Chi-Square Homogeneity or Independence (Setup) - AP Statistics

A company wants to compare whether customers from two different regions (Region 1 and Region 2) have the same distribution of preferred customer-service contact method (Phone, Email, Chat, In-person). Independent random samples of customers are taken from each region. The results are shown in a table. Which chi-square test is appropriate, and what are the correct hypotheses? State clearly whether this is one sample or multiple groups.
Answer: Chi-square test of homogeneity; $H_0$ : the distribution of preferred contact method is the same in Region 1 and Region 2; $H_a$ : the distributions differ 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.
A market analyst is comparing whether the distribution of preferred payment method (Credit, Debit, Cash, Mobile pay) is the same across two store locations (Mall, Downtown). The analyst takes separate random samples of customers from each location and records each customer's preferred payment method. The results are summarized in the two-way table. Which chi-square test is appropriate, and what are the correct hypotheses? Note: Multiple groups (locations) compared on one categorical response.
Answer: Chi-square test for homogeneity; $H_0$ : the distribution of payment preference is the same at both locations; $H_a$ : the distributions differ This tests the AP Statistics skill of choosing chi-square for homogeneity or independence via design. Separate samples per location, with one categorical response (payment method), warrant chi-square homogeneity to compare distributions. Choice C properly nulls same distributions and alternatives with difference. Choice B distracts as independence, needing one sample with two variables, not multiple samples. Mini-lesson: Goodness-of-fit tests one variable's fit; independence assesses association in one sample; homogeneity checks uniformity across groups. Choice A wrongly uses goodness-of-fit on methods overall. C is correct.
A public health team is comparing whether the distribution of vaccination status (Up to date, Not up to date) is the same across four clinics (A, B, C, D). They take separate random samples of patients from each clinic during the same week and record vaccination status. The two-way table shows the results. Which chi-square test is appropriate, and what are the correct hypotheses? Note: Multiple groups (clinics) with one categorical response.
Answer: Chi-square test for homogeneity; $H_0$ : the distribution of vaccination status is the same for all clinics; $H_a$ : at least one clinic differs The skill tested is setting up chi-square tests for homogeneity or independence in AP Statistics contexts. Separate random samples from each clinic, with one categorical response (vaccination status), indicate a chi-square test for homogeneity to compare distributions across clinics. Choice C's hypotheses correctly nullify that distributions are the same, with the alternative that at least one differs. A distractor is choice A, independence, which is unsuitable because it needs one sample with two variables per unit, not multiple samples. Mini-lesson on chi-square: Goodness-of-fit tests against a predefined distribution; independence checks two variables' relationship in one sample; homogeneity verifies uniform distributions across groups. Choice E limits to two proportions, ignoring the full categorical nature. Thus, C is right.
A university wants to determine whether major (STEM vs. Non-STEM) is related to preferred study location (Library, Dorm, Coffee shop). A single random sample of students is surveyed, and each student reports both variables. The counts are shown. Which test is appropriate, and what are the correct hypotheses?
Answer: Chi-square test of independence; $H_0$ : major and preferred study location are independent; $H_a$ : major and preferred study location are associated The skill assessed is choosing the correct chi-square test and hypotheses for investigating the relationship between two categorical variables in a single student sample for AP Statistics. The sampling design is one random sample where each student reports both major and preferred study location, indicating a single population with two variables. This requires the chi-square test of independence to test for an association between them. Brief mini-lesson: Chi-square goodness-of-fit is for one variable's fit to expectations, homogeneity compares distributions across separate samples, and independence tests variable independence in one sample. Option E is a distractor as it names homogeneity but uses independence hypotheses, confusing test types with sampling. Choice C properly states the independence test and hypotheses.
A political scientist wants to determine whether party affiliation differs among registered voters in four regions of a state. Separate random samples of registered voters are taken from each region, and each voter is classified as Democrat, Republican, or Independent. The results are shown in a table. Which chi-square test is appropriate, and what are the correct hypotheses? (Multiple groups/samples; one categorical variable compared across groups.)
Answer: Chi-square test of homogeneity; $H_0$ : the distribution of party affiliation is the same across the four regions; $H_a$ : at least one region has a different distribution The key phrase "separate random samples of registered voters are taken from each region" indicates a chi-square test of homogeneity. With four separate samples (one per region) and one categorical variable (party affiliation) being compared across these groups, we test whether the distribution of party affiliation is homogeneous across regions. Choice B correctly identifies this as homogeneity with proper hypotheses. Choice A incorrectly suggests independence - the separate sampling doesn't make it independence; rather, it's the hallmark of homogeneity. Choice E has the wrong test name but correct hypotheses wording. For homogeneity: H₀ states distributions are the same across groups, Hₐ states at least one differs.
A researcher wants to know whether pet ownership (Dog, Cat, Neither) is related to housing type (House, Apartment). One random sample of residents is surveyed, and each person reports both variables. The counts are shown. Which test is appropriate, and what are the correct hypotheses?
Answer: Chi-square test of independence; $H_0$ : pet ownership and housing type are independent; $H_a$ : pet ownership and housing type are associated The skill tested is determining the right chi-square test and hypotheses for assessing association between pet ownership and housing type in a single sample for AP Statistics. The sampling design surveys one random sample of residents, recording both variables for each, which indicates one population with two categorical variables. Thus, the chi-square test of independence is appropriate to check for a relationship. Mini-lesson: Goodness-of-fit examines one variable versus a model, homogeneity compares distributions across independent samples, and independence tests if two variables are unrelated in one sample. Option E is a distractor because it labels the test as independence but uses homogeneity-style distribution hypotheses, which don't fit the association goal. The marked answer C correctly identifies the test and hypotheses.
A university takes one random sample of 300 undergraduates and records each student's housing type (On-campus, Off-campus, With family) and class year (First-year, Sophomore, Junior/Senior). The results are summarized below. Which chi-square test is appropriate, and what are the correct hypotheses? (This is one sample with two categorical variables.)
Answer: Chi-square test of independence; $H_0$ : housing type and class year are independent in the undergraduate population; $H_a$ : they are associated. This scenario requires a chi-square test of independence. The university takes one random sample of 300 undergraduates and records two categorical variables for each student: housing type and class year. Option C correctly identifies this setup, with the null hypothesis stating that these two variables are independent in the undergraduate population. Option A incorrectly suggests homogeneity, but we don't have separate samples for each housing type - we have one sample where both variables are recorded. The distinction is crucial: one sample with two categorical variables measured on each unit leads to a test of independence, while multiple samples comparing distributions leads to a test of homogeneity.
A university wants to know whether class year (First-year, Sophomore, Junior, Senior) is related to preferred study location (Library, Dorm, Coffee shop). A single random sample of 240 students is surveyed, and each student reports both variables. The results are shown in the two-way table. Which test is appropriate, and what are the correct hypotheses? Note: One sample; two categorical variables.
Answer: Chi-square test of independence; $H_0$ : class year and preferred study location are independent; $H_a$ : they are associated This question evaluates the AP Statistics skill of chi-square test selection for homogeneity or independence. A single random sample of students, each reporting two categorical variables (class year and study location), calls for a chi-square test of independence to assess if they are related. Choice A accurately hypothesizes independence in the null and association in the alternative. Choice B distracts by suggesting homogeneity, but that's for multiple samples, not one sample like this. Mini-lesson: Distinguish chi-square as goodness-of-fit for one variable's fit to expectations, independence for two variables' linkage in one group, and homogeneity for comparing one variable across populations. Choice E resembles A but uses homogeneity phrasing, which isn't precise for independence. A is the correct setup.
A company tests whether three different training programs lead to different distributions of certification outcomes. They randomly assign 60 new employees to Program A, 55 to Program B, and 65 to Program C. After training, each employee's outcome is recorded as Pass, Fail, or No-show. Results are summarized below. Which chi-square test is appropriate, and what are the correct hypotheses? (There are multiple groups due to random assignment to programs.)
Answer: Chi-square test of homogeneity; $H_0$ : the distribution of outcomes (Pass/Fail/No-show) is the same for Programs A, B, and C; $H_a$ : at least one program has a different outcome distribution. This scenario calls for a chi-square test of homogeneity. The company randomly assigns employees to three different training programs, creating three separate groups, and then compares the distribution of outcomes (Pass/Fail/No-show) across these groups. Option B correctly identifies this setup, with the null hypothesis stating that all three programs have the same outcome distribution. Option A incorrectly suggests independence, but random assignment creates distinct groups rather than measuring two variables on the same individuals. Option C (goodness-of-fit) would test against theoretical proportions, not compare groups. Remember: when subjects are randomly assigned to different treatments and we compare categorical outcomes across treatments, use the test of homogeneity.
A food company wants to compare flavor preference among customers from three different cities. In each city, a separate random sample of customers is taken and each person chooses one favorite flavor (Chocolate, Vanilla, Strawberry). The counts are shown in a table. Which chi-square test is appropriate, and what are the correct hypotheses? (Multiple groups/samples; one categorical response variable.)
Answer: Chi-square test of homogeneity; $H_0$ : the distribution of favorite flavor is the same in all three cities; $H_a$ : at least one city has a different distribution of favorite flavor This scenario involves comparing the distribution of a single categorical variable (favorite flavor) across multiple groups, which is the classic setup for a chi-square test of homogeneity. The key indicator is that SEPARATE random samples were taken from each city - this multiple-sample design distinguishes homogeneity from independence. The null hypothesis states that the distribution of flavors is the same across all cities, while the alternative states at least one city differs. Choice A correctly identifies this as homogeneity with proper hypotheses. Choice B incorrectly suggests independence, which would require one sample with two variables measured on each person. The test of homogeneity compares distributions across independently sampled groups.
A marketing team wants to compare brand preference among four age groups (18–29, 30–44, 45–59, 60+). They take independent random samples from each age group and ask which of three brands (X, Y, Z) each person prefers. The results are shown. Which test is appropriate, and what are the correct hypotheses?
Answer: Chi-square test of homogeneity; $H_0$ : the distribution of brand preference is the same across the four age groups; $H_a$ : at least one age group has a different distribution This question evaluates the skill of choosing the chi-square test and hypotheses for comparing brand preference distributions across independent age group samples in AP Statistics. The design involves independent random samples from each age group (populations), with one categorical variable (brand preference) per sample. This setup warrants the chi-square test of homogeneity to test for identical distributions across groups. Quick mini-lesson: Chi-square goodness-of-fit is for one sample's match to expected, homogeneity for same distributions in independent multi-group samples, and independence for two variables in one sample. Distractor E names homogeneity but uses incorrect independence hypotheses, blending concepts erroneously. Option C is correct with proper test and hypotheses.
A company tests whether customer satisfaction ratings differ among three store locations. The company takes independent random samples of customers from each location (Downtown, Mall, Suburb). Each customer rates satisfaction as Low, Medium, or High. The results are shown. Which test is appropriate, and what are the correct hypotheses?
Answer: Chi-square test of homogeneity; $H_0$ : the distribution of satisfaction ratings is the same for all locations; $H_a$ : at least one location has a different distribution This problem evaluates the ability to choose the right chi-square test and hypotheses when comparing categorical distributions across independent samples from different locations in AP Statistics. The design uses independent random samples from each store location, treating them as separate populations, with one categorical variable (satisfaction rating) measured per sample. Thus, the chi-square test of homogeneity is the proper choice to test if satisfaction distributions are the same across locations. Quick mini-lesson: Goodness-of-fit is for one sample versus a hypothesized distribution, homogeneity for identical distributions in multiple groups via independent sampling, and independence for two variables' association in a single sample. Distractor E names the test correctly as homogeneity but errs by using independence-style hypotheses, which don't fit the multi-population context. Option C is correct in both test name and hypotheses.
An environmental group wants to compare recycling behavior across three neighborhoods. Independent random samples of households are taken from Neighborhood A, B, and C. Each household is classified as Recycles regularly (Yes/No). Results are shown. Which test is appropriate, and what are the correct hypotheses?
Answer: Chi-square test of homogeneity; $H_0$ : the distribution of recycling status is the same across neighborhoods; $H_a$ : at least one neighborhood has a different distribution This problem focuses on the skill of identifying the chi-square test and hypotheses for comparing a binary categorical variable across independent samples from neighborhoods in AP Statistics. The design takes independent random samples from each neighborhood (populations) and measures recycling status (yes/no) in each, fitting a multi-population comparison. The chi-square test of homogeneity is thus appropriate to check if recycling distributions are the same across neighborhoods. Mini-lesson: Use goodness-of-fit for single-sample distribution testing, homogeneity for equivalence across independent groups, and independence for two-variable association in one sample. Distractor E correctly names homogeneity but wrongly applies independence hypotheses, which doesn't align with the setup. Option D is accurate in test and hypotheses.
A company wants to compare the distribution of preferred device type for accessing its website across two age groups. A random sample of 120 users ages 18–34 and a separate random sample of 120 users ages 35+ are selected. Each user is classified by preferred device (Phone, Tablet, Computer). The results are shown in a table. Which chi-square test is appropriate, and what are the correct hypotheses? (Multiple groups/samples; one categorical response variable.)
Answer: Chi-square test of homogeneity; $H_0$ : the distribution of preferred device is the same for ages 18–34 and ages 35+; $H_a$ : the distributions differ The problem explicitly states "a random sample of 120 users ages 18-34 and a separate random sample of 120 users ages 35+," which is the classic setup for a chi-square test of homogeneity. With two separate samples and one categorical variable (preferred device) being compared, we test whether the distribution is the same across age groups. Choice B correctly identifies this as homogeneity with appropriate hypotheses. Choice A incorrectly suggests independence - having two samples doesn't make it independence; it's actually what defines homogeneity. Choice E has independence in the name but homogeneity-style hypotheses. Remember: separate samples compared on one variable = homogeneity test.
A researcher surveys a single random sample of 300 adults and records two categorical variables for each person: exercise level (Low, Moderate, High) and sleep quality (Poor, Fair, Good). The two-way table summarizes the counts. Which test is appropriate, and what are the correct hypotheses?
Answer: Chi-square test of independence; $H_0$ : exercise level and sleep quality are independent; $H_a$ : they are associated The skill being tested is identifying the correct chi-square test and hypotheses for examining the relationship between two categorical variables in a single sample for AP Statistics. The sampling design features one random sample of adults where both exercise level and sleep quality are recorded for each individual, indicating a single population with two variables. Therefore, the chi-square test of independence is suitable to check if these variables are associated. Mini-lesson: Chi-square goodness-of-fit tests a single distribution against expectations, homogeneity compares distributions across groups from independent samples, and independence assesses if two variables are related in one sample. Option E is a distractor because it names the test as independence but uses hypotheses worded like homogeneity, which mismatches the sampling design. The marked answer A correctly specifies the test and hypotheses.
A school district wants to know whether students in different grade levels prefer different after-school club types. A random sample of 80 ninth graders and a separate random sample of 90 twelfth graders are asked to choose their preferred club type (Arts, Sports, Academic). Results are summarized below. Which chi-square test is appropriate, and what are the correct hypotheses? (There are multiple groups: 9th grade vs 12th grade.)
Answer: Chi-square test of homogeneity; $H_0$ : the distribution of club preference is the same for 9th graders and 12th graders; $H_a$ : at least one grade has a different distribution. This question tests your ability to identify when to use a chi-square test of homogeneity. The key indicator is that we have two separate random samples (80 ninth graders and 90 twelfth graders) and we want to compare the distribution of a categorical variable (club preference) across these groups. Option C correctly identifies this as a test of homogeneity with the proper null hypothesis stating that the distribution of club preference is the same for both grade levels. Option B incorrectly suggests independence, but that's for when we have one sample with two variables measured on each individual. Option A (goodness-of-fit) is for comparing one sample to a theoretical distribution, and option D only compares one category rather than the entire distribution. Remember: multiple separate samples comparing distributions = homogeneity; one sample with two variables = independence.
A researcher wants to compare whether three different neighborhoods (North, Central, South) have the same distribution of primary commuting method (Car, Public transit, Bike/Walk). She takes separate random samples of adults from each neighborhood and records each person's commuting method. The results are shown in the two-way table. Which chi-square test is appropriate, and what are the correct hypotheses? Note: This study uses multiple groups (neighborhoods) and compares a single categorical response.
Answer: Chi-square test for homogeneity; $H_0$ : the distribution of commuting method is the same in all neighborhoods; $H_a$ : at least one neighborhood has a different distribution The skill here is identifying the correct chi-square test setup for homogeneity or independence based on study design in AP Statistics. This scenario involves separate random samples from each neighborhood, with one categorical response (commuting method) compared across groups, so a chi-square test for homogeneity is suitable to check if distributions are the same. The hypotheses in choice A correctly state that the distribution is the same for all neighborhoods under the null, with the alternative that at least one differs. A distractor like choice B, the test of independence, is wrong because independence requires one sample with two variables per unit, not multiple samples as here. For a mini-lesson: goodness-of-fit compares one variable to expected proportions; independence examines association in one sample with two variables; homogeneity tests if multiple populations have the same distribution for one variable. Choice E misapplies goodness-of-fit by assuming equal counts rather than proportions. Therefore, A is appropriate.
An online retailer wants to know whether device type is associated with whether a customer completes a purchase. From a single random sample of 500 site visits, the company records device type (Mobile, Tablet, Desktop) and outcome (Purchase, No purchase). The results appear in the two-way table below. Which test is appropriate, and what are the correct hypotheses? Note: One sample; two categorical variables measured on each visit.
Answer: Chi-square test of independence; $H_0$ : device type and purchase outcome are independent; $H_a$ : device type and purchase outcome are associated This AP Statistics question focuses on the skill of distinguishing chi-square test setups for homogeneity versus independence. With a single random sample of site visits where each visit records two categorical variables (device type and purchase outcome), the chi-square test of independence is correct to test for association. Choice C properly states the null as independence between variables and the alternative as association. Choice B is a distractor because it suggests homogeneity, which would require separate samples from groups like 'purchases' and 'no purchases,' but here it's one sample. Mini-lesson: Chi-square types include goodness-of-fit for one variable matching a model, independence for two variables in one sample, and homogeneity for one variable across multiple samples. Choice A incorrectly applies goodness-of-fit to device types alone. Overall, C fits the design.
A teacher wants to know whether participation in an optional review session is related to exam outcome. From a single random sample of 180 students enrolled in a course, the teacher records whether each student attended the review session (Yes/No) and whether they passed the exam (Pass/Fail). The results are shown in the two-way table. Which chi-square test is appropriate, and what are the correct hypotheses? Note: One sample; two categorical variables.
Answer: Chi-square test of independence; $H_0$ : review-session attendance and exam outcome are independent; $H_a$ : they are associated 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.
A beverage company is testing whether three advertising campaigns (A, B, C) lead to the same distribution of customer reactions (Positive, Neutral, Negative). The company runs each campaign in a different region and takes a separate random sample of customers exposed to that campaign, recording each customer's reaction. The results are in the two-way table. Which chi-square test is appropriate, and what are the correct hypotheses? Note: Multiple groups (campaigns) compared on one categorical response.
Answer: Chi-square test for homogeneity; $H_0$ : the distribution of reactions is the same for all campaigns; $H_a$ : at least one campaign differs In AP Statistics, this tests the skill of chi-square setup for homogeneity or independence based on sampling. Separate samples per campaign, each with one categorical response (reaction), suit a chi-square test for homogeneity to see if distributions match across campaigns. Choice C's hypotheses state the null as same distributions and alternative as at least one difference. Choice A is a distractor as independence requires one sample with dual variables, not separate groups here. Mini-lesson: Goodness-of-fit evaluates one variable against theory; independence tests two variables' connection in a single sample; homogeneity compares one variable's distribution over multiple samples. Choice E is too narrow, focusing on proportions for one reaction type. C aligns perfectly.

Start this quiz