Chi-Square Goodness of Fit (Setup) - AP Statistics

1. A city transportation department claims that riders' primary mode of commuting is distributed as Car 55%, Bus 20%, Train 15%, Bike/Walk 10%. A random sample of $n=250$ commuters is surveyed, and the observed counts are shown in a table. Which hypotheses are appropriate for a chi-square goodness-of-fit test of the department's claim?
   Answer: $H_0$: The distribution of commuting modes is Car 55%, Bus 20%, Train 15%, Bike/Walk 10%; $H_a$: The distribution of commuting modes differs from Car 55%, Bus 20%, Train 15%, Bike/Walk 10%. This question tests hypothesis setup for a chi-square goodness-of-fit test of commuting mode distribution. The null hypothesis correctly states the department's claim about population proportions (55% Car, 20% Bus, 15% Train, 10% Bike/Walk), while the alternative states the true distribution differs. Option B describes a test of association/independence which requires two categorical variables, Option C incorrectly uses observed counts instead of proportions, Option D uses sample proportion notation and only tests one category, and Option E tests a one-sided alternative for a single proportion rather than the entire distribution. Goodness-of-fit tests always use two-sided alternatives about the entire distribution. The key is recognizing that we're testing a claim about how all commuters in the city are distributed across modes, not just our sample.
2. A school cafeteria claims that students choose among 4 lunch options with probabilities 0.40 pizza, 0.30 salad, 0.20 sandwich, 0.10 pasta. On a randomly selected day, the first 250 students' choices are recorded, with observed counts shown in the table. Which hypotheses are appropriate for a chi-square goodness-of-fit test of the cafeteria's claim? Observed counts table (n = 250): Pizza 92, Salad 70, Sandwich 63, Pasta 25.
   Answer: $H_0$: The population distribution of lunch choices is (0.40, 0.30, 0.20, 0.10). $H_a$: The population distribution is not (0.40, 0.30, 0.20, 0.10). This question examines hypothesis setup for the chi-square goodness-of-fit test in AP Statistics, determining if categorical sample data matches a claimed population probability distribution for lunch choices. The null hypothesis posits the population follows 0.40 pizza, 0.30 salad, etc., while the alternative states it does not. Option B correctly uses this population-oriented language. Option A is a distractor, describing the chi-square independence test, which applies to relationships between two variables, not a single distribution. Mini-lesson: Always frame goodness-of-fit hypotheses around population distributions, steering clear of sample proportions (D), observed counts (C), or equal shares (E) unless that's the claim. Verification matches B as appropriate for the cafeteria's probabilities.
3. A die manufacturer claims a six-sided die is fair, so each face (1–6) has probability $\frac{1}{6}$. A student rolls the die 120 times and records the observed counts below. Which hypotheses are appropriate for a chi-square goodness-of-fit test of the manufacturer's claim?
   Answer: $H_0$: The population distribution of outcomes 1–6 is uniform (each $\frac{1}{6}$); $H_a$: The population distribution of outcomes is not uniform. This question involves testing whether a die is fair using chi-square goodness-of-fit. For a fair die, each face should have probability 1/6 in the population of all possible rolls. Option A incorrectly refers to sample proportions - hypotheses must be about population parameters. Option B correctly states the null hypothesis that the population distribution is uniform (each outcome has probability 1/6), with the alternative that it's not uniform. The goodness-of-fit test determines whether the observed frequencies from 120 rolls provide evidence against the claim of fairness.
4. A streaming service claims that the long-run distribution of subscription types among its customers is 55% Basic, 30% Standard, and 15% Premium. A random sample of 200 current customers is selected, and their subscription types are recorded as shown. Observed counts table: Basic 96, Standard 73, Premium 31 Claimed distribution: (0.55, 0.30, 0.15). Which hypotheses are appropriate for a chi-square goodness-of-fit test?
   Answer: $H_0$: The long-run distribution of subscription types is 55% Basic, 30% Standard, 15% Premium. $H_a$: The long-run distribution of subscription types differs from 55%, 30%, 15%. In AP Statistics, this question focuses on hypothesizing for a chi-square goodness-of-fit test. The test checks if sample data fits a hypothesized distribution of categories, here subscription types. The correct H0 is that the long-run distribution among all customers is 55% Basic, 30% Standard, and 15% Premium, with Ha stating it differs. This setup is appropriate as it targets the population claim, using the sample to test it. Choice B is a distractor because it wrongly frames the hypotheses around the sample distribution instead of the population. For a mini-lesson: Always ensure H0 references the population or long-run proportions, and avoid single-proportion tests like choice D, which only address one category. Hypotheses about independence, as in choice C, belong to the chi-square test of independence, not goodness-of-fit.
5. A candy company claims that the colors in its assorted bag occur in the following distribution: 30% red, 25% blue, 20% green, 15% yellow, and 10% orange. A student randomly samples 200 candies from recent bags and records the observed counts below. Which hypotheses are appropriate for a chi-square goodness-of-fit test of the company's claim?
   Answer: $H_0$: The distribution of candy colors in the population is 0.30 red, 0.25 blue, 0.20 green, 0.15 yellow, 0.10 orange; $H_a$: The distribution of candy colors in the population is not this distribution. This question tests understanding of proper hypothesis setup for a chi-square goodness-of-fit test. The null hypothesis should state the claimed population distribution (30% red, 25% blue, 20% green, 15% yellow, 10% orange), while the alternative states that the true population distribution differs from this claim. Option A incorrectly refers to sample proportions rather than population parameters - we never test hypotheses about sample statistics. Option B correctly states hypotheses about the population distribution, which is what we're testing. In goodness-of-fit tests, we compare observed sample data to a claimed population distribution, not to independence or equal proportions.
6. A city transit report claims that riders pay their fare using 55% card, 35% mobile app, and 10% cash. A random sample of 500 riders is observed, with counts shown in the table. Which hypotheses are appropriate for a chi-square goodness-of-fit test of the report's claim? Observed counts table (n = 500): Card 292, Mobile app 158, Cash 50.
   Answer: $H_0$: The population distribution of payment methods is 55% card, 35% app, 10% cash. $H_a$: The population distribution of payment methods is not 55%, 35%, 10%. This question evaluates hypothesis setup for the chi-square goodness-of-fit test in AP Statistics, to verify if payment method data fits the transit report's population distribution (55% card, etc.). The null asserts the population matches these percentages, with the alternative denying it. Option C correctly phrases this at the population level. A common distractor is option A, which is for chi-square independence, not applicable without a second variable like bus route. Mini-lesson: Focus hypotheses on population proportions matching the claim, avoiding observed counts (B), sample distributions (D), or equal probabilities (E). Independent verification supports C as the right choice.
7. A bookstore owner claims that customers pay using Cash 20%, Credit Card 50%, Debit Card 25%, and Mobile Pay 5%. A random sample of $n=400$ transactions is recorded, with observed counts shown in a table. Which hypotheses are appropriate for a chi-square goodness-of-fit test of the owner's claim?
   Answer: $H_0$: The distribution of payment methods is Cash 20%, Credit 50%, Debit 25%, Mobile 5%; $H_a$: The distribution of payment methods is different from Cash 20%, Credit 50%, Debit 25%, Mobile 5%. This question asks for the correct hypothesis setup to test a bookstore owner's claim about payment method distribution. The null hypothesis should state the owner's claim about population proportions (20% Cash, 50% Credit, 25% Debit, 5% Mobile), while the alternative states these proportions are incorrect. Option B describes a test of independence which would require a second categorical variable, Option C incorrectly uses observed counts from the sample, Option D uses sample proportion notation and only tests credit cards, and Option E tests a one-sided alternative for mobile pay only. Chi-square goodness-of-fit tests always examine the entire distribution with a two-sided alternative. The test determines whether the observed payment distribution significantly differs from the owner's claimed distribution across all customers.
8. A museum claims that visitors' favorite exhibit is distributed as 30% Ancient History, 25% Modern Art, 20% Science, 15% Nature, and 10% Technology. A random sample of 400 visitors is surveyed, and the observed counts are shown in the table. Which hypotheses are appropriate for a chi-square goodness-of-fit test? Observed counts table (n = 400): Ancient 132, Modern Art 88, Science 70, Nature 66, Technology 44.
   Answer: $H_0$: The population distribution of favorite exhibits is 30%, 25%, 20%, 15%, 10%. $H_a$: The population distribution is not 30%, 25%, 20%, 15%, 10%. This question assesses setting up hypotheses for the chi-square goodness-of-fit test in AP Statistics, checking if visitor preferences align with the museum's claimed population distribution (30% Ancient, etc.). The null should state the population follows these percentages, alternative that it does not. Option B properly defines this. Option A distracts with independence hypotheses, which test associations, not distribution fit. Mini-lesson: Goodness-of-fit demands population parameter statements, not sample proportions (C), observed counts (D), or equal divisions (E) if unequal in the claim. Confirmation shows B is correct for the survey data.
9. A bagel shop claims that the distribution of daily customer orders is 40% plain, 35% everything, 15% cinnamon raisin, and 10% sesame. A manager records a random sample of 120 bagel orders and obtains the observed counts shown below. Which hypotheses are appropriate for a chi-square goodness-of-fit test of the shop's claim?
   Answer: $H_0:$ The distribution of bagel types ordered by all customers is 40% plain, 35% everything, 15% cinnamon raisin, 10% sesame; $H_a:$ The distribution is different from the claimed distribution. For a chi-square goodness-of-fit test, we need hypotheses about the population distribution, not sample statistics. The null hypothesis states that the distribution of bagel types ordered by all customers matches the claimed percentages (40% plain, 35% everything, 15% cinnamon raisin, 10% sesame). Choice A correctly states this using population language. Choice B incorrectly uses sample proportion notation (p-hat), which describes what we observe rather than what we're testing. Choice C describes a test of independence between two variables, Choice D tests equal proportions rather than the specific claimed values, and Choice E incorrectly focuses on the sample matching rather than the population distribution. The key insight: we use sample data to test claims about populations, not about samples themselves.
10. A school counselor claims that the long-run distribution of students' preferred study times is 25% morning, 45% afternoon, and 30% evening. A random sample of 120 students is surveyed with results shown. Observed counts table: Morning 22, Afternoon 61, Evening 37 Claimed distribution: (0.25, 0.45, 0.30). Which hypotheses are appropriate for a chi-square goodness-of-fit test?
    Answer: $H_0$: The long-run distribution of preferred study times for students at this school is 25% morning, 45% afternoon, 30% evening. $H_a$: The long-run distribution is not 25%, 45%, 30%. In AP Statistics, this tests hypothesis setup for the chi-square goodness-of-fit test. The test assesses if categorical data from a sample fits a proposed population distribution, here study times. The correct H0 is that the long-run distribution for students is 25% morning, 45% afternoon, and 30% evening, with Ha that it is not. This is accurate because it evaluates the counselor's claim about the population. Choice A distracts by applying to the sample, not the population. Mini-lesson: Frame H0 with population proportions and Ha generally; avoid independence hypotheses (choice B) or single-proportion tests (choice D). Choice E incorrectly focuses on exact count equality, overlooking the role of expected counts in the test.
11. A local election official claims that the distribution of voters arriving at a polling place by hour is 10% from 7–8am, 25% from 8–10am, 35% from 10am–1pm, and 30% from 1–5pm. A random sample of 300 voters from election day is recorded with observed counts shown in the table. Which hypotheses are appropriate for a chi-square goodness-of-fit test? Observed counts table (n = 300): 7–8am 24, 8–10am 90, 10am–1pm 108, 1–5pm 78.
    Answer: $H_0$: The population distribution of arrival times matches 10%, 25%, 35%, 30%. $H_a$: The population distribution of arrival times does not match 10%, 25%, 35%, 30%. This question assesses hypothesis formulation for the chi-square goodness-of-fit test in AP Statistics, which examines whether sample data conforms to a hypothesized population distribution of categories. Appropriate hypotheses involve a null stating the population arrival times follow the official's claim (10%, 25%, 35%, 30%) and an alternative indicating mismatch in the population. Option A correctly specifies this, focusing on population parameters. Option C is a common distractor, representing the chi-square independence test, which is inappropriate here as there's only one categorical variable. In a mini-lesson, emphasize that goodness-of-fit setups must target population distributions, avoiding references to samples (D), observed counts (B), or equal proportions unless claimed (E). Verification shows A aligns with the test's requirements for the given claim.
12. A political analyst claims that support for three candidates in a district is: 45% Candidate A, 35% Candidate B, and 20% Candidate C. A random sample of 500 registered voters is polled and the observed counts are shown below. Which hypotheses are appropriate for a chi-square goodness-of-fit test of the analyst's claim?
    Answer: $H_0$: The population distribution of candidate support is 0.45 A, 0.35 B, 0.20 C; $H_a$: The population distribution of candidate support differs from 0.45/0.35/0.20. This question tests understanding of chi-square goodness-of-fit hypothesis setup for political polling. The null hypothesis should state the analyst's claimed population distribution of support (45% Candidate A, 35% Candidate B, 20% Candidate C). Option C incorrectly refers to the sample distribution - we test population parameters, not sample statistics. Option A correctly states the null hypothesis about the population distribution matching the claim, with the alternative that it differs. Goodness-of-fit tests specifically examine whether observed sample data provides evidence against a claimed population distribution.
13. A wildlife biologist claims that a certain species of bird nests in four habitat types in the following proportions: Forest 35%, Grassland 30%, Wetland 20%, and Urban 15%. The biologist observes $n=160$ nests and records the counts by habitat type shown in a table. Which hypotheses are appropriate for a chi-square goodness-of-fit test of the claimed habitat distribution?
    Answer: $H_0$: The distribution of nests across habitats is 35% forest, 30% grassland, 20% wetland, 15% urban; $H_a$: The distribution of nests across habitats is different from 35%, 30%, 20%, 15%. This question requires setting up hypotheses to test a biologist's claim about bird nesting habitat distribution. The null hypothesis states the claimed population distribution (35% forest, 30% grassland, 20% wetland, 15% urban), and the alternative states the true distribution differs from these proportions. Option B incorrectly describes a test of independence between two variables rather than goodness-of-fit, Option C uses observed counts instead of population proportions, Option D only tests one habitat type rather than the entire distribution, and Option E incorrectly uses sample proportions in the hypotheses. In ecology studies using goodness-of-fit tests, we test whether observed habitat use matches theoretical predictions. The hypotheses must be about the true proportions across all nests of this species, not just the observed sample.
14. A game developer claims that a spinner in an app lands on four colors with probabilities 0.40 (Red), 0.30 (Blue), 0.20 (Green), and 0.10 (Yellow). A user records the results of 150 spins, shown in a table. Which hypotheses are appropriate for a chi-square goodness-of-fit test to assess the developer's claim?
    Answer: $H_0$: The population distribution is 40% Red, 30% Blue, 20% Green, 10% Yellow. $H_a$: The population distribution is not 40%, 30%, 20%, 10%. For a chi-square goodness-of-fit test, we need to test whether the population distribution matches the claimed probabilities. The null hypothesis states that the population distribution is 40% Red, 30% Blue, 20% Green, and 10% Yellow, as claimed by the developer. The alternative hypothesis states that the population distribution differs from these percentages. Option A correctly expresses both hypotheses in terms of the population distribution. Option B is too vague and doesn't specify the claimed proportions. Option C incorrectly refers to sample proportions instead of population parameters. Option D suggests a test of independence, which is inappropriate here. Option E only tests one color rather than the entire distribution.
15. A candy company claims that a bag of its assorted candies contains 20% red, 20% blue, 20% green, 20% yellow, and 20% orange candies. A student counts the colors in a randomly selected bag containing 100 candies; results are shown in a table. Which hypotheses are appropriate for a chi-square goodness-of-fit test of the company's claim?
    Answer: $H_0$: The population distribution of candy colors is 20% each for red, blue, green, yellow, and orange. $H_a$: The population distribution of candy colors is not 20% each. For testing the candy company's claim about color distribution, we need hypotheses about the population distribution. The null hypothesis should state that each color comprises 20% of the population (equal distribution), while the alternative states that the population distribution is not 20% for each color. Option A correctly expresses this by stating the population distribution in both hypotheses. Option B incorrectly focuses on exact counts in the sample rather than population proportions. Option C describes a test of independence between two variables, which doesn't apply here. Option D only tests one color rather than the entire distribution. Option E incorrectly refers to observed proportions rather than population parameters.
16. A survey firm claims that, in a certain county, political affiliation is distributed as 48% Independent, 42% Democrat, and 10% Republican. A random sample of 250 registered voters is selected and affiliation is recorded, with observed counts shown in the table. Which hypotheses are appropriate for a chi-square goodness-of-fit test? Observed counts table (n = 250): Independent 110, Democrat 112, Republican 28.
    Answer: $H_0$: The population distribution of political affiliation is 48% Independent, 42% Democrat, 10% Republican. $H_a$: The population distribution of political affiliation differs from 48%, 42%, 10%. This question examines hypothesis setup for the chi-square goodness-of-fit test in AP Statistics, testing if voter affiliations fit the survey firm's population claims (48% Independent, etc.). The null should affirm the population distribution matches, alternative that it differs. Option C correctly specifies this. Option A distracts by using independence language, inappropriate for a single categorical variable. Mini-lesson: Ensure goodness-of-fit hypotheses target population proportions, not observed counts (B), sample proportions (D), or equal thirds (E). Independent solving confirms C as the appropriate choice.
17. A city's transportation office claims that commuters use the following primary modes of transportation: 50% drive alone, 20% carpool, 15% public transit, 10% bike, and 5% walk. A random sample of 400 commuters is taken and the observed counts are shown below. Which hypotheses are appropriate for a chi-square goodness-of-fit test of the office's claim?
    Answer: $H_0$: The population distribution of primary commute mode is 0.50 drive alone, 0.20 carpool, 0.15 public transit, 0.10 bike, 0.05 walk; $H_a$: The population distribution is not as claimed. This question tests understanding of chi-square goodness-of-fit hypothesis setup for transportation mode claims. The null hypothesis should state the claimed population distribution of commute modes (50% drive alone, 20% carpool, 15% public transit, 10% bike, 5% walk). Option B incorrectly refers to sample proportions - we test population parameters, not sample statistics. Option C correctly states the null hypothesis about the population distribution matching the claim, with the alternative that it differs. Options about independence or equal proportions are inappropriate for goodness-of-fit tests, which specifically test whether data fits a claimed distribution.
18. A wildlife biologist claims that a certain species of bird builds nests in three types of trees in the following proportions: 50% oak, 30% pine, and 20% maple. The biologist observes 120 nests and records the tree type for each nest; results are shown in a table. Which hypotheses are appropriate for a chi-square goodness-of-fit test of the biologist's claim?
    Answer: $H_0$: The population proportions of nests are 0.50 oak, 0.30 pine, 0.20 maple. $H_a$: The population proportions of nests are not 0.50, 0.30, 0.20. For testing the biologist's claim about nest distribution across tree types, we need to set up hypotheses about the population proportions. The null hypothesis states that 50% of nests are in oak trees, 30% in pine, and 20% in maple, while the alternative states that the population proportions differ from these values. Option A correctly expresses both hypotheses in terms of population parameters. Option B incorrectly refers to sample proportions rather than population proportions. Option C describes a test of independence between two variables, which is not the goal here. Option D only tests one tree type rather than the entire distribution. Option E misunderstands the nature of hypothesis testing by focusing on exact matching.
19. A biology textbook states that a particular species of beetle has a 9:3:3:1 phenotypic ratio (four categories) in offspring. A student observes 320 offspring from controlled crosses and records the observed counts shown in the table. Which hypotheses are appropriate for a chi-square goodness-of-fit test? Claimed distribution: $\frac{9}{16},\frac{3}{16},\frac{3}{16},\frac{1}{16}$. Observed counts table (n = 320): Type I 190, Type II 64, Type III 50, Type IV 16.
    Answer: $H_0$: The population distribution of phenotypes is $9/16,3/16,3/16,1/16$. $H_a$: The population distribution of phenotypes differs from $9/16,3/16,3/16,1/16$. This question probes the hypothesis setup for a chi-square goodness-of-fit test in AP Statistics, used to test if observed frequencies fit an expected population ratio like the 9:3:3:1 for beetle phenotypes. The null should assert the population distribution matches 9/16, 3/16, 3/16, 1/16, with the alternative claiming a deviation in the population. Option B accurately does this by specifying population proportions. A typical distractor is option C, which confuses goodness-of-fit with the independence test, irrelevant without two variables. Mini-lesson: Hypotheses in goodness-of-fit must describe population parameters, not sample data (A), observed counts (E), or equal probabilities unless specified (D). Independent solving confirms B as the correct choice for this genetic claim.
20. A restaurant owner claims that the long-run distribution of entrée orders during dinner is 50% pasta, 20% burger, 15% salad, and 15% tacos. Over a randomly selected set of 180 dinner orders, the counts are recorded. Observed counts table: Pasta 84, Burger 44, Salad 22, Tacos 30 Claimed distribution: (0.50, 0.20, 0.15, 0.15). Which hypotheses are appropriate for a chi-square goodness-of-fit test?
    Answer: $H_0$: The long-run distribution of entrée orders is (0.50, 0.20, 0.15, 0.15). $H_a$: The long-run distribution of entrée orders is different from (0.50, 0.20, 0.15, 0.15). The skill here in AP Statistics is formulating hypotheses for the chi-square goodness-of-fit test. This test is designed to see if the distribution of a categorical variable in a population matches specified proportions, based on sample observations like entrée orders. The correct H0 is that the long-run distribution is 50% pasta, 20% burger, 15% salad, and 15% tacos, with Ha that it differs. This fits because it directly tests the owner's population claim. Choice B distracts by focusing on the sample distribution, not the population. Mini-lesson: Set H0 to the claimed population proportions and Ha as the negation; distinguish from independence tests (choice C) or single-category tests (choice D). Avoid hypotheses about expected counts directly (choice E), as the test compares observed to expected under H0.

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