In Section 10.2, we tested hypotheses regarding a population proportion using a z-test. However, we can also use the chi-square goodness-of-fit test to test hypotheses with k = 2 possible outcomes. In Problems 25 and 26, we test hypotheses with the use of both methods.
Living Alone? In 2000, 25.8% of Americans 15 years of age or older lived alone, according to the Census Bureau. A sociologist, who believes that this percentage is greater today, conducts a random sample of 400 Americans 15 years of age or older and finds that 164 are living alone.
b. Test the sociologist’s belief at the alpha=0.05 level of significance using the goodness-of-fit test.

Verified step by step guidance

1. Identify the null and alternative hypotheses for the goodness-of-fit test. The null hypothesis \(H_0\) states that the proportion of Americans living alone is still 25.8%, i.e., \(p = 0.258\). The alternative hypothesis \(H_a\) is that the proportion is greater than 25.8%, i.e., \(p > 0.258\).
2. Calculate the expected counts for each category based on the null hypothesis. Since there are two categories (living alone and not living alone), the expected count for living alone is \(E_1 = n \times p = 400 \times 0.258\), and for not living alone is \(E_2 = n \times (1 - p) = 400 \times (1 - 0.258)\).
3. Determine the observed counts from the sample. The observed count for living alone is given as \(O_1 = 164\), and for not living alone is \(O_2 = 400 - 164 = 236\).
4. Compute the chi-square test statistic using the formula: \(\chi^2 = \sum_{i=1}^2 \frac{(O_i - E_i)^2}{E_i}\) where \(O_i\) are observed counts and \(E_i\) are expected counts for each category.
5. Find the critical value from the chi-square distribution table with degrees of freedom \(df = k - 1 = 1\) at the significance level \(\alpha = 0.05\). Compare the calculated \(\chi^2\) statistic to this critical value to decide whether to reject or fail to reject the null hypothesis.

Key Concepts

Hypothesis Testing for Population Proportion

Hypothesis testing for a population proportion involves assessing whether the observed sample proportion significantly differs from a claimed population proportion. It starts with formulating null and alternative hypotheses, then calculating a test statistic to decide if the evidence supports the alternative claim at a given significance level.

Chi-Square Goodness-of-Fit Test

The chi-square goodness-of-fit test evaluates whether observed categorical data fit an expected distribution. For two categories, it compares observed counts to expected counts under the null hypothesis, using the chi-square statistic to determine if deviations are due to chance or indicate a significant difference.

Significance Level and Decision Rule

The significance level (alpha) is the threshold for rejecting the null hypothesis, commonly set at 0.05. If the test statistic's p-value is less than alpha, we reject the null hypothesis, concluding there is sufficient evidence to support the alternative hypothesis; otherwise, we fail to reject it.