In exercise 7 a sales manager collected the following data on y = annual sales (
ID: 3298459 • Letter: I
Question
In exercise 7 a sales manager collected the following data on y= annual sales ($1,000s) for new customer accounts and x = number of years of experience for the sample of 10 salespersons provided the estimated regression equation = 80 + 4x. For these data x = 7, (Xi -x )^2 = 142, and s = 4.6098
Salesperson
Years of Experience
Annual Sales ($1000s)
1
1
80
2
3
97
3
4
92
4
4
102
5
6
103
6
8
111
7
10
119
8
10
123
9
11
117
10
13
136
a. develop a 95% confidence interval for the means annual sales for the 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).
Salesperson
Years of Experience
Annual Sales ($1000s)
1
1
80
2
3
97
3
4
92
4
4
102
5
6
103
6
8
111
7
10
119
8
10
123
9
11
117
10
13
136
Explanation / Answer
we have
= 80 + 4x
for x=9
Y^=80+4(9)
=80+36
=116
SolutionA:
using R code:
DS.lm <- lm(AnnualSales ~ YearsofExperience, data = DS)
summary(DS.lm)
new.dat <- data.frame(YearsofExperience = 9)
predict(DS.lm, newdata = new.dat, interval = 'confidence')
fit lwr upr
1 116 112.1943 119.8057
lower limit=112.1943
upper limit=119.8057
SolutionB
predict(DS.lm, newdata = new.dat, interval = 'prediction')
fit lwr upr
1 116 104.7092 127.2908
lower limit=104.7092
upper limit=127.2908
SolutionC:
Prediction intervals range from 104.7092 to127.2908
Confidence intervals range from 112.1943 to 119.8057
Prediction intervals are wider than confidence intervals
his is due to the additional term in the standard error of prediction. It should be noted prediction and confidence intervals are similar in that they are both predicting a response, however, they differ in what is being represented and interpreted. The best predictor of a random variable (assuming the variable is normally distributed) is the mean . The best predictor for an observation from a sample of xx data points, x1,x2,,xnx1,x2,,xn and error is x¯x¯. Since the prediction interval must take into account the variability of the estimators for and , the interval will be wider.