Mandate Perceptions. L. Grossback et al. examined mandate...

Mandate Perceptions. L. Grossback et al. examined mandate perceptions and their causes in the paper "Comparing Competing Theories on the Causes of Mandate Perceptions" (American Joumal of Political Science, Vol. 49, Issue $2,$ pp. $406-419$ ). Following are data on the percentage of members in each chamber of Congress who reacted to mandates in various years. At the $10 \%$ significance level, do the data provide sufficient evidence to conclude that, on average, the percentage of senators who react to mandate perceptions each year exceeds that of representatives?

Key Concepts

One-Tailed Test

A one-tailed test is used when the research hypothesis specifies a direction of the expected difference between groups. In this case, the focus is specifically on whether the average percentage for senators is greater than that for representatives, which requires testing the hypothesis in one direction only. This affects the calculation and interpretation of the p-value and critical values in the statistical test.

Significance Level

The significance level, often denoted as alpha (?), represents the threshold for rejecting the null hypothesis. It is the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. In this question, a 10% significance level (? = 0.10) is used, meaning there is a 10% risk of incorrectly concluding that senators react more than representatives when they truly do not.

p-value

The p-value is the probability of obtaining test results at least as extreme as the ones observed during the study, under the assumption that the null hypothesis is correct. In hypothesis testing, if the p-value is less than or equal to the significance level, the evidence is considered strong enough to reject the null hypothesis in favor of the alternative hypothesis.

Hypothesis Testing

Hypothesis testing is a statistical procedure used to evaluate whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. In the context of this question, the process involves setting up a null hypothesis (that there is no difference or that senators do not exceed representatives in reacting to mandate perceptions) and an alternative hypothesis (that senators do exceed representatives), and then using sample data to assess which hypothesis is more supported by the evidence.

Null and Alternative Hypotheses

The null hypothesis (H?) typically states that there is no difference or that the difference is equal to zero between the groups being compared, while the alternative hypothesis (H?) asserts that there is a difference in a specific direction. For this scenario, the alternative hypothesis specifically posits that, on average, a higher percentage of senators react to mandates compared to representatives, making it a one-tailed test.

Transcript

00:01 All right, we're doing a man -witney test on the comparison of reactions, whether senators have more reactions than the other representatives.

00:09 Since they are our first demographic or first sample, we're going to say that mu -1 is greater than mu -2.

00:18 At the nine sample size of nine and nine, at the 10 % significance level, we go in the back of the book and find out that our critical value is 101.

00:28 One.

00:30 So the next step that we have to do is make that stem and leaf plot of the senator reactions.

00:39 So we can start that off by starting at the v10s.

00:45 Yep, the tens place.

00:47 So we're going to go one, two, and up to three.

00:55 Oh, even four.

00:56 Okay, so it's 40s.

00:57 All right.

01:00 So we'll go ahead and copy those numbers in there with this back -to -back stem and leaf.

01:05 Plot.

01:10 So we'll start from the lowest number.

01:12 That's going to be 17 on this left hand side.