A researcher wanted to determine if the distribution of educational attainment of Americans today is different from the distribution in 2000. The distribution of educational attainment in 2000 was as follows:



Source: Statistical Abstract of the United States.


The researcher randomly selects 500 Americans, learns their levels of education, and obtains the data shown in the table. Do the data suggest that the distribution of educational attainment has changed since 2000? Use the α = 0.1 level of significance.

Verified step by step guidance

1. Step 1: Identify the type of hypothesis test to use. Since we want to compare the observed frequencies of educational attainment in the sample to the expected frequencies based on the 2000 distribution, we will use a Chi-square goodness-of-fit test.
2. Step 2: State the null and alternative hypotheses. The null hypothesis \(H_0\) is that the distribution of educational attainment has not changed since 2000, meaning the observed frequencies follow the same distribution as in 2000. The alternative hypothesis \(H_a\) is that the distribution has changed.
3. Step 3: Calculate the expected frequencies for each education category using the relative frequencies from 2000 and the sample size of 500. For each category, the expected frequency \(E_i\) is calculated as \(E_i = 500 \times p_i\), where \(p_i\) is the relative frequency from the 2000 distribution.
4. Step 4: Compute the Chi-square test statistic using the formula: \[\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}\] where \(O_i\) is the observed frequency and \(E_i\) is the expected frequency for each category.
5. Step 5: Determine the degrees of freedom, which is the number of categories minus 1 (i.e., \(df = 6 - 1 = 5\)). Then, compare the calculated Chi-square statistic to the critical value from the Chi-square distribution table at \(\alpha = 0.1\) and \(df = 5\). If the test statistic exceeds the critical value, reject the null hypothesis; otherwise, do not reject it.

Key Concepts

Chi-Square Goodness-of-Fit Test

This test evaluates whether the observed frequency distribution of a categorical variable differs from a specified expected distribution. It compares observed counts to expected counts under the null hypothesis, using the chi-square statistic to measure discrepancies. A significant result suggests the distributions are different.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) represents no change or difference—in this case, that the educational attainment distribution remains the same as in 2000. The alternative hypothesis (Ha) states that the distribution has changed. The test determines which hypothesis the data support.

Level of Significance (α) and Decision Rule

The significance level α (here, 0.1) is the threshold for rejecting the null hypothesis. If the p-value from the test is less than α, we reject H0, indicating evidence of change. This controls the probability of a Type I error, or falsely detecting a difference when none exists.