According to a 2018 report by the CDC, the mean body mass index (BMI) for American adult men is 29.1 kg/m², while the mean for women is 29.6 kg/m². Suppose these are population averages.

a. Let X̅ be the sample average BMI of 50 randomly selected American adult men Y̅ be the sample average BMI of 75 randomly selected American adult women. What is the expected value of X̅ – Y̅? How does your answer depend on the specified sample sizes?

b. Suppose the population standard deviations of BMI are 4.7 for men and 6.2 for women (these values are consistent with the study). With the sample sizes given in part (a), what is the variance of the statistic X̅ – Y̅, and what is its standard deviation?

c. For the sample sizes given in part (a), what is the approximate distribution of X̅ – Y̅, and why?

d. Would the shape of the distribution in part (c) necessarily be the same for sample sizes of 5 men and 7 women? Explain.