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When discussing voter turnout, we want to understand the broader picture of participation in elections within different groups. The population proportion, denoted as \( p \), represents the fraction of the total population demonstrating a particular characteristic—in this case, voting in an election. If we consider the population of registered voters in each state, the sample proportion (\( \hat{p} \)) acts as an estimate of the true population proportion. It is calculated by dividing the number of voters who actually cast their vote by the total number of registered voters. In our exercise, the sample proportion for California is \( \hat{p}_1 = \frac{141}{288} \approx 0.4896 \) and for Colorado, it's \( \hat{p}_2 = \frac{125}{216} \approx 0.5787 \). These figures help us estimate the actual voting proportions in the entire voter populations of each state.

The significance level, often denoted by \( \alpha \), is a threshold set by the researcher to determine when to reject the null hypothesis \( H_0 \). It essentially represents the probability of making a Type I error, which is incorrectly rejecting a true null hypothesis. In our example, a 5% significance level (\( \alpha = 0.05 \)) was used, indicating we are willing to accept a 5% risk of this type of error. This level is crucial when testing for differences or changes in statistics as it guides the threshold for making decisions. If our calculated test statistic exceeds the critical value corresponding to this significance level, we have enough evidence to reject \( H_0 \) and conclude in favor of the alternative hypothesis \( H_a \).

Standard error is a measure of the variability or uncertainty in a statistic. It indicates how much a sample proportion will likely deviate from the true population proportion. In hypothesis testing, especially when comparing two population proportions, the standard error takes into account the combined variance from both samples. The formula used is \( SE = \sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)} \).

• In this exercise, \( \hat{p} \) is the pooled sample proportion, \( \hat{p} = \frac{266}{504} = 0.5278 \).
• The calculated standard error \( SE \approx 0.0441 \) informs us about the reliability of our estimate of the difference between two proportions.

This value helps us determine the distribution of our test statistic.

A test statistic is a standardized value derived from sample data to determine the degree of compatibility with the null hypothesis. In our case of comparing population proportions, we use the \( Z \)-test for hypothesis testing. The test statistic \( Z \) is computed using the formula: \( Z = \frac{\hat{p}_2 - \hat{p}_1}{SE} \).

• Here, \( \hat{p}_2 \) and \( \hat{p}_1 \) are the sample proportions for Colorado and California respectively, while \( SE \) is the standard error.
• The calculated \( Z \) value in our example is approximately \( 2.02 \).

This \( Z \) value is compared to the critical value from the standard normal distribution table (here, 1.645 for a 5% significance level and one-tailed test) to make a decision. If \( Z \) exceeds this critical value, as it does in our example, we reject the null hypothesis, suggesting a significant difference favoring the proportion in Colorado over California.