A working paper (Mark Aguiar and Erik Hurst, “Measuring Trends in Leisure: The Allocation of Time over Five Decades,” 2006) for the Federal Reserve Bank of Boston concluded that average leisure time spent per week by women in 2003 was 33.80 hours and 37.56 hours for men. The sample standard deviations were 40 and 70, respectively. These results were obtained from samples of women and men of size 8,492 and 6,752, respectively. In this study, leisure refers to the time individuals spent socializing, in passive leisure, in active leisure, volunteering, in pet care, gardening, and recreational child care. Assume that the amount of leisure time spent by men and women have normal distributions with equal population variances.
a. Determine the pooled estimate of the common populations’ standard deviation.
b. Produce the margin of error to estimate the difference of the two population means with a confidence level of 95%.
c. Calculate a 95% confidence interval for the difference in the average leisure time between women and men.
d. Do your results in part c indicate that the average amount of men’s leisure time was larger than that of women in 2003? Support your assertions.
e. Would your conclusion in part d change if you did not assume the population variances were equal?