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In hypothesis testing, the null and alternative hypotheses serve as the foundation for statistical decision-making. The **null hypothesis** ( H_0 ) typically assumes that there is no effect or no difference between groups. It's a position of skepticism or the status quo. In our example, the null hypothesis is that the proportion of men who believe the division of household duties is fair is equal to the proportion of women who feel the same: \( H_0: p_1 = p_2 \). The **alternative hypothesis** ( H_1 ) represents what we aim to support through evidence. It's a statement of change or effect. Here, it suggests that the proportion of men believing the division is fair is larger than that of women: \( H_1: p_1 > p_2 \). The outcome of the test, based on data, will either allow us to reject the null hypothesis in favor of the alternative or fail to do so.

Sampling proportions are calculated when we want to estimate the percentage of a population that shares a specific characteristic. They give us a way to understand what part of the sample holds a particular view or trait. In this problem, the sampling proportion for men ( \( \hat{p_1} \) ) is calculated as \( \frac{990}{1500} = 0.66 \). This means 66% of the sampled men believe the division of household duties is fair. Similarly, the sampling proportion for women ( \( \hat{p_2} \) ) is \( \frac{970}{1600} \approx 0.60625 \), indicating approximately 60.63% of women feel the division is fair. These calculations are crucial as they convey the actual data we are comparing in hypothesis testing.

A pooled proportion is needed when comparing two population proportions. This provides a combined estimate from both sample groups, offering a standardized measure under the null hypothesis where both proportions are believed to be equal. For our problem, the pooled proportion ( \( \hat{p} \) ) is computed as:

• Combine favorable outcomes from both groups: \( 990 + 970 = 1960 \)
• Combine sizes of both samples: \( 1500 + 1600 = 3100 \)
• Calculate \( \hat{p} = \frac{1960}{3100} \approx 0.63226 \)

This pooled proportion represents a weighted overall proportion and is pivotal for calculating the standard error when testing differences between sample proportions.

The standard error (SE) quantifies the variability or "margin of error" of the sampling distribution of a statistic. It is used to understand how much the sample proportion will vary from the actual population proportion. For our case, the standard error is calculated based on the pooled proportion:

• Formula: \( SE = \sqrt{\hat{p}(1 - \hat{p}) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)} \)
• Substitute \( \hat{p} \approx 0.63226 \), \( n_1 = 1500 \), and \( n_2 = 1600 \)
• Standard Error \( \approx 0.01978 \)

This standard error becomes instrumental in calculating the test statistic, allowing for comparison across different sample sizes.

A Z-test is used to decide whether to reject the null hypothesis by determining how far the observed sample proportion is from the hypothesized population proportion in terms of standard errors. To undertake a Z-test, we compute the Z-test statistic:

• Calculate the difference in sample proportions: \( \hat{p_1} - \hat{p_2} \)
• Divide by the standard error: \( \frac{0.66 - 0.60625}{0.01978} \approx 2.72 \)

This Z-score tells us how many standard deviations away our observation is from the null hypothesis. For this test, with a significance level of 0.01, the critical Z-value is around 2.33. Since the observed Z-value of 2.72 exceeds this, we reject the null hypothesis. This conclusion is also supported by the p-value (\( \approx 0.0033 \)), confirming a statistically significant result.