This question involves comparing the distribution of satisfaction ratings across three restaurant locations, where managers take separate random samples from each location. Since we have distinct samples from different populations (the three locations) and want to test if the categorical distribution is the same across these populations, we use a chi-square test of homogeneity. The null hypothesis states that the distribution of satisfaction ratings is the same for all locations. Option A incorrectly suggests independence (which requires one sample), Option B is inappropriate because ratings are categorical not numerical, Option D tests equal likelihood within each location rather than comparing across locations, and Option E only examines one rating category rather than the full distribution.

Assessing AP Statistics skill in chi-square homogeneity versus independence setups, this uses one random sample of students with two categorical variables (attendance and outcome), ideal for chi-square test of independence on association. Choice C hypothesizes null independence and alternative association correctly. Choice B is a distractor, suggesting homogeneity, but that requires separate samples, not one here. Mini-lesson: Chi-square includes goodness-of-fit for single-variable proportions, independence for two-variable ties in one group, and homogeneity for multi-group comparisons. Choice A misapplies goodness-of-fit to outcomes. C is accurate.

This scenario involves one random sample of 200 adults where two categorical variables (gender and viewing preference) are recorded for each person. Since we have a single sample and want to test whether these two variables are related, we use a chi-square test of independence. The null hypothesis states that gender and viewing preference are independent in the population, meaning knowing someone's gender doesn't help predict their viewing preference. Option B incorrectly treats this as comparing separate samples, Option C focuses on overall preferences rather than the relationship between variables, Option D only examines streaming preference rather than the full association, and Option E misinterprets the goal as testing for uniform distribution.

This scenario describes one random sample of 240 students where two categorical variables (residence status and transportation mode) are recorded for each student. Since we have a single sample and want to test whether these two variables are associated, we use a chi-square test of independence. The null hypothesis states that residence status and transportation mode are independent in the population. Option A wrongly treats this as comparing separate samples, Option B tests for equal likelihood of transportation modes overall rather than their relationship with residence, Option D only examines walking rather than the full association, and Option E misinterprets the goal as testing for uniform distribution.

This question requires a chi-square test of homogeneity. The health department takes two separate random samples (140 residents from County X and 160 from County Y) and compares the distribution of vaccination status across these counties. Option B correctly identifies this, with the null hypothesis stating that both counties have the same distribution of vaccination status. Since vaccination status has only two categories (Vaccinated/Not vaccinated), option A's two-proportion z-test would also be appropriate and give the same result, but option B is the more general chi-square approach that extends to multiple categories. Option C incorrectly suggests independence, but we have separate samples from different populations, not one sample with two variables.

This question asks about comparing diet type distributions between Forest and Grassland habitats, where separate random samples of animals are taken from each habitat. Since we have distinct samples from different populations (the two habitats) and want to test if the categorical distribution is the same across these populations, we use a chi-square test of homogeneity. The null hypothesis states that the distribution of diet type is the same in both habitats. Option A incorrectly suggests independence (which requires one sample), Option B tests against a specified model rather than comparing habitats, Option D only examines carnivores rather than the full distribution, and Option E focuses on sample sizes rather than diet distributions.

The problem states "a single random sample of 180 adults" with two categorical variables recorded for each person: smoking status and exercise frequency. This is a one-sample design where we examine the association between two categorical variables within that sample. A chi-square test of independence is appropriate, testing whether smoking status and exercise frequency are independent in the population. The null hypothesis states these variables are independent, while the alternative states they are associated. Choice C correctly identifies this setup. The single sample with two variables measured on each individual distinguishes this from a homogeneity test, which would require separate samples from different populations.

The problem states "Independent random samples of customers are taken from each region," clearly indicating two separate samples (one from Region 1, one from Region 2). This is a multiple groups design comparing distributions across independent samples, requiring a chi-square test of homogeneity. We're testing whether the distribution of preferred contact methods is the same between the two regions. The null hypothesis states that both regions have the same distribution of preferences, while the alternative states the distributions differ. Choice A correctly identifies this as a test of homogeneity with appropriate hypotheses. The independent sampling from each region creates separate groups for comparison, distinguishing this from an independence test.

The problem states "a single random sample of 250 students" with two categorical variables recorded for each student: class year and campus residence status. This is a one-sample design examining the relationship between two categorical variables within that sample. A chi-square test of independence is appropriate, testing whether class year and living on campus are independent in the population. The null hypothesis states these variables are independent, while the alternative states they are associated. Choice C correctly identifies this setup. The single sample with two variables measured on each unit distinguishes this from a homogeneity test, which would require separate samples from different populations.

The problem explicitly states "a single random sample of 220 adults" with two categorical variables recorded for each adult: employment status and education level. This is a one-sample design examining the relationship between two categorical variables within that sample. A chi-square test of independence is appropriate, testing whether employment status and education level are independent in the population. The null hypothesis states these variables are independent, while the alternative states they are associated. Choice C correctly identifies this setup. The single sample with two variables measured on each unit is the key indicator for independence testing, as opposed to homogeneity testing which requires multiple independent samples.