A specialty manufacturer wants to purchase remnants of sheet aluminum foil.

The foil, all of which is the same thickness, is stored on 1,462 rolls, each containing a varying amount of foil. To obtain an estimate of the total number of square feet of foil on all the rolls, the manufacturer randomly sampled 100 rolls and measured the number of square feet on each roll. The sample mean was 47.4, and the sample standard deviation was 12.4.

a. Find an approximate 95% confidence interval for the mean amount of foil on the 1,462 rolls.

b. Estimate the total number of square feet of foil on all the rolls by multiplying the confidence interval, part a, by 1,462. Interpret the result.