In: Statistics and Probability
A firm that operates a large, direct-to-consumer sales force would like to put in place a system to monitor the progress of new agents. A key task for agents is to open new accounts; an account is a new customer to the business. The goal is to identify"superstar agents" as rapidly as possible, offer them incentives, and keep them with the company. To build such a system, the firm has been monitoring sales of new agents over the past two years. The response of interest is the profit to the firm (in dollars) of contracts sold by agents over their first year. Among the possible predictors of this performance is the number of new accounts developed by the agent during the first three months of work. Formulate the SRM with Y given by the natural log of Profit from Sales and X given by the natural log of Number of Accounts.
Data Table:
Profit_from_Sales Number_of_Accounts
23248.39 46
35347.43 10
27384.29 83
19946.44 7
8201.63 12
35757.84 169
19386.39 10
39931.42 22
18165.13 16
17813.03 15
16141.71 14
33833.27 23
35746.25 33
17043.17 19
25808.56 28
25910.96 72
33035.56 21
10157.41 13
59928.04 489
30807.71 136
25407.95 18
4508.74 2
40696.58 37
18013.02 6
14866.23 10
35936.01 31
12825.21 7
48987.22 25
33015.75 26
32611.79 42
If a sales representative opens 150 accounts, what level of sales would you predict for this individual? Express your answer as an interval. (Use 95%.)
__ to __ dollars (Round to two decimals places as needed)
Column1 | Column2 | log sales | log account | |
23248.39 | 46 | 10.05399 | 3.828641 | |
35347.43 | 10 | 10.47298 | 2.302585 | |
27384.29 | 83 | 10.21772 | 4.418841 | |
19946.44 | 7 | 9.900806 | 1.94591 | |
8201.63 | 12 | 9.012088 | 2.484907 | |
35757.84 | 169 | 10.48452 | 5.129899 | |
19386.39 | 10 | 9.872327 | 2.302585 | |
39931.42 | 22 | 10.59492 | 3.091042 | |
18165.13 | 16 | 9.807259 | 2.772589 | |
17813.03 | 15 | 9.787685 | 2.70805 | |
16141.71 | 14 | 9.689162 | 2.639057 | |
33833.27 | 23 | 10.4292 | 3.135494 | |
35746.25 | 33 | 10.4842 | 3.496508 | |
17043.17 | 19 | 9.743505 | 2.944439 | |
25808.56 | 28 | 10.15846 | 3.332205 | |
25910.96 | 72 | 10.16242 | 4.276666 | |
33035.56 | 21 | 10.40534 | 3.044522 | |
10157.41 | 13 | 9.225959 | 2.564949 | |
59928.04 | 489 | 11.0009 | 6.192362 | |
30807.71 | 136 | 10.33552 | 4.912655 | |
25407.95 | 18 | 10.14282 | 2.890372 | |
4508.74 | 2 | 8.413773 | 0.693147 | |
40696.58 | 37 | 10.6139 | 3.610918 | |
18013.02 | 6 | 9.79885 | 1.791759 | |
14866.23 | 10 | 9.606847 | 2.302585 | |
35936.01 | 31 | 10.4895 | 3.433987 | |
12825.21 | 7 | 9.459168 | 1.94591 | |
48987.22 | 25 | 10.79931 | 3.218876 | |
33015.75 | 26 | 10.40474 | 3.258097 | |
32611.79 | 42 | 10.39243 | 3.73767 |
SUMMARY OUTPUT | |||||
Regression Statistics | |||||
Multiple R | 0.702472246 | ||||
R Square | 0.493467257 | ||||
Adjusted R Square | 0.475376801 | ||||
Standard Error | 0.401853136 | ||||
Observations | 30 | ||||
ANOVA | |||||
df | SS | MS | F | Significance F | |
Regression | 1 | 4.404976259 | 4.404976259 | 27.27776903 | 1.50845E-05 |
Residual | 28 | 4.521606407 | 0.161485943 | ||
Total | 29 | 8.926582666 | |||
Coefficients | Standard Error | t Stat | P-value | Lower 95% | |
Intercept | 8.946916302 | 0.226362488 | 39.52473032 | 4.1491E-26 | 8.483233764 |
log account | 0.355405202 | 0.068048625 | 5.222812368 | 1.50845E-05 | 0.216013912 |
ln sales ^= 8.9469 + 0.3554 ln account
= 8.9469 + 0.3554*ln(150)
= 10.7277
ata = read.csv("sales.csv")
y = data$ï..log.sales
x=data$log.account
model = lm (y~x)
predict(model,data.frame(x=log(150)) ,data=
data,conf.level=0.95,interval= "confidence")
fit lwr upr 1 10.72772 10.4276 11.02785
exp(10.4276) [1] 33779.18 > exp(11.02785) [1] 61565.07
hence confidence interval
( 33779.18 , 61565.07 )