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In hypothesis testing, we often compare proportions from different samples to draw conclusions about a population. A sample proportion represents the fraction or percentage of the sample that exhibits a certain characteristic. In the example given, we calculate the sample proportions of urban and rural dwellers who agree with a statement about regional preferences.

To find the sample proportion, divide the number of positive responses by the total number of respondents in each group. For instance:- Urban dwellers: 417 agreed out of 646, giving a proportion of \( p_1 = \frac{417}{646} \approx 0.645 \).- Rural dwellers: 78 agreed out of 154, giving a proportion of \( p_2 = \frac{78}{154} \approx 0.506 \).

These proportions provide the basis for testing whether there's a significant difference in preferences between the two groups.

When comparing two proportions, we often calculate a pooled proportion. This helps us better estimate the common population proportion under the null hypothesis that both groups are alike.

To find the pooled proportion, you must sum the total successes and divide by the total number of respondents from both samples. In this exercise, this is calculated as:- Total successes = 417 (urban) + 78 (rural) = 495- Total respondents = 646 (urban) + 154 (rural) = 800

Thus, the pooled proportion \( \hat{p} = \frac{495}{800} \approx 0.619 \).

The pooled proportion assumes that the combined samples follow the same underlying distribution, and it assists in calculating the standard error.

The standard error quantifies the variability of the sample statistic (like the difference between two proportions) from the actual population parameter. It provides a measure of how much we expect the sample statistic to fluctuate due to sampling variability.

In the context of two sample proportions, the standard error is calculated as follows:\[ SE = \sqrt{ \hat{p}(1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right) } \]Insert the values obtained: \( \hat{p} = 0.619 \), \( n_1 = 646 \), and \( n_2 = 154 \).

This results in: \[ SE = \sqrt{ 0.619 \times 0.381 \left( \frac{1}{646} + \frac{1}{154} \right) } \approx 0.051 \]

The standard error is crucial for calculating the test statistic, which in turn helps determine whether the observed difference is statistically significant.

The p-value is a probability measure that helps us decide whether to reject the null hypothesis. It represents the probability of observing a test statistic at least as extreme as the one calculated, assuming the null hypothesis is true.

In hypothesis testing with two proportions, once we calculate the test statistic (z-score), we reference a standard normal distribution to find the p-value. For this problem, the z-score calculated was approximately 2.725.

By checking this z-score against standard normal distribution tables or using computational tools, we find the p-value \( \approx 0.0064 \) for a two-tailed test.

This p-value indicates how strongly the sample data contradict the null hypothesis. Typically, if the p-value is less than the chosen significance level (commonly \( \alpha = 0.05 \)), we reject the null hypothesis, suggesting a real difference between the groups.