The Correct Answer and Explanation is:
The correct answer is: A two-sample z-test for a difference in proportions.
The most appropriate inference procedure to assess Mr. Wilcox’s claim is a two-sample z-test for a difference in proportions. This conclusion is based on three key aspects of the study’s design: the number of samples, the type of samples, and the nature of the data being collected.
First, the investigation aims to compare two distinct populations: the weather in Michigan and the weather in Mississippi. Because two separate groups are being analyzed to see if they differ, a two-sample test is required. This immediately rules out any one-sample procedure.
Second, the problem explicitly states that Mr. Wilcox and Mrs. Cornelius took “separate random samples.” This means the 30 days chosen to represent Michigan are independent of the 30 days chosen to represent Mississippi. The selection of days for one state did not influence the selection for the other. This independence is a crucial condition for a two-sample test and eliminates the possibility of using a matched pairs t-test, which is designed for dependent, or paired, data. For example, a paired test would have been appropriate if they had observed the weather in both states on the same 30 days.
Finally, the data collected is categorical. For each day, the recorded outcome is a qualitative label: either “cloudy” or “not cloudy.” When analyzing categorical data, we are interested in proportions, not means. The goal is to compare the proportion of cloudy days in Michigan to the proportion of cloudy days in Mississippi. Statistical tests involving proportions rely on the z-distribution, not the t-distribution. T-tests are reserved for comparing the means of quantitative data, such as average daily temperature.
Since the objective is to compare the proportions of a categorical variable from two independent samples, the two-sample z-test for a difference in proportions is the correct and most suitable method.
