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The mean difference is a way to compare the average values between two groups. In this exercise, the two groups are Republicans and Democrats. The mean wasted cents for Republicans is 54, while for Democrats it is 41. To find out how much more Republicans think is wasted compared to Democrats, we calculate the mean difference. We simply subtract the Democrat mean from the Republican mean: 54 - 41 = 13 cents. This tells us that on average, Republicans believe 13 more cents per tax dollar are wasted compared to Democrats. Understanding the mean difference is helpful in comparing groups and identifying any significant variations in their responses.

Standard error measures how much the sample mean of a group is expected to vary. In this problem, we need to determine the standard error of the difference in mean wasted cents between Republicans and Democrats. The formula we use is: \[ SE = \sqrt{ \frac{s_R^2}{n_R} + \frac{s_D^2}{n_D} } \]. Here, \(s_R\) and \(s_D\) are the standard deviations for Republicans and Democrats, while \(n_R\) and \(n_D\) are their respective sample sizes.
Calculating step-by-step:
1. Square the standard deviations:
For Republicans: 2.9^2 = 8.41
For Democrats: 2.6^2 = 6.76
2. Divide by the sample sizes:
For Republicans: 8.41/513
For Democrats: 6.76/513
3. Sum these two values and take the square root:
\[ SE = \sqrt{ \frac{8.41}{513} + \frac{6.76}{513} } = \sqrt{0.0296} \approx 0.172 \]
This combined standard error helps us understand how much the mean difference (13 cents) might vary due to sampling variability.

The critical value is a threshold number that helps us determine the extent to which we can be confident in our interval. For the 95% confidence level used in the exercise, the critical value for a Z-distribution is 1.96. This means, statistically, that there is a 95% chance that the true mean difference lies within 1.96 standard errors of the observed mean difference.
To construct our 95% confidence interval, we multiply the critical value by the standard error and add/subtract this product from our mean difference:
\[ \text{Confidence interval} = (\bar{X}_R - \bar{X}_D) \pm z^* \times SE = 13 \pm 1.96 \times 0.172 = 13 \pm 0.337 \]
Thus, our confidence interval is (12.663, 13.337). This range tells us that we can be 95% confident that the true mean difference in perceived government waste between Republicans and Democrats lies within this interval. Interpreting this, we understand that there is a statistically significant difference was believed to be wasted.