AP Statistics Chi-Square Tests Test Solutions and Scoring (PDF)

AP Statistics math test solutions: Chi-Square Tests – Maths PDF

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This chi square test answers for AP Statistics covers chi-square tests (Common Core AP STAT 8.1, AP STAT 8.2, AP STAT 8.3, AP STAT 8.4, AP STAT 8.5, AP STAT 8.6, AP STAT 8.7), with a free PDF to print.

Each solution is followed by a scoring guide. Award points for correct hypotheses, expected counts, the chi-square statistic with df, the p-value, and a conclusion linked to the context.

AP Statistics · Common Core State Standards: AP STAT 8.1, AP STAT 8.2, AP STAT 8.3, AP STAT 8.4, AP STAT 8.5, AP STAT 8.6, AP STAT 8.7

Chi square test answers: worked solutions

Question 1 – Goodness-of-fit test [5 pts]

Right-tail p-value.

\(H_0: p_A=p_B=p_C=p_D=0.25\). Random sample; expected counts all 20, at least 5. \(\chi^{2}=6.00,\ df=3\), p-value \(=0.1116\).

Fail to reject H0: not convincing evidence that the categories are not equally likely.

Scoring: 1 point each for hypotheses, conditions, chi-square and df, p-value, and conclusion in context.

Question 2 – Expected counts [3 pts]

Observed (expected).

Expected = row total x column total / 150: 28.0, 32.0, 42.0, 48.0. \(df=(2-1)(2-1)=1\).

Expected counts as shown; df = 1.

Scoring: 2 points for expected counts (1 if one error), 1 point for df.

Question 3 – Chi-square statistic [3 pts]

\(\chi^{2}=\sum\frac{(O-E)^{2}}{E}=4.02,\ df=1\), p-value \(=0.0450\).

chi-square = 4.02; p = 0.0450.

Scoring: 1 point for the sum, 1 point for the statistic, 1 point for the p-value.

Question 4 – Which test [2 pts]

(a) One sample, two variables: independence. (b) Two populations, one variable: homogeneity.

(a) independence; (b) homogeneity.

Scoring: 1 point each.

Question 5 – Conditions [2 pts]

The smallest expected count is 4, which is less than 5.

No, because one expected count is below 5.

Scoring: 1 point for the answer, 1 point for the reason.

Question 6 – Interpretation [2 pts]

Interpreting chi-square.

The p-value 0.0074 is less than 0.05.

There is convincing evidence of an association between type of music and age group.

Scoring: 1 point for rejecting H0, 1 point for the conclusion in context.

Question 7 – Degrees of freedom [1 pt]

\(4-1=3\) and \((4-1)(3-1)=6\).

3 and 6.

Scoring: 1 point for both.

Question 8 – Choose a procedure [2 pts]

(a) One-sample t interval. (b) Two-sample z test (or interval) for the difference of proportions.

(a) one-sample t interval; (b) two-proportion z procedure.

Scoring: 1 point each.

Total score

Total

20 points

Key facts behind this chi square test answers

Key facts

chi-square = sum of (O – E)^2 / E; right-tailed p-value; distribution skewed right.
Goodness-of-fit: one categorical variable; df = number of categories – 1.
Two-way table: E = row total x column total / grand total; df = (r – 1)(c – 1).
Conditions: random sample or random assignment, and all expected counts at least 5.
Homogeneity: compare distributions across groups; independence: two variables in one sample.

How to use this chi square test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

All AP Statistics chapters: AP Statistics math.

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