In each exercise,
b. find the critical value and identify the rejection region,


In Exercises 1 and 2, use the table, which lists the distribution of educational achievement for people in the United States ages 25 and older. It also lists the results of a random survey for two additional age groups. (Adapted from U.S. Census Bureau)



Use the data for 30- to 34-year-olds and 65- to 69-year-olds to test whether age and educational attainment are related. Use α=0.01.

Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). H₀: Age and educational attainment are independent. H₁: Age and educational attainment are related.
2. Step 2: Choose the significance level (α = 0.01) and identify the test statistic to be used. In this case, use the chi-square test for independence.
3. Step 3: Calculate the expected frequencies for each cell in the table using the formula: E = (row total × column total) / grand total. This ensures the expected frequencies are based on the assumption of independence.
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.
5. Step 5: Determine the critical value from the chi-square distribution table using the degrees of freedom (df = (number of rows - 1) × (number of columns - 1)) and the significance level α = 0.01. Compare the test statistic to the critical value to decide whether to reject or fail to reject the null hypothesis.

Key Concepts

Critical Value

The critical value is a point on the scale of the test statistic beyond which we reject the null hypothesis. It is determined based on the significance level (α) and the type of test being conducted (one-tailed or two-tailed). For example, with α = 0.01 in a two-tailed test, the critical values correspond to the extreme 1% of the distribution, indicating where the test statistic must fall to reject the null hypothesis.

Rejection Region

The rejection region is the range of values for the test statistic that leads to the rejection of the null hypothesis. It is defined by the critical values and represents the area in the tails of the distribution where the observed data would be considered statistically significant. In hypothesis testing, if the calculated test statistic falls within this region, we conclude that there is enough evidence to reject the null hypothesis.

Chi-Square Test of Independence

The Chi-Square Test of Independence is a statistical method used to determine if there is a significant association between two categorical variables. In this context, it can be applied to assess whether age groups and educational attainment levels are related. The test compares the observed frequencies in each category to the expected frequencies under the assumption of independence, providing a p-value to help make decisions about the null hypothesis.