AP Statistics Practice Quiz: Setting Up a Chi-Square Goodness of Fit Test

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 16 questions to check your progress.

Question 1 of 16

Which of the following best describes the general shape of chi-square distributions?

A)Symmetric and bell-shaped, centered at 0.B)Positive and skewed to the left.C)Positive and skewed to the right.D)Uniformly distributed between 0 and 1.

All Questions (16)

Which of the following best describes the general shape of chi-square distributions?

A) Symmetric and bell-shaped, centered at 0.

B) Positive and skewed to the left.

C) Positive and skewed to the right.

D) Uniformly distributed between 0 and 1.

Correct Answer: C

Based on the provided content, 'Chi-square distributions are positive and skewed right.' This means the values cannot be negative and the tail of the distribution extends to the right.

A researcher believes the distribution of M&M's colors is 20% blue, 20% orange, 20% green, 15% red, 15% yellow, and 10% brown. What is the correct null hypothesis (H0) for a chi-square goodness-of-fit test?

A) At least one of the color proportions is different from the claimed distribution.

B) The observed counts for each color will be equal to the expected counts.

C) The proportions of colors are p_blue=0.20, p_orange=0.20, p_green=0.20, p_red=0.15, p_yellow=0.15, and p_brown=0.10.

D) The sample proportions of colors are equal to the claimed proportions.

Correct Answer: C

The content states that for a goodness-of-fit test, 'H0 specifies proportions for each category.' Option C correctly lists the specific, hypothesized population proportions for all categories.

A marketing analyst wants to determine if the observed distribution of customer preferences for five different packaging designs matches the company's expected distribution. Which statistical test is most appropriate for this purpose?

A) A two-sample t-test for a difference in means.

B) A chi-square test for goodness of fit.

C) A chi-square test for independence.

D) A one-sample z-test for a proportion.

Correct Answer: B

The content specifies that 'For one categorical variable's distribution of proportions, the appropriate test is the chi-square test for goodness of fit.' Customer preference among five designs is a single categorical variable.

A school district claims that the distribution of its high school students across four schools (North, South, East, West) is 30%, 25%, 25%, and 20%, respectively. If a random sample of 200 students is taken, what is the expected count for North High School?

A) 30

B) 50

C) 60

D) 200

Correct Answer: C

The content states that 'Expected counts for a goodness-of-fit test are (sample size) * (null proportion).' For North High School, the calculation is 200 * 0.30 = 60.

Before conducting a chi-square goodness-of-fit test, which of the following conditions must be verified to ensure the validity of statistical inferences?

A) The sample size must be greater than 30 and the population distribution must be approximately Normal.

B) All observed counts must be greater than or equal to 5.

C) The data must come from a random sample or randomized experiment, and all expected counts must be at least 5.

D) The number of categories must be exactly two.

Correct Answer: C

The provided content explicitly states that for a chi-square goodness-of-fit test, one must 'check for independence and large counts (all expected counts >= 5).' Random sampling addresses independence, and the Large Counts condition is that all expected, not observed, counts are at least 5.

What does the chi-square statistic for a goodness-of-fit test fundamentally measure?

A) The probability that the null hypothesis is true.

B) The difference between the largest and smallest observed counts.

C) The sum of the expected counts for all categories.

D) A measure of the distance between the observed counts and the expected counts.

Correct Answer: D

According to the content, 'The chi-square statistic measures the distance between observed and expected counts relative to expected counts.' It quantifies how much the sample data deviate from what would be expected under the null hypothesis.

A city planner hypothesizes that the modes of transportation used by commuters are 60% personal vehicle, 25% public transit, 10% walking, and 5% other. What is the correct alternative hypothesis (Ha) for a chi-square goodness-of-fit test?

A) The proportions of commuters using each mode of transportation are all different from what the planner hypothesizes.

B) At least one of the proportions for the modes of transportation is different from the planner's hypothesis.

C) The observed counts from a sample will not match the expected counts.

D) The sample proportions are different from the hypothesized proportions.

Correct Answer: B

The content states, 'Ha states at least one proportion is different.' This is the correct formulation, as it does not require all proportions to be different, only that the null hypothesis as a whole is incorrect because one or more proportions differ.

In the context of a chi-square goodness-of-fit test, what do the 'expected counts' represent?

A) The number of observations that should be in each category if the alternative hypothesis were true.

B) The average number of observations across all categories in the collected sample.

C) The number of observations that would be expected in each category if the null hypothesis were true.

D) The actual number of observations recorded in each category from the sample data.

Correct Answer: C

The content defines expected counts as 'counts consistent with the null hypothesis.' They are calculated based on the sample size and the proportions specified in H0.

A biologist claims that four species of fish in a lake are distributed in the ratio 10:5:2:1. In a random sample of 60 fish, the researcher wants to perform a chi-square goodness-of-fit test. Which problem arises when checking the conditions?

A) The independence condition is violated because the fish were sampled from the same lake.

B) The Large Counts condition is violated because one of the observed counts might be less than 5.

C) The Large Counts condition is violated because at least one expected count is less than 5.

D) The data on fish species are not categorical.

Correct Answer: C

The total ratio is 10+5+2+1 = 18. The null proportions are 10/18, 5/18, 2/18, and 1/18. With a sample size of 60, the expected count for the fourth species is 60 * (1/18) ≈ 3.33. The content states that 'all expected counts >= 5'. Since 3.33 < 5, the Large Counts condition is violated.

How does the shape of a chi-square distribution change as its degrees of freedom increase?

A) It becomes more skewed to the right.

B) It becomes less skewed to the right.

C) It becomes more skewed to the left.

D) It remains unchanged, as all chi-square distributions have the same shape.

Correct Answer: B

The content states that for chi-square distributions, the 'skew lessens as degrees of freedom increase.' Since the distributions are inherently skewed right, a lessening of skew means they become less skewed to the right and more symmetric.

A casino manager wants to test if a six-sided die is fair. The null hypothesis is that the die is fair, meaning each outcome (1, 2, 3, 4, 5, 6) is equally likely. If the die is rolled 150 times, what is the expected count for any given outcome?

A) 1/6

B) 25

C) 30

D) 150

Correct Answer: B

If the die is fair, the null proportion for each of the six outcomes is 1/6. According to the content, the expected count is (sample size) * (null proportion). Therefore, the expected count for any outcome is 150 * (1/6) = 25.

A student is performing a chi-square goodness-of-fit test. They calculate the expected counts for five categories to be 24, 18, 12, 6, and 4. Why is it inappropriate to proceed with the test?

A) The sum of the expected counts is not equal to the sample size.

B) The independence condition is violated because there are five categories.

C) The Large Counts condition is not met because one of the observed counts might be less than 5.

D) The Large Counts condition is not met because one of the expected counts is less than 5.

Correct Answer: D

The content specifies that to make statistical inferences for a chi-square goodness-of-fit test, one must verify the condition of 'large counts (all expected counts >= 5)'. Since one of the calculated expected counts is 4, this condition is violated.

Which of the following is a correctly stated alternative hypothesis (Ha) for a chi-square goodness-of-fit test with four categories?

A) Ha: p1 ≠ 0.25, p2 ≠ 0.25, p3 ≠ 0.25, p4 ≠ 0.25

B) Ha: The observed counts are not equal to the expected counts.

C) Ha: At least one of the population proportions is different from the value stated in H0.

D) Ha: The sample proportions are not equal to the population proportions.

Correct Answer: C

The content states that 'Ha states at least one proportion is different.' Option A is incorrect because it requires all proportions to be different. Option B refers to sample statistics (counts) rather than population parameters (proportions). Option D confuses sample and population. Option C is the correct, general form for the alternative hypothesis.

In a chi-square goodness-of-fit test, a very small chi-square statistic (a value close to 0) would indicate which of the following?

A) The observed counts are very different from the expected counts, providing strong evidence against H0.

B) The observed counts are very close to the expected counts, providing weak evidence against H0.

C) The sample size was too small to meet the conditions for the test.

D) The null hypothesis is definitively proven to be true.

Correct Answer: B

The content explains that the chi-square statistic 'measures the distance between observed and expected counts.' A small distance (a small statistic) implies that the observed data fit the null hypothesis model well, thus providing little or no evidence to reject H0.

A company claims its cereal boxes contain prizes with the following distribution: 50% stickers, 30% rings, and 20% temporary tattoos. A consumer group buys 40 boxes of cereal to test this claim. What is the first step they must take after collecting the data but before calculating the chi-square statistic?

A) Formulate the null hypothesis that at least one proportion is different.

B) Calculate the expected counts for each prize category and verify that they are all at least 5.

C) Calculate the p-value and compare it to the significance level.

D) Assume the conditions are met because the sample size is greater than 30.

Correct Answer: B

After collecting data and stating hypotheses, the next crucial step is to check conditions. This involves calculating the expected counts to verify the Large Counts condition. The expected counts are: Stickers = 40 * 0.50 = 20; Rings = 40 * 0.30 = 12; Tattoos = 40 * 0.20 = 8. Since all are >= 5, the condition is met. This verification must happen before calculating the test statistic.

What is the formula used to calculate expected counts for a chi-square test for goodness of fit?

A) Sample size divided by the number of categories.

B) The observed count for that category.

C) Sample size times the null proportion for that category.

D) The square root of the sample size.

Correct Answer: C

The provided content explicitly states that 'Expected counts for a goodness-of-fit test are (sample size) * (null proportion).'