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How do you state the hypotheses, compute expected counts, and check conditions for a chi-square goodness-of-fit test?

Topic 8.2 Setting Up a Chi-Square Goodness of Fit Test: state the hypotheses for a goodness-of-fit test, compute expected counts from a claimed distribution, and verify the conditions.

A focused answer to AP Statistics Topic 8.2, on stating the hypotheses for a goodness-of-fit test, computing expected counts from a claimed distribution, and checking the random, large-counts (expected at least 5), and 10% conditions.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this topic is asking
Distributional hypotheses
Computing expected counts
Checking the conditions
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What this topic is asking

The College Board (Topic 8.2) wants you to set up a chi-square goodness-of-fit test: state the distributional hypotheses, compute the expected counts from a claimed distribution, and check the conditions (random, all expected counts at least $5$ , and $10\%$ ).

Distributional hypotheses

These hypotheses are about the entire distribution, not a single proportion. Write $H_0$ as the claimed split, "equal across categories," or "the $9:3:3:1$ ratio," or specific percentages, and $H_a$ as "the distribution is not as claimed" (it is enough that some category's proportion differs; you do not specify which). Stating $H_a$ as "all proportions differ" is wrong; "at least one differs" is correct.

Computing expected counts

Expected counts are what the null predicts you would see on average. For an "equal" claim with $k$ categories, each $E_i = n/k$ . For a ratio like $2:3:5$ , the parts sum to $10$ , so the proportions are $0.2, 0.3, 0.5$ and $E_i = 0.2n, 0.3n, 0.5n$ . The expected counts need not be whole numbers, and they should sum to $n$ , a useful check. They are the backbone of both the condition check and the test statistic in Topic 8.3.

Checking the conditions

The large-counts condition for chi-square is "all expected counts $\ge 5$ ," which differs from the proportion test's " $np_0 \ge 10$ and $n(1-p_0) \ge 10$ ." Use the expected counts you just computed; if any falls below $5$ , the chi-square approximation is unreliable and categories may need combining. Checking expected (not observed) counts is the distinctive requirement here and a common slip.

Setting up a goodness-of-fit test

A manager claims customers arrive equally across the five weekdays. A random sample of $250$ arrivals is classified by weekday: Mon $40$ , Tue $55$ , Wed $45$ , Thu $50$ , Fri $60$ . Set up the goodness-of-fit test.

step 1 Hypotheses

Let the categories be the five weekdays.

H_0: \text{arrivals are equally distributed across the five weekdays } (p_i = 0.2 \text{ each}).

H_a: \text{the distribution of arrivals is not equal across the weekdays (at least one differs).}

step 2 Expected counts

Under "equal," each weekday expects $\dfrac{1}{5} \times 250 = 50$ arrivals. Expected counts: $50, 50, 50, 50, 50$ (summing to $250$ , as required).

step 3 Check the conditions

Random: stated random sample. Large counts: every expected count is $50 \ge 5$ . $10\%$ : $250$ arrivals is plausibly under $10\%$ of all arrivals in the relevant period.

step 4 Readiness

Conditions met; a chi-square goodness-of-fit test is appropriate, with $df = k - 1 = 5 - 1 = 4$ for the P-value in Topic 8.3.

Try this

Q1. A $1:2:1$ claim is tested with $n = 120$ . Find the three expected counts. [2 points]

Cue. Parts sum to $4$ , so proportions are $0.25, 0.5, 0.25$ ; expected counts $30, 60, 30$ .

Q2. Which counts must be at least $5$ for the condition, observed or expected? [1 point]

Cue. Expected counts; every expected count must be at least $5$ .

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2018 (style)1 marksSection I (multiple choice). A goodness-of-fit test claims a

1:1:1:1:1

distribution across

5

categories with

n = 200

. The expected count in each category is (A)

5

(B)

40

(C)

200

(D)

1

Show worked answer →

The correct answer is (B).

Under an equal distribution, each category's expected count is $\dfrac{1}{5} \times 200 = 40$ .

(A) is the minimum-expected-count condition value, not the expected count. (C) is the total. (D) is the ratio number, not a count. The expected count is $40$ .

AP 2021 (style)3 marksSection II (free response). A geneticist claims offspring appear in a

9:3:3:1

ratio across four phenotypes. A random sample of

160

offspring is classified. (a) State the hypotheses for a goodness-of-fit test. (b) Compute the expected count for each phenotype. (c) Check the conditions.

Show worked answer →

A 3-point set-up question.

(a) (1 point) $H_0$ : the true distribution of phenotypes follows the $9:3:3:1$ ratio. $H_a$ : the true distribution is not the $9:3:3:1$ ratio (at least one proportion differs).
(b) (1 point) Total ratio parts $= 9 + 3 + 3 + 1 = 16$ . Expected counts: $\dfrac{9}{16}(160) = 90$ , $\dfrac{3}{16}(160) = 30$ , $\dfrac{3}{16}(160) = 30$ , $\dfrac{1}{16}(160) = 10$ .
(c) (1 point) Random: stated random sample. Large counts: all expected counts ( $90, 30, 30, 10$ ) are at least $5$ . $10\%$ : $160$ is plausibly under $10\%$ of all offspring.

Markers reward distributional hypotheses (not about a single proportion), correct expected counts from the ratio, and the all-expected-at-least-5 check.

Related dot points

Sources & how we know this

AP Statistics Course and Exam Description — College Board (2020)