Performing a Chi-Square Independence Test In Exercises 13–28, perform the indicated chi-square independence test by performing the steps below.
a. Identify the claim and state H₀ and Hₐ


b. Determine the degrees of freedom, find the critical value, and identify the rejection region.


c. Find the chi-square test statistic.


d. Decide whether to reject or fail to reject the null hypothesis.


e. Interpret the decision in the context of the original claim.


Choosing a College The contingency table shows the results of a survey asking 1858 parents and students of different incomes what their deciding factor was in choosing a college. At α=0.01, can you conclude that the deciding factor in choosing a college is related to the income of the family? (Adapted from Sallie Mae)

Verified step by step guidance

1. Step 1: Identify the claim and state the hypotheses. The claim is that the deciding factor in choosing a college is related to the family income. Set the null hypothesis H₀: The deciding factor is independent of family income. Set the alternative hypothesis Hₐ: The deciding factor is related to family income.
2. Step 2: Determine the degrees of freedom (df). Use the formula df = (number of rows - 1) × (number of columns - 1). Here, there are 3 income groups (rows) and 4 deciding factors (columns), so df = (3 - 1) × (4 - 1) = 2 × 3 = 6. Then, find the critical value from the chi-square distribution table at α = 0.01 and df = 6. Identify the rejection region as chi-square values greater than this critical value.
3. Step 3: Calculate the expected counts for each cell in the contingency table using the formula: $\mathrm{Expected}=\frac{\mathrm{Row\; total}\times \mathrm{Column\; total}}{\mathrm{Grand\; total}}$. Do this for all 12 cells.
4. Step 4: Compute the chi-square test statistic using the formula: $\chi 2^{}={\sum }_{\mathrm{all\; cells}}^{}\frac{{(}^{\mathrm{Observed}}}{2}\mathrm{Expected}$. Sum the squared differences between observed and expected counts divided by expected counts for all cells.
5. Step 5: Compare the calculated chi-square statistic to the critical value. If the statistic is greater than the critical value, reject the null hypothesis; otherwise, fail to reject it. Finally, interpret the decision in context: state whether there is sufficient evidence at the 0.01 significance level to conclude that the deciding factor in choosing a college is related to family income.

Key Concepts

Chi-Square Test of Independence

This test determines whether two categorical variables are related or independent. It compares observed frequencies in a contingency table to expected frequencies calculated under the assumption of independence. A significant result suggests a relationship between the variables.

Degrees of Freedom and Critical Value

Degrees of freedom for a chi-square test of independence are calculated as (number of rows - 1) × (number of columns - 1). The critical value is found using the chi-square distribution table at a chosen significance level (α). It defines the rejection region for the test.

Hypothesis Testing and Interpretation

Hypothesis testing involves stating a null hypothesis (H₀: variables are independent) and an alternative hypothesis (Hₐ: variables are related). After calculating the test statistic, compare it to the critical value to decide whether to reject H₀. The conclusion should be interpreted in the context of the problem.