Question

I mostly need help with question 2(I put Q1 on here for reference) 1. The Gallup organization periodically polls adults living in the U.S. about their approval or disapproval of the job the President of the U.S. is doing. At the end of President Trump’s first month in office, January 2017, his approval rating was 45%, while at the end of last month, March, 2020, his approval rating was 49%. For the purpose of this problem, suppose that the sample size was 200 in both months, that each represented a simple random sample of adults living in the U.S. at that time, and that the two samples were chosen independently of each other. Is there evidence for a real change in opinion among U.S. adults, or could this result be due just to chance? 2. Consider the same setup as in problem 1, but suppose the participants who were randomly selected in March, 2020 were also asked whether they approved of the job being done by U.S. Speaker of the House, Nancy Pelosi, and the percentage was 39%. Would it be appropriate to use the sample values of 49% and 39% to construct a two-sample z test statistic, as we did in problem 1, for the null hypothesis that the approval ratings between President Trump and Speaker Pelosi were the same in March, 2020? YES or NO (circle one) Explain your answer: When do you use a Z test for paired samples over two sample Z statistic test?

Answer 1

Problem 1:

Let denote the approval rating of of the job the President of the U.S. is doing by two independent samples of size (=n) 200, recorded in January 2017 and March 2020 respectively.

By Normal approximation of the data,

Hence, the two sample means are obtained as:

,

Let denote the mean approval rating of of the job the President of the U.S. is doing by two independent samples of size (=n) 200 each, recorded in January 2017 and March 2020 respectively.

To test: Vs

Hence, the appropriate statistical test to test the above hypothesis would be an independent sample test. Assuming that the population standard deviations () are known, we may go for a two-sample z statistic test.

Problem 2:

Here, the same sample of size 200, chosen in in March 2020 who had provided their approval rating of of the job the President of the U.S. is doing were also asked to provide their approval rating of of the job the U.S. Speaker of the House is doing. Here, we are studying the 2 dependent groups of observations, since recording of the values are done on the same sample.

Let denote the approval ratings for President Trump and Speaker Pelosi respectively. Let denote the difference in their rating for the their jobs.

To test: Vs

where

Hence, here, the appropriate statistical test to test the above hypothesis would be a Z test for paired samples.

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Would it be appropriate to use the sample values of 49% and 39% to construct a two-sample z test statistic, as we did in problem 1, for the null hypothesis that the approval ratings between President Trump and Speaker Pelosi were the same in March, 2020? NO

When do you use a Z test for paired samples over two sample Z statistic test?

A Z test for paired samples is used over two sample Z statistic test when the data recorded in the two groups are paired / dependent. (As mentioned in the explanation above.)