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Proportions are a way to describe how a part relates to the whole. In statistics, when we talk about proportions, we often refer to the fraction of a population or sample that possesses a particular trait. For example, in the exercise, we have two specific groups - urban and rural residents - and we wish to know what fraction of each group favors increasing the highway speed limit. This is represented by the proportions \( p_1 \) and \( p_2 \) for urban and rural residents, respectively.
To find these proportions, you divide the number of people with the trait (favoring the speed limit increase) by the total number in the sample for each group. For instance, the sample proportion for urban residents is calculated as \( \hat{p_1} = \frac{63}{300} = 0.21 \), indicating that 21% of the urban residents sampled favor the increase. Understanding proportions is crucial because they help us to make predictions and decisions based on sample data.

When comparing two proportions, a pooled proportion can be very useful, especially in hypothesis testing. The pooled proportion is essentially the overall proportion of a trait across combined groups. This approach assumes that the proportions of interest are equal, hence pooling them gives a single estimate of population proportion.
In the context of our problem, we combine data from both urban and rural groups to calculate the pooled proportion. The formula used is \( \hat{p} = \frac{x_1 + x_2}{n_1 + n_2} \). For this exercise, \( \hat{p} = \frac{63 + 75}{300 + 180} = 0.2875 \), indicating about 28.75% overall favor the speed limit increase. This pooled proportion is vital for computing the standard error and, later, the test statistic needed to decide whether the sentiment differs significantly between the groups.

The Z-Test is a statistical method that allows researchers to compare sample percentages to find evidence supporting or refuting a hypothesis about populations. It is often used when dealing with large sample sizes. For proportions, a Z-Test helps to ascertain if there exists a significant difference between two proportions from different populations.
To carry out a Z-Test in this exercise, we need to calculate the test statistic, commonly noted as the z-score. It is derived using the formula \( z = \frac{\hat{p_1} - \hat{p_2}}{SE} \), where \( SE \) is the standard error. Given \( \hat{p_1} = 0.21 \), \( \hat{p_2} = 0.4167 \), and \( SE = 0.0463 \), the calculated z-score is approximately -4.45. A large z-score, far from zero, suggests that the difference in sample proportions is not likely to be from random chance.
Next, we check this z-score against critical z-values that define a range of values expected under the null hypothesis. With \( \alpha = 0.05 \), which is typically for two-tailed tests, the critical values are \( \pm 1.96 \). Since our z-score falls outside this range, we have enough evidence to reject the null hypothesis \( H_0 \), suggesting that the sentiment does differ between urban and rural residents.

The power of a statistical test is its capability to correctly reject the null hypothesis when it is false. A high power means the test is likely to detect significant effects. Typically, a power of 0.80 (or 80%) is considered adequate in research.
In this case study, the power is determined for the scenario where the true proportions differ: the true proportion of urban residents favoring the speed increase is 0.20, and for rural residents, it's 0.40. After computing the relevant standard error, we find the effect size, subsequently determining the z-score (approximately -4.14) for the difference in true proportions.
To find the power, we compute the probability of acquiring a test statistic as extreme as the critical value given the truth of the alternative hypothesis. If we look at the standard normal distribution, the power equals 1 minus the probability of committing a Type II error (\( \beta \)). The calculations here lead us to a significant power of approximately 0.9854, or 98.54%, indicating a very high likelihood of detecting the true difference.