chi-square tests for two-way tables

Review chi-square tests for two-way tables for AP Statistics and try 2 free AP-style practice questions. 10 questions on this topic in AP Statistics · Unit 3 · Inference for Categorical Data: Proportions.

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Question 1

A psychologist surveys random samples of 100 students from each of three types of school (public, private and charter) about their level of academic pressure (low, moderate, high or overwhelming). A chi-square test for homogeneity gives $\chi^2 = 18.44$ with a p-value of about 0.005. Which of the following statements are true?

I. The test has $(3 - 1)(4 - 1) = 6$ degrees of freedom.
II. At the 0.05 level, there is convincing evidence that the distribution of pressure levels differs for at least one type of school.
III. A confidence interval for the difference between two school types in the proportion reporting overwhelming pressure could help show where the distributions differ.

Question 2

A government analyst is conducting a chi-square test for independence on a $3 \times 4$ contingency table (3 rows, 4 columns). Which of the following correctly describes the degrees of freedom and the shape of the chi-square distribution for this test?

Sample free-response question

A town tests samples from its water supply. Each sample is either contaminated or not. The safe standard is that at most 5 percent of samples are contaminated. The town tests H0: p = 0.05 against Ha: p > 0.05 every month.

(a) Describe a Type I error and a Type II error in this context.
(b) Which error is more serious for the town's residents? Explain.
(c) Should the town use a significance level of 0.01 or 0.10? Explain using part (b).
(d) Describe two ways the town could increase the power of its test.
(e) The test has power 0.70 if the true rate is 8 percent. Interpret this value.

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Answer, then see why

The price of gasoline rises 10% and the quantity demanded falls 5%. What happens to total revenue?

Not quite. The answer is B.

Why the other options are wrong

AI grading for free-response

Write your FRQ answer and get it scored point by point against the AP rubric, with the line from your answer that earned or missed each point.

Question 13 / 4 pts

Part a1 / 1 pts

Point 1: Earned

94% sure

Reason: Correctly identifies demand for gasoline as inelastic.

Evidence: "The price of gasoline rises and the quantity falls only a little."

Part b2 / 3 pts

Point 1: Earned

91% sure

Reason: Graph shows relatively inelastic demand.

Evidence: Draws a steep demand curve.

Point 2: Earned

88% sure

Reason: Correct new equilibrium price.

Evidence: Shifts supply left, P1 → P2.

Point 3: Not earned

82% sure

Reason: With inelastic demand, a price increase raises total revenue.

Evidence: "Revenue goes down."

Chi-square tests for two-way tables: AP Statistics Practice Questions | S&D Academy