Chi-Square Goodness of Fit (Setup)
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AP Statistics › Chi-Square Goodness of Fit (Setup)
A manufacturer claims that defects in its products fall into 4 categories with probabilities 0.50 cosmetic, 0.20 packaging, 0.20 functional, and 0.10 missing parts. An auditor inspects 150 defective products and records the observed counts shown in the table. Which hypotheses are appropriate for a chi-square goodness-of-fit test of the claim?
Observed counts table (n = 150): Cosmetic 68, Packaging 34, Functional 36, Missing parts 12.
$H_0$: $p_{\text{cos}}=p_{\text{pack}}=p_{\text{func}}=p_{\text{miss}}=0.25$. $H_a$: Not all four proportions are equal.
$H_0$: Defect type and production line are independent. $H_a$: Defect type and production line are not independent.
$H_0$: The observed counts are 75, 30, 30, 15. $H_a$: The observed counts are not exactly 75, 30, 30, 15.
$H_0$: The sample distribution of defect types is (0.50, 0.20, 0.20, 0.10). $H_a$: The sample distribution differs.
$H_0$: The population distribution of defect types is (0.50, 0.20, 0.20, 0.10). $H_a$: The population distribution of defect types is not (0.50, 0.20, 0.20, 0.10).
Explanation
This question tests hypothesis formulation for the chi-square goodness-of-fit test in AP Statistics, to see if defect types match the manufacturer's population probabilities (0.50 cosmetic, etc.). The null claims the population distribution is as stated, alternative that it's not. Option A accurately captures this. Option B is a distractor, suggesting independence, which isn't relevant without another variable like production line. Mini-lesson: Hypotheses must refer to population distributions, not observed counts (C), sample distributions (D), or equal proportions (E) unless claimed. Verification aligns with A as the proper setup.
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.
$H_0$: The observed counts equal 30, 75, 105, 90. $H_a$: The observed counts are not exactly 30, 75, 105, 90.
$H_0$: 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%.
$H_0$: The sample proportions are 10%, 25%, 35%, 30%. $H_a$: At least one sample proportion differs.
$H_0$: $p_{7-8}=p_{8-10}=p_{10-1}=p_{1-5}=0.25$. $H_a$: Not all four proportions equal 0.25.
$H_0$: Voter arrival time and political party are independent. $H_a$: Voter arrival time and political party are not independent.
Explanation
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.
A die manufacturer claims its 6-sided die is fair, so each face (1–6) has probability $1/6$. A quality-control inspector rolls one die 120 times and records the observed counts shown in the table. Which hypotheses are appropriate for a chi-square goodness-of-fit test?
Claimed distribution: each face 1/6.
Observed counts table (n = 120): 1: 15, 2: 18, 3: 22, 4: 17, 5: 25, 6: 23.
$H_0$: The sample probabilities for faces 1–6 are each $1/6$. $H_a$: At least one sample probability differs from $1/6$.
$H_0$: The die is biased toward larger numbers. $H_a$: The die is not biased toward larger numbers.
$H_0$: The population probabilities for faces 1–6 are each $1/6$. $H_a$: At least one population probability for faces 1–6 differs from $1/6$.
$H_0$: The observed counts are exactly 20 for each face. $H_a$: At least one observed count is not 20.
$H_0$: Outcome and trial number are independent. $H_a$: Outcome and trial number are not independent.
Explanation
This question tests hypothesis configuration for the chi-square goodness-of-fit test in AP Statistics, assessing if a die's outcomes fit a fair population model with each face at 1/6 probability. The null should claim equal population probabilities of 1/6 for faces 1-6, and the alternative that at least one differs. Option A precisely states this for the population. Option C distracts by suggesting an independence test, unsuitable for testing a single variable's distribution across rolls. Mini-lesson: Goodness-of-fit requires population-based hypotheses, not sample-based (B), count equality (D), or directional alternatives (E) unless specified. My check confirms A fits the fairness claim perfectly.
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?
$H_0$: The observed counts must be exactly 20 for each face; $H_a$: At least one observed count is not 20.
$H_0$: $p(6)=\frac{1}{6}$; $H_a$: $p(6)\ne\frac{1}{6}$.
$H_0$: Outcome is independent of roll number; $H_a$: Outcome is not independent of roll number.
$H_0$: The sample proportions for faces 1–6 are each $\frac{1}{6}$; $H_a$: The sample proportions are not all $\frac{1}{6}$.
$H_0$: The population distribution of outcomes 1–6 is uniform (each $\frac{1}{6}$); $H_a$: The population distribution of outcomes is not uniform.
Explanation
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.
A school cafeteria manager claims that students choose among four lunch options in the following proportions: 35% pizza, 30% salad, 20% sandwich, and 15% pasta. On a randomly selected day, a random sample of 200 students is recorded with the observed counts below.
Which hypotheses are appropriate for a chi-square goodness-of-fit test of the manager’s claim?
$H_0$: The sample distribution of lunch choices is 0.35, 0.30, 0.20, 0.15; $H_a$: The sample distribution is not 0.35, 0.30, 0.20, 0.15.
$H_0$: Each lunch option is equally likely (25% each); $H_a$: At least one option has a different probability.
$H_0$: The observed counts are 70 pizza, 60 salad, 40 sandwich, 30 pasta; $H_a$: The observed counts are not those values.
$H_0$: The population distribution of lunch choices is 0.35 pizza, 0.30 salad, 0.20 sandwich, 0.15 pasta; $H_a$: The population distribution differs from these proportions.
$H_0$: Lunch choice is independent of whether a student was sampled; $H_a$: Lunch choice is not independent of whether a student was sampled.
Explanation
This question tests proper hypothesis formulation for a cafeteria manager's claim about lunch preferences. The null hypothesis must state the claimed population distribution (35% pizza, 30% salad, 20% sandwich, 15% pasta), not sample values or observed counts. Option C incorrectly refers to the sample distribution - hypotheses are about population parameters. Option B correctly states that the population distribution follows the claimed proportions, with the alternative being that it differs. The chi-square goodness-of-fit test helps determine if observed sample data provides sufficient evidence against the claimed population distribution.
A school district claims that the distribution of students’ primary mode of transportation to school is 50% bus, 30% car, 15% walk, and 5% bike. A random sample of 160 students is surveyed, and the observed counts are shown below. Which hypotheses are appropriate for a chi-square goodness-of-fit test of the district’s claim?
$H_0:$ Transportation mode and grade level are independent; $H_a:$ Transportation mode and grade level are not independent.
$H_0:$ The distribution of transportation modes for all students in the district is 50% bus, 30% car, 15% walk, 5% bike; $H_a:$ The distribution of transportation modes for all students is different from this claim.
$H_0: p_{bus}=p_{car}=p_{walk}=p_{bike}$; $H_a:$ At least one population proportion differs.
$H_0: \hat{p}{bus}=0.50,\ \hat{p}{car}=0.30,\ \hat{p}{walk}=0.15,\ \hat{p}{bike}=0.05$; $H_a:$ at least one sample proportion differs.
$H_0:$ The sample distribution is 50% bus, 30% car, 15% walk, 5% bike; $H_a:$ The sample distribution differs.
Explanation
This question requires setting up hypotheses for testing whether the school district's claimed distribution is accurate. The null hypothesis should state that the population distribution (all students in the district) follows the claimed percentages: 50% bus, 30% car, 15% walk, 5% bike. Choice C correctly expresses this by referring to "all students in the district." Choice A incorrectly sets up a test of independence between two variables, Choice B and E incorrectly reference sample distributions or proportions rather than population parameters, and Choice D tests for equal proportions rather than the specific claimed values. Remember that in goodness-of-fit tests, we're testing whether the true population distribution matches a specific claimed distribution, using sample data as evidence.
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?
$H_0$: $p_{drive}=p_{carpool}=p_{transit}=p_{bike}=p_{walk}$; $H_a$: At least one proportion differs.
$H_0$: The sample proportions are 0.50, 0.20, 0.15, 0.10, 0.05 for the five modes; $H_a$: The sample proportions differ from these values.
$H_0$: 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.
$H_0$: Mode of transportation is independent of commuter; $H_a$: Mode of transportation is not independent of commuter.
$H_0$: The observed counts match the claim exactly; $H_a$: The observed counts do not match the claim exactly.
Explanation
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.
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?
$H_0$: Candidate support is independent of voter; $H_a$: Candidate support is not independent of voter.
$H_0$: The sample distribution is 0.45/0.35/0.20; $H_a$: The sample distribution is not 0.45/0.35/0.20.
$H_0$: 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.
$H_0$: The observed counts equal the expected counts; $H_a$: The observed counts do not equal the expected counts.
$H_0$: Each candidate has equal support (1/3 each); $H_a$: Support is not equal for all candidates.
Explanation
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.
An online retailer claims that orders are shipped using three carriers in these proportions: 50% Carrier A, 30% Carrier B, and 20% Carrier C. A random sample of 250 recent orders is selected and the observed counts are shown below.
Which hypotheses are appropriate for a chi-square goodness-of-fit test of the retailer’s claim?
$H_0$: The sample distribution of shipping carriers is 0.50/0.30/0.20; $H_a$: The sample distribution is not 0.50/0.30/0.20.
$H_0$: Each carrier is equally likely (1/3 each); $H_a$: At least one carrier has a different probability.
$H_0$: Shipping carrier is independent of order number; $H_a$: Shipping carrier is not independent of order number.
$H_0$: The population distribution of shipping carriers is 0.50 A, 0.30 B, 0.20 C; $H_a$: The population distribution of shipping carriers differs from 0.50/0.30/0.20.
$H_0$: The observed counts are exactly 125, 75, and 50; $H_a$: The observed counts are not exactly 125, 75, and 50.
Explanation
This question tests understanding of hypothesis setup for testing a retailer's shipping carrier claim. The null hypothesis should state the claimed population distribution (50% Carrier A, 30% Carrier B, 20% Carrier C), not sample statistics or observed counts. Option D incorrectly refers to the sample distribution rather than population parameters. Option A correctly states that the population distribution of shipping carriers follows the claimed proportions, with the alternative that it differs. Chi-square goodness-of-fit tests whether observed sample data provides evidence against a specific claimed population distribution.
A museum claims that visitors’ favorite exhibit among four options is distributed as follows: 10% Exhibit 1, 20% Exhibit 2, 30% Exhibit 3, and 40% Exhibit 4. A random sample of 150 visitors is surveyed, producing the observed counts below.
Which hypotheses are appropriate for a chi-square goodness-of-fit test of the museum’s claim?
$H_0$: The population distribution of favorite exhibit is 0.10, 0.20, 0.30, 0.40 for Exhibits 1–4; $H_a$: The population distribution is not 0.10, 0.20, 0.30, 0.40.
$H_0$: Each exhibit is equally likely (25% each); $H_a$: The exhibits are not equally likely.
$H_0$: Favorite exhibit is independent of whether a visitor is in the sample; $H_a$: Favorite exhibit is not independent of whether a visitor is in the sample.
$H_0$: The sample proportions are 0.10, 0.20, 0.30, 0.40; $H_a$: The sample proportions are not 0.10, 0.20, 0.30, 0.40.
$H_0$: The observed counts match the expected counts exactly; $H_a$: They do not match exactly.
Explanation
This question assesses proper hypothesis formulation for testing a museum's claim about exhibit preferences. The null hypothesis must state the claimed population distribution (10% Exhibit 1, 20% Exhibit 2, 30% Exhibit 3, 40% Exhibit 4). Option C incorrectly refers to sample proportions - hypotheses are always about population parameters. Option B correctly states that the population distribution of favorite exhibits follows the claimed proportions, with the alternative being that it differs. The chi-square goodness-of-fit test determines whether the observed visitor preferences provide evidence against the museum's claimed distribution.