An economist was interested in studying the impact of the recession on dining out, including drive thru meals at fast food restaurants. A random sample of forty-eight families of four with discretionary incomes between $300 and $400 per week indicated that they reduced their spending on dining out by an average of $31.47 per week, with a sample standard deviation of $10.95. Another random sample of 42 families of five with discretionary incomes between $300 and $400 per week reduced their spending on dining out by an average $35.28 per week, with a sample standard deviation of $12.37. Assume that the distributions of reductions in weekly dining-out spendings for the two groups have the same population standard deviation.
a. Construct a 90% confidence interval for the difference in the mean weekly reduction in dining out spending levels for the two populations.
b. Using a 5% significance level, can you conclude that the average weekly spending reduction for all families of four with discretionary incomes between $300 and $400 per week is less than the average weekly spending reduction for all families of five with discretionary incomes between $300 and $400 per week?