3. The heights of men and women are approximately normally distributed. Suppose we wish to estimate the average difference in heights between men and women attending a certain university. A random sample of seven men has average height 68.5 inches with standard deviation 3.0 inches, while a random sample of seven women has average height 65.5 inches with standard deviation 2.5 inches. (a) Explain why we would use the t-distribution and not the standard normal distribution to calculate P-values and confidence intervals in this case. (b) According to Welch's approximation, the number of degrees of freedom is 11.6. Using the fact that the R code qt (. 975, df=11. 62) gives the value 2.187, find an approximate 95% confidence interval for the average difference in heights between men and women at the university. (c) The confidence interval you calculated in part (b) contains the value 0. Should you conclude that there is no difference between the average height of men and the average height of women at the university? Explain why or why not