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In hypothesis testing, sample proportions provide an estimate of the population proportions based on sampled data.
For our example, the researcher observed two different groups: Republicans and Democrats. Calculating sample proportions involves dividing the number of favorable responses by the total number of respondents in each sample.

• Republicans: 46 out of 200 sampled, giving a sample proportion, \( \hat{p}_1 = \frac{46}{200} = 0.23 \).
• Democrats: 34 out of 200 sampled, giving a sample proportion, \( \hat{p}_2 = \frac{34}{200} = 0.17 \).

The sample proportions \( \hat{p}_1 \) and \( \hat{p}_2 \) represent the fraction of each group that strongly favors the death penalty, allowing us to make statistical inferences about the entire population of Republicans and Democrats.

A pooled proportion is used in hypothesis testing when comparing two independent sample proportions.
It provides a combined estimate assuming no difference between the groups under the null hypothesis.Combining data from both groups, we calculate the pooled proportion \( \hat{p} \) as follows:\[ \hat{p} = \frac{46 + 34}{200 + 200} = \frac{80}{400} = 0.20 \]

• Numerator: Total number of successes from both samples (46 Republicans + 34 Democrats).
• Denominator: Sum of both sample sizes (200 Republicans + 200 Democrats).

The pooled proportion \( 0.20 \) reflects a combined estimate of those strongly favoring the death penalty under the hypothesis that there is no actual distinction between the two populations.

Standard Error (SE) measures the variability of the sampling distribution of a statistic, particularly concerning the difference between two sample proportions.
It helps gauge the precision of our sample estimates.The formula for calculating SE in this context is:\[ SE = \sqrt{\hat{p} (1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \]With our data:

• \( \hat{p} = 0.20 \): the pooled proportion.
• \( n_1 \) and \( n_2 \): size of the Republican and Democrat samples, respectively.

Substituting the values results in:\[ SE = \sqrt{0.20 \times (1 - 0.20) \left( \frac{1}{200} + \frac{1}{200} \right)} = \sqrt{0.20 \times 0.80 \times \frac{1}{100}} = \sqrt{0.0016} = 0.04 \]The SE of 0.04 indicates the spread of the sample mean differences from the true population mean, informing us about the reliability of our hypothesis test.

The test statistic quantifies how far the sample statistic is from the null hypothesis, measured in units of the standard error.
It's a crucial part in determining whether or not we can reject the null hypothesis.In this scenario, the test statistic \( z \) is critical for comparing the two group proportions. The formula for \( z \) is:\[ z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \]Where:

• \( \hat{p}_1 = 0.23 \): the sample proportion for Republicans.
• \( \hat{p}_2 = 0.17 \): the sample proportion for Democrats.
• \( SE = 0.04 \): the previously calculated standard error.

With these values, our calculation becomes:\[ z = \frac{0.23 - 0.17}{0.04} = \frac{0.06}{0.04} = 1.5 \]The test statistic of 1.5 is compared against a critical value (usually from a z-table) to decide whether to reject the null hypothesis. In this case, we compare it to the critical value of 1.645 and conclude whether there's supporting evidence for the researcher's claim.