What are the degrees of freedom?

Answer: 6 df = (r-1)(c-1) = (4-1)(3-1) = 6.

Which test is appropriate?

Answer: Chi-square test of homogeneity Two separate samples (men, women) are compared on a single categorical variable (brand). This is a chi-square test of homogeneity.

To check for an association, the most informative comparison is:

Answer: Conditional distributions across rows or columns Comparing conditional distributions reveals whether the response distribution depends on the explanatory category.

AP Statistics · Topic 8.5

Setting Up a Chi-Square Test for Homogeneity or Independence Practice

Part of Inference for Categorical Data: Chi-Square.(VAR-8.D)

Practice questions

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Sample questions

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Sample 1difficulty 1/5
$A 4 \times 3 two-way table is analyzed for independence.$
$What are the degrees of freedom?$
- A
  $6$
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- B
  $7$
- C
  $12$
- D
  $11$
Why
$df = (r-1)(c-1) = (4-1)(3-1) = 6.$
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Sample 2difficulty 2/5
$Researchers take separate random samples of 100 men and 100 women, asking each their preference among three brands (A, B, C). They want to determine if preference distributions are the same for men and women.$
$Which test is appropriate?$
- A
  $Two-sample t-test$
- B
  $Chi-square test of independence$
- C
  $Chi-square goodness-of-fit$
- D
  $Chi-square test of homogeneity$
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Why
$Two separate samples (men, women) are compared on a single categorical variable (brand). This is a chi-square test of homogeneity.$
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Sample 3difficulty 2/5
$Marginal distributions of two categorical variables tell us about each variable separately.$
$To check for an association, the most informative comparison is:$
- A
  $Just the row totals$
- B
  $Just the grand total$
- C
  $Joint frequencies only$
- D
  $Conditional distributions across rows or columns$
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Why
$Comparing conditional distributions reveals whether the response distribution depends on the explanatory category.$
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Sample 4difficulty 2/5
$A researcher tests whether eye color and dominant hand are independent in a single random sample.$
$Which is the correct null hypothesis?$
- A
  $Distributions of hand are the same across eye colors$
- B
  $There is an association$
- C
  $Eye color and hand are independent$
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- D
  $All eye colors are equally likely$
Why
$For a chi-square test of independence with one sample, H0 states the two variables are independent (no association).$
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Sample 5difficulty 2/5
$For a chi-square test of independence, which condition must be checked using expected counts?$
$The large counts condition requires...$
- A
  $Sample size n >= 30$
- B
  $All counts >= 10$
- C
  $All observed counts >= 5$
- D
  $All expected counts >= 5$
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Why
$The large-counts condition for chi-square inference is that all expected counts are at least 5.$
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