College Statistics Quiz: Categorical Association
20 questions · exam conditions
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Categorical AssociationQuestion 1 of 20

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?

Temperature must be negatively associated with either ice cream sales or drowning incidents.
Temperature must be associated with ice cream sales, but not necessarily with drowning incidents.
Temperature must be associated with drowning incidents, but not necessarily with ice cream sales.
Temperature must be associated with both ice cream sales and drowning incidents.
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College Statistics Quiz

College Statistics Quiz: Categorical Association

Practice Categorical Association in College Statistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Categorical Association, giving you a quick way to practice the rules, question types, and explanations that matter most for College Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

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?

  1. Temperature must be negatively associated with either ice cream sales or drowning incidents.
  2. Temperature must be associated with ice cream sales, but not necessarily with drowning incidents.
  3. Temperature must be associated with drowning incidents, but not necessarily with ice cream sales.
  4. Temperature must be associated with both ice cream sales and drowning incidents. (correct answer)
Explanation: 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.

Question 2

In a clinical trial with two treatments, A and B, an analyst computes a value k=a/bc/dk = \frac{a/b}{c/d}, where aa is the number of successes on A, bb is failures on A, cc is successes on B, and dd is failures on B. The analyst finds that k>1k > 1. What is the correct interpretation of this result?

  1. The probability of success with Treatment A is greater than the probability of success with Treatment B.
  2. The ratio of successes to failures is greater for Treatment A than it is for Treatment B. (correct answer)
  3. The total number of successes with Treatment A is greater than the total number of successes with Treatment B.
  4. Treatment A is the cause of a higher success rate compared to Treatment B.
Explanation: The value kk is the odds ratio. The numerator, a/ba/b, is the odds of success for Treatment A. The denominator, c/dc/d, is the odds of success for Treatment B. Therefore, kk is the ratio of these odds. If k>1k > 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)a/(a+b) vs c/(c+d)c/(c+d)), which is related but not the same as comparing odds. If k>1k>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.

Question 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?

  1. The number of categories for variable A must be equal to the number of categories for variable B.
  2. The proportion of observations in any given category of variable B is the same for every category of variable A. (correct answer)
  3. The marginal distribution for variable A must be identical to the marginal distribution for variable B.
  4. The frequency count in each cell of the table must be identical.
Explanation: 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∣A1)=P(B∣A2)=...=P(B∣Ak)P(B|A_1) = P(B|A_2) = ... = P(B|A_k) for all categories AiA_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.

Question 4

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?

  1. The 60% figure, as it represents the conditional rate of app usage given smartphone ownership. (correct answer)
  2. The 90% figure, as it represents the conditional rate of smartphone ownership given app usage.
  3. Both figures are needed to calculate the desired proportion using Bayes' theorem.
  4. Neither figure is directly relevant without knowing the overall proportion of people who are smartphone users.
Explanation: The goal is to find the proportion of smartphone users who use the app. This is the conditional probability P(extAppUser∣SmartphoneUser)P( ext{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(extSmartphoneUser∣AppUser)P( ext{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.

Question 5

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?

  1. The sample size of 5,000 accidents is not large enough to draw a valid conclusion.
  2. The study should have compared the accident rates per 1,000 cars on the road for each color, not just data from accidents.
  3. The association between car color and accident rate might be coincidental and not statistically significant.
  4. 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. (correct answer)
Explanation: 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.

Question 6

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?

  1. Calculating the total number of people who reported high trust and comparing it to the total number who reported low trust.
  2. Comparing the proportion of television viewers who have high trust to the proportion of online readers who have high trust. (correct answer)
  3. Calculating the proportion of all surveyed adults who primarily use online sources for news.
  4. Comparing the marginal distribution of news sources to the marginal distribution of trust levels.
Explanation: 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.

Question 7

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?

  1. The three bars corresponding to the education levels are all of different heights.
  2. The segments for 'Drive' are the largest segment in all three bars.
  3. The total area for the 'Public Transit' segments across all bars is less than the total area for the 'Drive' segments.
  4. The relative heights of the segments for each commuting method are approximately the same across all three education bars. (correct answer)
Explanation: 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.

Question 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?

  1. It is a response variable, as it is affected by the sales training.
  2. It is an independent variable, controlled by the researchers to test its effect on sales.
  3. It is a confounding variable, as it is associated with both the training and sales performance. (correct answer)
  4. It is an irrelevant factor, since the study is only concerned with the training program.
Explanation: 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.

Question 9

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?

  1. There are twice as many Humanities majors as STEM majors in the sample, and one-third of Humanities majors plan to attend graduate school. (correct answer)
  2. The proportion of Humanities majors who plan to attend graduate school is greater than the proportion of STEM majors who do.
  3. Half of all students who plan to attend graduate school are Humanities majors.
  4. The number of STEM majors who plan to attend graduate school is equal to the number of Humanities majors who do not.
Explanation: 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.

Question 10

A university analyzed the graduation rates of students in two colleges: the College of Engineering and the College of Arts & Sciences. The overall data showed that the graduation rate in the College of Arts & Sciences was higher than in the College of Engineering. However, when the data was broken down by students' incoming GPA (categorized as High or Moderate), they found that for both High GPA students and for Moderate GPA students, the College of Engineering had a higher graduation rate.

Which of the following is the most likely explanation for this apparent contradiction?

  1. The overall data is more reliable, and the engineering college's graduation rate is indeed lower.
  2. A calculation error must have occurred; it is mathematically impossible for the disaggregated trend to be reversed in the aggregated data.
  3. The College of Engineering enrolls a much higher proportion of students with moderate incoming GPAs, who have a lower graduation rate overall. (correct answer)
  4. The association between college and graduation rate is negative, but the association between GPA and graduation rate is positive.
Explanation: 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.

Question 11

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?

  1. The three bars corresponding to the education levels are all of different heights.
  2. The segments for 'Drive' are the largest segment in all three bars.
  3. The total area for the 'Public Transit' segments across all bars is less than the total area for the 'Drive' segments.
  4. The relative heights of the segments for each commuting method are approximately the same across all three education bars. (correct answer)
Explanation: 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.

Question 12

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?

  1. The 60% figure, as it represents the conditional rate of app usage given smartphone ownership. (correct answer)
  2. The 90% figure, as it represents the conditional rate of smartphone ownership given app usage.
  3. Both figures are needed to calculate the desired proportion using Bayes' theorem.
  4. Neither figure is directly relevant without knowing the overall proportion of people who are smartphone users.
Explanation: The goal is to find the proportion of smartphone users who use the app. This is the conditional probability P(extAppUser∣SmartphoneUser)P( ext{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(extSmartphoneUser∣AppUser)P( ext{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.

Question 13

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?

  1. It is a response variable, as it is affected by the sales training.
  2. It is an independent variable, controlled by the researchers to test its effect on sales.
  3. It is a confounding variable, as it is associated with both the training and sales performance. (correct answer)
  4. It is an irrelevant factor, since the study is only concerned with the training program.
Explanation: 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.

Question 14

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?

  1. There are twice as many Humanities majors as STEM majors in the sample, and one-third of Humanities majors plan to attend graduate school. (correct answer)
  2. The proportion of Humanities majors who plan to attend graduate school is greater than the proportion of STEM majors who do.
  3. Half of all students who plan to attend graduate school are Humanities majors.
  4. The number of STEM majors who plan to attend graduate school is equal to the number of Humanities majors who do not.
Explanation: 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.

Question 15

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?

  1. Temperature must be negatively associated with either ice cream sales or drowning incidents.
  2. Temperature must be associated with ice cream sales, but not necessarily with drowning incidents.
  3. Temperature must be associated with drowning incidents, but not necessarily with ice cream sales.
  4. Temperature must be associated with both ice cream sales and drowning incidents. (correct answer)
Explanation: 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.

Question 16

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?

  1. The sample size of 5,000 accidents is not large enough to draw a valid conclusion.
  2. The study should have compared the accident rates per 1,000 cars on the road for each color, not just data from accidents.
  3. The association between car color and accident rate might be coincidental and not statistically significant.
  4. 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. (correct answer)
Explanation: 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.

Question 17

In a clinical trial with two treatments, A and B, an analyst computes a value k=a/bc/dk = \frac{a/b}{c/d}, where aa is the number of successes on A, bb is failures on A, cc is successes on B, and dd is failures on B. The analyst finds that k>1k > 1. What is the correct interpretation of this result?

  1. The probability of success with Treatment A is greater than the probability of success with Treatment B.
  2. The ratio of successes to failures is greater for Treatment A than it is for Treatment B. (correct answer)
  3. The total number of successes with Treatment A is greater than the total number of successes with Treatment B.
  4. Treatment A is the cause of a higher success rate compared to Treatment B.
Explanation: The value kk is the odds ratio. The numerator, a/ba/b, is the odds of success for Treatment A. The denominator, c/dc/d, is the odds of success for Treatment B. Therefore, kk is the ratio of these odds. If k>1k > 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)a/(a+b) vs c/(c+d)c/(c+d)), which is related but not the same as comparing odds. If k>1k>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.

Question 18

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?

  1. Calculating the total number of people who reported high trust and comparing it to the total number who reported low trust.
  2. Comparing the proportion of television viewers who have high trust to the proportion of online readers who have high trust. (correct answer)
  3. Calculating the proportion of all surveyed adults who primarily use online sources for news.
  4. Comparing the marginal distribution of news sources to the marginal distribution of trust levels.
Explanation: 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.

Question 19

A university analyzed the graduation rates of students in two colleges: the College of Engineering and the College of Arts & Sciences. The overall data showed that the graduation rate in the College of Arts & Sciences was higher than in the College of Engineering. However, when the data was broken down by students' incoming GPA (categorized as High or Moderate), they found that for both High GPA students and for Moderate GPA students, the College of Engineering had a higher graduation rate.

Which of the following is the most likely explanation for this apparent contradiction?

  1. The overall data is more reliable, and the engineering college's graduation rate is indeed lower.
  2. A calculation error must have occurred; it is mathematically impossible for the disaggregated trend to be reversed in the aggregated data.
  3. The College of Engineering enrolls a much higher proportion of students with moderate incoming GPAs, who have a lower graduation rate overall. (correct answer)
  4. The association between college and graduation rate is negative, but the association between GPA and graduation rate is positive.
Explanation: 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.

Question 20

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?

  1. 30
  2. 37.5 (correct answer)
  3. 40
  4. 50
Explanation: To find the expected frequency for a cell under independence, we use the formula: E=(Row Total)×(Column Total)(Grand Total)E = \frac{(\text{Row Total}) \times (\text{Column Total})}{(\text{Grand Total})}. The row total for Category A is 30+70=10030 + 70 = 100. The column total for Category P is 30+45=7530 + 45 = 75. The grand total is 200. Expected frequency = (100×75)/200=7500/200=75/2=37.5(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.