You decide to sample from two random variables X and Y. You take 61 observations from X and find it has a sample mean of 9.8 with sample variance of 1.2. You take 81 observations from Y and find it has a sample mean of 10.2 with sample variance of 1.4. Assume you believe the mean of X is smaller than the mean of Y.

a) Solve for the two sided 95% confidence interval for the true difference in means between X and Y. Assume you know they have unequal variance.

b) Solve for the approximate one sided 80% confidence interval for the true difference in means between X and Y. Assume you know they have equal variance and the direction is based on your beliefs.

c) Suggest which confidence interval you prefer. Justify.