In a recent conversation with Mrs. Cornelius, Mr. Wilcox claimed that the weather in Michigan was much cloudier than the weather in Mississippi. To investigate Mr. Wilcox took a random sample of 30 days from the past year and recorded whether or not it was cloudy in Michigan. Mrs. Cornelius took a separate random sample of 30 days from the past year and recorded whether or not it was cloudy in Mississippi. Which inference procedure is most appropriate to assess Mr. Wilcox’s claim?
In a recent conversation with Mrs. Cornelius, Mr. Wilcox claimed that the weather in Michigan was much cloudier than the weather in Mississippi. To investigate Mr. Wilcox took a random sample of 30 days from the past year and recorded whether or not it was cloudy in Michigan. Mrs. Cornelius took a separate random sample of 30 days from the past year and recorded whether or not it was cloudy in Mississippi. Which inference procedure is most appropriate to assess Mr. Wilcox’s claim
The Correct Answer and Explanation is:
Correct Answer:
Two-sample z-test (or two-proportion z-test) for comparing proportions
Explanation
To assess Mr. Wilcox’s claim that Michigan is cloudier than Mississippi, we are comparing the proportion of cloudy days between two independent samples — one from Michigan and one from Mississippi. Each observation in the samples is binary (cloudy or not cloudy), so we are dealing with proportions rather than means.
The most appropriate inference procedure in this case is the two-proportion z-test, which tests the difference between two population proportions. Here’s why:
- Binary Outcomes: The variable recorded is whether the day was cloudy — a categorical variable with only two outcomes (cloudy or not). Therefore, we are working with proportions, not averages.
- Two Independent Samples: The data comes from two separate, randomly selected samples — one from Michigan and one from Mississippi. Since the samples are independent and not paired (i.e., not the same days in both states), a two-sample test is appropriate.
- Objective: Mr. Wilcox’s claim is directional — he believes Michigan is cloudier than Mississippi. So, the hypothesis test would be:
- Null Hypothesis (H₀): p₁ = p₂ (the proportion of cloudy days is the same in both states)
- Alternative Hypothesis (H₁): p₁ > p₂ (the proportion of cloudy days in Michigan is greater than in Mississippi)
- Sample Size: Each sample includes 30 days, which is reasonably large for approximate normality if the success-failure condition is met (at least 10 cloudy and 10 non-cloudy days in each sample).
Using a two-proportion z-test, one can calculate the test statistic and p-value to assess whether the observed difference in cloudiness between the two states is statistically significant. If the p-value is sufficiently small, we would reject the null hypothesis and conclude that Michigan is indeed cloudier than Mississippi, supporting Mr. Wilcox’s claim.
