Question

In exercise 7, the data on y = annual sales ($1000s) for new customer accounts and
x = number of years of experience for a sample of 10 salespersons provided the estimated
regression equation yˆ = 80 + 4x. For these data x = 7, o(xi − x)2 = 142, and s = 4.6098.
a. Develop a 95% confidence interval for the mean annual sales for all salespersons with
nine years of experience.
b. The company is considering hiring Tom Smart, a salesperson with nine years of experience.
Develop a 95% prediction interval of annual sales for Tom Smart.
c. Discuss the differences in your answers to parts (a) and (b).

Answer 1

a. To find the 95% confidence interval for the mean annual sales for salespersons with nine years of experience:

First, calculate the mean annual sales (¯) using the estimated regression equation: ¯=80+4(7)=108.

Then, use the formula: ¯±/2(/√).

Given =10 and =4.6098, and with degrees of freedom −2=8, find /2 (approximately 2.306).

Plug in the values: 108±2.306(4.6098/√10).

This gives the 95% confidence interval: approximately (104.205,111.795).

b. To determine the 95% prediction interval of annual sales for Tom Smart:

Calculate the predicted annual sales (^) using the estimated regression equation: ^=80+4(9)=116.

Use the formula: ^±/2(√1+1/+(−¯)2/∑(−¯)2).

Given =9 and the provided data, plug in the values to get the prediction interval: approximately (108.858,123.142).

c. The confidence interval estimates the mean annual sales for all salespersons with nine years of experience, while the prediction interval predicts the sales for an individual with nine years of experience. The prediction interval accounts for individual variation, while the confidence interval is focused on estimating the population mean.