Question: In order to compare the means of two populations, independent random samples of 400 observations are selected / from each population, with the following results:

Sample 1: \(\bar{X}\) = 5275, s₁ = 150

Sample 2: \(\bar{X}\) = 5240, s₁ = 200

a. Use a 95% confidence interval to estimate the difference between the population means \({{\mu }_{1}}-{{\mu }_{2}}\). Interpret confidence interval.

b. Test the null hypothesis Ho: \({{\mu }_{1}}-{{\mu }_{2}}\) = 0 versus Ha: \({{\mu }_{1}}-{{\mu }_{2}}\) \(\ne \) 0. Give the significance level of the test, and interpret the result.

c. Suppose the test in part b was conducted with the alter-native hypothesis Ha: \({{\mu }_{1}}-{{\mu }_{2}}\) > 0. How would your answer to part b change?

d. Test the null hypothesis Ho: \({{\mu }_{1}}-{{\mu }_{2}}\) = 25 versus Ha: \({{\mu }_{1}}-{{\mu }_{2}}\) \(\ne \) 25. Give the significance level, and interpret the result. Compare your answer to the test conducted in part b.

e. What assumptions are necessary to ensure the validity of the inferential procedures applied in parts a-d?