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.


Ages and Goals You are investigating the relationship between the ages of U.S. adults and what aspect of career development they consider to be the most important. You randomly collect the data shown in the contingency table. At α=0.10, is there enough evidence to conclude that age is related to which aspect of career development is considered to be most important? (Adapted from The Harris Poll)

Verified step by step guidance

1. Step 1: Identify the claim and state the null hypothesis (H₀) and alternative hypothesis (Hₐ). The claim is that age is related to the aspect of career development considered most important. H₀: Age is not related to the aspect of career development considered most important (independence). Hₐ: Age is related to the aspect of career development considered most important (dependence).
2. Step 2: Determine the degrees of freedom (df), find the critical value, and identify the rejection region. Degrees of freedom are calculated using the formula df = (number of rows - 1) × (number of columns - 1). Here, there are 3 rows (age groups) and 3 columns (career development aspects), so df = (3 - 1) × (3 - 1) = 4. Using α = 0.10, find the critical value from the chi-square distribution table corresponding to df = 4. The rejection region is where the test statistic exceeds the critical value.
3. Step 3: Calculate the expected frequencies for each cell in the contingency table using the formula E = (row total × column total) / grand total. For example, for the cell corresponding to '18–26 years' and 'Learning new skills', calculate E = (row total for 18–26 years × column total for Learning new skills) / grand total. Repeat this for all cells in the table.
4. Step 4: Compute the chi-square test statistic using the formula χ² = Σ((O - E)² / E), where O represents the observed frequency and E represents the expected frequency for each cell. Sum the values for all cells to obtain the test statistic.
5. Step 5: Compare the calculated chi-square test statistic to the critical value. If the test statistic exceeds the critical value, reject the null hypothesis (H₀). Otherwise, fail to reject H₀. Interpret the decision in the context of the original claim: If H₀ is rejected, conclude that there is enough evidence to suggest that age is related to the aspect of career development considered most important. If H₀ is not rejected, conclude that there is not enough evidence to support the claim.

Key Concepts

Chi-Square Independence Test

The Chi-Square Independence Test is a statistical method used to determine if there is a significant association between two categorical variables. It compares the observed frequencies in each category of a contingency table to the frequencies expected if the variables were independent. A significant result indicates that the variables are related, while a non-significant result suggests independence.

Null and Alternative Hypotheses (H₀ and Hₐ)

In hypothesis testing, the null hypothesis (H₀) represents the default position that there is no effect or relationship between the variables being studied. The alternative hypothesis (Hₐ) posits that there is a significant effect or relationship. In the context of the Chi-Square test, H₀ would state that age and career development aspects are independent, while Hₐ would suggest they are related.

Degrees of Freedom and Critical Value

Degrees of freedom in a Chi-Square test are calculated based on the number of categories in the variables being analyzed, typically as (rows - 1) * (columns - 1). The critical value is a threshold that determines the rejection region for the null hypothesis. If the calculated Chi-Square statistic exceeds the critical value at a specified significance level (e.g., α=0.10), the null hypothesis is rejected, indicating a significant relationship between the variables.