To obtain the sampling distribution of the difference between two sample means for independent samples, as stated in Key Fact 10.1 on page 394, we need to know that, for independent observations, the difference of two normally distributed variables is also a normally distributed variable. In this exercise, you are to perform a computer simulation to make that fact plausible.
a. Simulate 2000 observations from a normally distributed variable with a mean of 100 and a standard deviation of 16.
b. Repeat part (a) for a normally distributed variable with a mean of 120 and a standard deviation of 12.
c. Determine the difference between each pair of observations in parts (a) and (b).
d. Obtain a histogram of the 2000 differences found in part (c). Why is the histogram bell shaped?