is an association between news source and trust level. Which of the following calculations would provide the most direct evidence to answer this question?
   Answer: Comparing the proportion of television viewers who have high trust to the proportion of online readers who have high trust. To assess an association between two categorical variables, we must compare the conditional distributions. Choice B directly compares the conditional probability of having high trust given the news source for two different news sources. If these proportions are different, it suggests an association. Choice A uses marginal totals and ignores the news source. Choice C calculates a single marginal proportion, which doesn't help assess association. Choice D compares two marginal distributions, which describes the overall sample but does not examine the relationship between the variables.
3. If two categorical variables, A and B, are perfectly independent, which of the following statements must be true for the two-way table of their frequencies?
   Answer: The proportion of observations in any given category of variable B is the same for every category of variable A. Independence between two categorical variables means that the conditional distribution of one variable is the same across all categories of the other variable. Choice B is a correct statement of this definition. For any category of B, $P(B|A_1) = P(B|A_2) = ... = P(B|A_k)$ for all categories $A_i$ of A. Choice D is a very strong condition that is almost never true unless all marginal distributions are uniform and the grand total is a specific number; it is not a requirement of independence. Choice A and C are not related to the definition of independence.
4. A survey asked employees about their job satisfaction and also included a 'Prefer not to answer' option. The table shows the results. When calculating the proportion of managers who are 'Satisfied', how should the 'Prefer not to answer' responses be handled for the most accurate representation of those with a stated opinion?
   Answer: They should be included in the denominator but not the numerator. The question asks for the proportion of managers who are 'Satisfied'. The total number of managers surveyed is the sum of all outcomes for that row: $60 + 30 + 10 = 100$. The number of satisfied managers is 60. The proportion is therefore $60/100 = 0.6$. The 'Prefer not to answer' group is a valid response group from the total sample of managers. Excluding them (Choice C, which would give $60/90$) would calculate the proportion of managers who expressed an opinion that are satisfied, which is a different question. The most standard approach is to treat all surveyed individuals as the base for the proportion, so the denominator should be the total number of managers surveyed. Thus, they are included in the denominator. They are clearly not in the numerator as they are not 'Satisfied'.
5. A marketing report states that among smartphone users, 60% use the brand's messaging app. It also states that among users of the brand's messaging app, 90% are smartphone users. A separate survey is planned to determine the proportion of smartphone users who use the brand's messaging app. Which piece of information from the report is directly relevant for this goal?
   Answer: The 60% figure, as it represents the conditional rate of app usage given smartphone ownership. The goal is to find the proportion of smartphone users who use the app. This is the conditional probability $P(\text{App User | Smartphone User})$. The first statement, "among smartphone users, 60% use the brand's messaging app," directly provides this value. The second statement provides $P(\text{Smartphone User | App User})$, which is the reverse conditional probability and not what is being sought. Therefore, only the 60% figure is directly relevant. Choices C and D are incorrect because the question asks for a value that is explicitly given in the report.
6. A segmented bar chart is created to visualize the relationship between a person's highest educational attainment (High School, Bachelor's, Graduate) and their preferred method of commuting to work (Drive, Public Transit, Other). What feature of the chart would indicate a lack of association between education level and commute method?
   Answer: The relative heights of the segments for each commuting method are approximately the same across all three education bars. A lack of association (independence) means the conditional distributions are the same. In a segmented bar chart, the relative height of each segment within a bar represents a conditional proportion. If these proportions are the same across all bars (i.e., for all education levels), it indicates that the distribution of commute method does not depend on education level. Choice A reflects unequal sample sizes for the education levels, not a lack of association. Choice B indicates that driving is the most popular mode in all groups, but the proportions could still differ, indicating an association. Choice C relates to marginal popularity, not association.
7. In a clinical trial with two treatments, A and B, an analyst computes a value $k = \frac{a/b}{c/d}$, where $a$ is the number of successes on A, $b$ is failures on A, $c$ is successes on B, and $d$ is failures on B. The analyst finds that $k > 1$. What is the correct interpretation of this result?
   Answer: The ratio of successes to failures is greater for Treatment A than it is for Treatment B. The value $k$ is the odds ratio. The numerator, $a/b$, is the odds of success for Treatment A. The denominator, $c/d$, is the odds of success for Treatment B. Therefore, $k$ is the ratio of these odds. If $k > 1$, it means the odds of success for Treatment A are greater than the odds of success for Treatment B, which is stated in choice B. Choice A compares probabilities ($a/(a+b)$ vs $c/(c+d)$), which is related but not the same as comparing odds. If $k>1$, A is also true, but B is the direct interpretation of the calculation given. Choice C compares raw counts, which can be misleading if group sizes differ. Choice D makes a causal claim, which is not justified by a statistical measure alone.
8. A company studies the effectiveness of a sales training program. They find that employees who attended the training have higher average sales than those who did not. However, it is also known that the training was offered only to employees who had been with the company for at least one year. Employee tenure is also positively associated with sales. In this context, what role does employee tenure play?
   Answer: It is a confounding variable, as it is associated with both the training and sales performance. When evaluating research studies, you need to identify variables that might create alternative explanations for observed relationships. This question tests your ability to recognize confounding variables—factors that are associated with both the treatment and the outcome, potentially explaining away what appears to be a causal relationship. Employee tenure is a confounding variable because it satisfies both conditions for confounding: it's linked to who receives the training (only employees with at least one year of tenure were eligible) and it's independently associated with the outcome (sales performance increases with tenure). This creates a problem because the higher sales among trained employees might be due to their longer tenure rather than the training itself. Looking at the wrong answers: Choice A incorrectly suggests tenure is a response variable affected by training, but tenure (time with company) cannot be changed by attending a training session. Choice B treats tenure as an independent variable controlled by researchers, but the researchers didn't manipulate tenure—they simply restricted training eligibility based on existing tenure levels. Choice D dismisses tenure as irrelevant, which ignores the fundamental principle that any variable associated with both treatment assignment and outcomes must be considered. The key study tip: whenever you see an observational study where treatment assignment isn't random, immediately ask "what other factors might differ between the treatment and control groups?" Variables that differ between groups AND affect the outcome are potential confounders that threaten the validity of causal conclusions.
9. The table shows observed counts for a sample of 200 people classified by two variables, X and Y. If these variables were perfectly independent, what would be the expected frequency for the cell corresponding to Category A and Category P?
   Answer: 37.5 To find the expected frequency for a cell under independence, we use the formula: $E = \frac{(\text{Row Total}) \times (\text{Column Total})}{(\text{Grand Total})}$. The row total for Category A is $30 + 70 = 100$. The column total for Category P is $30 + 45 = 75$. The grand total is 200. Expected frequency = $(100 \times 75) / 200 = 7500 / 200 = 75 / 2 = 37.5$. The observed frequency is 30, but the question asks for the expected frequency under independence.
10. A mosaic plot displays the relationship between a student's major (on the x-axis) and their intention to attend graduate school (on the y-axis). The width of the bar for 'Humanities' is twice the width of the bar for 'STEM'. Within the 'Humanities' bar, the segment for 'Yes' (intends to attend) is half the height of the segment for 'No'. Which of the following is a correct interpretation?
    Answer: There are twice as many Humanities majors as STEM majors in the sample, and one-third of Humanities majors plan to attend graduate school. When interpreting mosaic plots, you need to understand that bar widths represent the relative frequencies of categories, while segment heights within bars show conditional proportions. The Humanities bar being twice as wide as the STEM bar tells us there are twice as many Humanities majors as STEM majors in the sample. Within the Humanities bar, if the "Yes" segment is half the height of the "No" segment, this means the conditional proportion splits 1:2, so one-third of Humanities majors plan to attend graduate school while two-thirds do not. Choice A correctly captures both pieces of information: twice as many Humanities majors exist in the sample, and one-third of them plan graduate school. Choice B is incorrect because we have no information about the proportions within the STEM bar - we can't compare conditional proportions between majors without knowing how the STEM bar is divided. Choice C misinterprets what the plot shows. We know the relative size of each major and the conditional proportions within Humanities, but we can't determine what fraction of all "Yes" responses come from Humanities majors without additional information about STEM proportions. Choice D makes a specific numerical comparison that requires knowing the exact breakdown of both bars. While we know Humanities has twice the sample size and splits 1:2 for Yes:No, we don't know how STEM students are distributed between Yes and No responses. Remember: in mosaic plots, bar width = marginal frequency, segment height = conditional proportion within that category. Always distinguish between these two types of information.
11. The table shows the genre preference for a sample of 200 moviegoers, categorized by age group. Let $p_1$ be the proportion of teenagers who prefer Action, and let $p_2$ be the proportion of action movie fans who are teenagers. What are the values of $p_1$ and $p_2$?
    Answer: $p_1 = 0.375$ and $p_2 = 0.75$ This question tests the difference between $P(A|B)$ and $P(B|A)$. $p_1$ is the proportion of teenagers who prefer Action. This is $P(\text{Action | Teenager})$. The denominator is the total number of teenagers. $p_1 = 60 / (60 + 20) = 60 / 80 = 0.75$. $p_2$ is the proportion of action movie fans who are teenagers. This is $P(\text{Teenager | Action})$. The denominator is the total number of action fans. $p_2 = 60 / (60 + 100) = 60 / 160 = 6/16 = 3/8 = 0.375$. Therefore, the correct choice is D. The other choices swap the values or use incorrect denominators (like the grand total of 200).
12. The table displays data on the political affiliation and primary residence type for 1,000 registered voters. To investigate a possible association, a statistician compares the conditional distributions of residence type for each political affiliation. Which of the following comparisons provides the strongest evidence of an association between these two variables?
    Answer: Comparing the proportion of Party A members who live in apartments ($100/400$) with the proportion of Party B members who live in apartments ($200/600$). Evidence of an association comes from differences in the conditional distributions. Choice A compares $P(\text{Apartment | Party A}) = 100/400 = 0.25$ with $P(\text{Apartment | Party B}) = 200/600 \approx 0.33$. This is a valid comparison of conditional probabilities. The other necessary comparison is for houses: $P(\text{House | Party A}) = 300/400 = 0.75$ and $P(\text{House | Party B}) = 400/600 \approx 0.67$. Because $0.25 \neq 0.33$ (and $0.75 \neq 0.67$), there is evidence of an association. Choice B is a comparison of marginal counts, not conditional distributions. Choice C uses the wrong denominator (column total instead of row total), so it is not calculating valid conditional probabilities for the distribution of residence given party. Choice D compares two marginal proportions, which does not test for association.
13. The table shows the results of a clinical trial for a new medication. Patients were randomly assigned to either the medication or a placebo. Which statement accurately interprets the data regarding the medication's effectiveness?
    Answer: The medication was more effective because the proportion of patients who improved was 0.80, compared to 0.70 for the placebo. To evaluate effectiveness, we must compare the success rates (conditional probabilities), not the raw counts, because the group sizes are different. The proportion who improved with the medication is $P(\text{Improvement | Medication}) = 160 / (160 + 40) = 160/200 = 0.80$. The proportion who improved with the placebo is $P(\text{Improvement | Placebo}) = 210 / (210 + 90) = 210/300 = 0.70$. Since $0.80 > 0.70$, the medication was more effective in this trial. Distractor A makes the common error of comparing raw counts. Distractor C uses marginal totals, which is irrelevant for comparing the groups. Distractor D incorrectly claims the groups are incomparable; using proportions makes them comparable.
14. A researcher finds that a city's ice cream sales and drowning incidents are positively associated. A critic suggests that temperature is a confounding variable. For this criticism to be valid, which of the following must be true?
    Answer: Temperature must be associated with both ice cream sales and drowning incidents. For a variable to be a confounder, it must be associated with both the explanatory variable (ice cream sales) and the response variable (drowning incidents). In this classic example, high temperatures cause people to buy more ice cream and also cause more people to go swimming, which leads to more drowning incidents. Because temperature is associated with both, it confounds the relationship between them. If temperature were only associated with one of the variables, it would not be a confounder that could explain the observed association.
15. A study of 5,000 recent car accidents found a statistical association between the color of the car and the likelihood of being in an accident, with brightly colored cars being involved in slightly fewer accidents. The study was published with the headline, 'Want to Avoid Accidents? Paint Your Car Yellow!' Which of the following represents the most significant flaw in the headline's conclusion?
    Answer: The observed association may be due to confounding variables, and does not imply that car color is the cause of the difference in accident rates. The headline makes a causal claim ('Paint Your Car Yellow!' to 'Avoid Accidents'). The study only found an association. A fundamental principle of statistics is that association does not imply causation. There could be confounding variables. For instance, people who choose to buy brightly colored cars might also be safer, more cautious drivers. Choice D correctly identifies this flaw. Choice A is unlikely to be the most significant flaw, as 5,000 is a large sample. Choice C is a possibility, but the stem says an association was found. Choice B points to a potential methodological issue (sampling bias) but the primary flaw in the headline is the leap from association to causation.
16. The partially completed table shows the distribution of 400 products by supplier and defect status. If the supplier and defect status are known to be independent, how many products from Supplier B are expected to be defective?
    Answer: 36 First, complete the marginal totals. Total Defective = 60. Total from Supplier A = 160. Grand Total = 400. This means Total from Supplier B = 400 - 160 = 240. If the variables are independent, the expected count in any cell is (Row Total × Column Total) / (Grand Total). For the cell (Supplier B, Defective), the expected count is $(240 \times 60) / 400$. $(240 \times 60) / 400 = 14400 / 400 = 144 / 4 = 36$. Distractor C (48) comes from $(160 \times 120) / 400$, the wrong cell. Distractor A (30) might come from assuming the 60 defects are split proportionally to the non-defective items (120 vs 180 is a 2:3 ratio, 2/5 of 60 is 24, 3/5 is 36, so this is a potential path to the answer, but a misstep could lead elsewhere). Distractor D (60) is the total number of defective items.
17. The table summarizes the results of a survey asking students their primary year in college and whether they have an internship. What is the absolute difference between the proportion of juniors who have an internship and the proportion of sophomores who have an internship?
    Answer: 0.2 This requires calculating two conditional probabilities and finding their difference. First, the proportion of juniors with an internship: $P(\text{Internship | Junior}) = 75 / (75 + 75) = 75/150 = 0.50$. Second, the proportion of sophomores with an internship: $P(\text{Internship | Sophomore}) = 30 / (30 + 70) = 30/100 = 0.30$. The absolute difference is $|0.50 - 0.30| = 0.20$. Distractor A (0.15) might come from $75/500 - 30/500 = 0.09$ or a similar miscalculation. Distractor C (0.25) might result from miscalculating one of the proportions, for example $75/150 - 30/120 = 0.5-0.25=0.25$. Distractor D (0.40) might come from comparing the wrong groups or using wrong totals.
18. The table shows the results of a screening test for a certain disease relative to the actual disease status of 1000 individuals. What proportion of individuals who test positive are actually disease-free (a false positive)?
    Answer: 0.333 This question asks for a specific conditional probability: $P(\text{No Disease} | \text{Test Positive})$. First, find the total number of individuals who tested positive. This is the column total for 'Test Positive', which is $90 + 45 = 135$. This is our denominator. Next, find the number of individuals in that group who are actually disease-free. This is the value in the cell (No Disease, Test Positive), which is 45. This is our numerator. The proportion is $45 / 135 = 1/3 \approx 0.333$. Distractor A (0.050) is the false positive rate, $P(\text{Test Positive} | \text{No Disease}) = 45 / 900$. Distractor B (0.053) is $45/865$, using the wrong column total. Distractor D (0.950) is $855/900$, the specificity of the test.
19. Consider the table showing results for 100 patients. Suppose 10 patients who were originally in the (Treatment A, Success) cell were misclassified and actually belong in the (Treatment A, Failure) cell. How would correcting this error affect the evidence for an association between treatment and outcome?
    Answer: It would weaken the evidence for an association by decreasing the success rate difference between the treatments. Initially, $P(\text{Success|A}) = 40/50 = 0.8$ and $P(\text{Success|B}) = 25/50 = 0.5$. The difference is 0.3. After correction, the (Treatment A, Success) cell becomes $40 - 10 = 30$, and the (Treatment A, Failure) cell becomes $10 + 10 = 20$. The row total for Treatment A is still 50. The new success rate for A is $P(\text{Success|A}) = 30/50 = 0.6$. The success rate for B remains $25/50 = 0.5$. The new difference is $0.6 - 0.5 = 0.1$. Since the difference in conditional probabilities decreased from 0.3 to 0.1, the evidence for an association is weaker.
20. The table summarizes pet ownership and allergy status for a group of 300 office workers. What proportion of pet owners in this group report having allergies?
    Answer: 0.3 This question requires a multi-step calculation of a conditional probability. First, we must identify the subgroup of 'pet owners'. This includes both cat and dog owners. Total pet owners = (Total Cat Owners) + (Total Dog Owners) = 50 + 100 = 150. Second, we need the number of people within this subgroup who have allergies. This is (Cat Owners with Allergies) + (Dog Owners with Allergies) = 15 + 30 = 45. Finally, the proportion is $45 / 150 = 9/30 = 3/10 = 0.300$. Distractor A (0.150) is $45/300$, using the grand total as the denominator. Distractor C (0.500) is $150/300$, the proportion of all people who are pet owners. Distractor D (0.563) is $45/80$, the proportion of allergy sufferers who own a pet, which is $P(\text{Pet Owner | Allergies})$, not what was asked.

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