Question

Perform these steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. In a specific year \(53.7 \%\) of men in the United States were married and \(50.3 \%\) of women were married. Two independent random samples of 300 men and 300 women found that 178 men and 139 women were married (not to each other). At the 0.05 level of significance, can it be concluded that the proportion of men who were married is greater than the proportion of women who were married?

Short answer

The data supports the claim that a greater proportion of men are married compared to women.

Step-by-step solution

State the Hypotheses

We need to set up the null and alternative hypotheses. The null hypothesis ( H_0 ) states that the proportion of men who are married ( p_m ) is equal to the proportion of women who are married ( p_w ). In symbols: H_0: p_m = p_w . The alternative hypothesis ( H_a ) suggests that the proportion of men who are married is greater than the proportion of women who are married. In symbols: H_a: p_m > p_w .

Identify the Claim

The claim in this scenario is that the proportion of men who are married is greater than that of women ( p_m > p_w ). This matches our alternative hypothesis ( H_a ).

Find the Critical Value

Using a significance level of 0.05 and a one-tailed test (since the test is looking for 'greater than'), we find the critical value for the z-distribution. For a 0.05 significance level in a one-tailed test, the critical value is 1.645 .

Compute the Test Value

To find the test statistic, use the formula for the difference between two proportions:\[z = \frac{(\hat{p}_m - \hat{p}_w)}{\sqrt{ \hat{p}(1-\hat{p})(\frac{1}{n_m}+\frac{1}{n_w}) }}\]Where \hat{p}_m = \frac{178}{300}, \hat{p}_w = \frac{139}{300}, and overall sample proportion \hat{p} = \frac{178+139}{600}.Calculating these:\hat{p}_m = 0.5933,\hat{p}_w = 0.4633, and \hat{p} = 0.5283.Put these values back into the formula to find z:\[z = \frac{(0.5933 - 0.4633)}{\sqrt{0.5283(1-0.5283)(\frac{1}{300}+\frac{1}{300})}} = 3.6212\].

Make the Decision

Since the calculated test statistic value ( 3.6212 ) is greater than the critical value ( 1.645 ), we reject the null hypothesis ( H_0 ).

Summarize the Results

We have sufficient evidence at the 0.05 level of significance to reject the null hypothesis and support the claim that the proportion of men who were married is greater than the proportion of women who were married.

Key concepts

Null and Alternative Hypotheses

In hypothesis testing, the first step is to establish the null and alternative hypotheses. These hypotheses are simply statements about the data or population parameters that we want to test.

The **null hypothesis** (\(H_0\)) typically represents the "no effect" or "no difference" scenario, serving as a default position that we aim to test against. In this exercise, the null hypothesis \(H_0: p_m = p_w\) indicates that there is no difference in marriage proportions between men and women.

On the other hand, the **alternative hypothesis** (\(H_a\)) represents what we are testing for and usually suggests some difference or change. Here, the alternative hypothesis \(H_a: p_m > p_w\) suggests that a greater proportion of men are married compared to women. This hypothesis aligns with the claim we are investigating.

These hypotheses are the foundation for the hypothesis test, determining the aim and focus of the analysis. The test will ultimately assess whether there's enough evidence in the sample data to reject the null hypothesis.

Critical Value

A critical part of hypothesis testing is determining the critical value, which aids in deciding whether to reject the null hypothesis. The **critical value** is a point on the test distribution that divides the acceptance region from the rejection region.

The critical value depends on the chosen significance level (often denoted as \(\alpha\), and here it is 0.05) and the nature of the test (one-tailed or two-tailed). Since our hypothesis test explores whether one proportion is greater than another, it's a one-tailed test.

To find the critical value for a one-tailed test with a 0.05 significance level, we look at the z-distribution (standard normal distribution). For this setting, the critical value is approximately **1.645**. This number tells us that if our calculated test statistic is greater than 1.645, we have enough evidence to reject the null hypothesis.

Thus, determining the critical value is essential in hypothesis testing as it sets the threshold our test statistic needs to surpass to suggest a significant result.

Test Statistic

Once we've established the hypotheses and determined the critical value, the next step is to calculate the **test statistic**. This value measures how far our sample statistic diverges from the null hypothesis. For comparing two proportions, the test statistic is calculated using the z-test formula for proportions:

\[z = \frac{(\hat{p}_m - \hat{p}_w)}{\sqrt{ \hat{p}(1-\hat{p})(\frac{1}{n_m}+\frac{1}{n_w}) }}\]

Here, \(\hat{p}_m\) and \(\hat{p}_w\) are the sample proportions of married men and women, respectively. \(\hat{p}\) is the overall sample proportion. In our situation:

• \(\hat{p}_m = 0.5933\)
• \(\hat{p}_w = 0.4633\)
• \(\hat{p} = 0.5283\)

After substituting these values into the formula, we find the test statistic \(z = 3.6212\).

Calculating the test statistic is crucial because it quantifies the evidence against the null hypothesis by representing how much the sample data deviate from what the null hypothesis predicts.

Decision Rule for Hypothesis Testing

The **decision rule** for hypothesis testing provides a guideline for making the conclusion based on the calculated test statistic and the critical value.

In this scenario, we have a rule: if the test statistic is greater than the critical value, we reject the null hypothesis. Here:

• The test statistic we calculated is **3.6212**
• The critical value for a one-tailed test with a 0.05 significance level is **1.645**

Since **3.6212** is greater than **1.645**, our test statistic falls into the rejection region, meaning we have sufficient evidence to reject the null hypothesis.

Consequently, we conclude that the proportion of men who are married is indeed greater than the proportion of women who are married, with a confidence level of 0.05 (or 95% confidence).

The decision rule is integral in hypothesis testing as it translates the statistical findings into a clear and understandable decision, helping to affirm or refute the initial claim.