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In hypothesis testing involving proportions, we examine the ratio of a specific characteristic within a population. For instance, in this problem, we aim to find out if conservative voters have a higher proportion of preference for paintings with fully clothed figures in comparison to liberal voters.

A proportion is denoted as \( p \), which represents the number of favorable outcomes divided by the total number of observations.

• For conservative voters: \( \hat{p}_1 = \frac{r_1}{n_1} = \frac{45}{59} \)
• For liberal voters: \( \hat{p}_2 = \frac{r_2}{n_2} = \frac{36}{62} \)

The "hat" symbol indicates an estimate from the sample, as opposed to the true unknown population proportion. Understanding these sample proportions is crucial for proceeding with the hypothesis test. These sample proportions summarize our sample data and help us move forward in testing whether there's a significant difference based on the hypothesis set.

The standard error (SE) provides a measure of the variability or "spread" of a sample statistic. It is critical in hypothesis testing, as it allows us to gauge how much the sample proportion differs from the true population proportion.

Specifically, in this problem, the standard error helps us understand the variability in the difference between two sample proportions. The calculation of SE for two proportions is given by the formula: \[ SE = \sqrt{ \hat{p}(1 - \hat{p}) \left(\frac{1}{n_1} + \frac{1}{n_2}\right) } \] This formula uses a pooled sample proportion \( \hat{p} \), which is a combined estimate from both groups:

• \( \hat{p} = \frac{r_1 + r_2}{n_1 + n_2} \)

Using the pooled proportion allows us to obtain a single measure of variability when comparing two samples. The SE essentially tells us how much variability there is in our estimation of the difference between the two proportions.

A test statistic is a standardized value that helps us determine whether to reject the null hypothesis. This value is computed from the sample data. For comparing two proportions, the test statistic used is the Z-score, which tells us how many standard errors the sample proportion difference is from the null hypothesis proportion difference.

The formula for the test statistic in this problem is: \[ Z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \] where \( \hat{p}_1 \) and \( \hat{p}_2 \) are the sample proportions, and SE is the standard error of the difference.

• If \( Z \) is large and positive, it suggests that \( \hat{p}_1 \) is significantly greater than \( \hat{p}_2 \).
• If \( Z \) is negative, it suggests the opposite.

In hypothesis testing, we compare this calculated \( Z \) value to a critical value to make inferences about the population proportions.

A critical value in hypothesis testing is a threshold that determines whether the null hypothesis should be rejected. For a one-tailed test, such as the one in this problem where we are testing if one proportion is greater, the critical value is derived from the significance level \( \alpha \).

At a significance level of \( \alpha = 0.05 \), the critical Z-value is 1.645 for a one-tailed test. If the computed test statistic \( Z \) is greater than 1.645, we reject the null hypothesis, suggesting that the proportion of conservative voters who prefer paintings with fully clothed people is indeed higher than that of liberal voters.

• This critical value represents the point beyond which only 5% of the distribution falls, indicating a rare event under the null hypothesis.

Thus, comparing the test statistic with the critical value helps us make a decision on whether the observed sample result could have happened by random chance or suggests a real difference in the populations.