Question

Question 1

A soft drink bottler is analyzing the vending machine service routes in his distribution system. He is interested in predicting the amount of time required by the route driver to service the vending machines in an outlet. The industrial engineer responsible for the study has suggested that the two most important variables affecting the delivery time (Y) are the number of cases of product stocked (X₁) and the distance walked by the route driver (X₂). The engineer has collected 24 observations on the delivery times, which are shown on the Excel file.

Observation	Delivery Time, y	Number of Cases, x1	Distance, x2 (ft)
1	16.68	7	560
2	11.5	3	220
3	12.03	3	340
4	14.88	4	80
5	13.75	6	150
6	18.11	7	330
7	8	2	110
8	17.83	7	210
9	21.5	5	605
10	40.33	16	688
11	21	10	215
12	13.5	4	255
13	19.75	6	462
14	24	9	448
15	29	10	776
16	15.35	6	200
17	19	7	132
18	9.5	3	36
19	35.1	17	770
20	17.9	10	140
21	52.32	26	810
22	18.75	9	450
23	19.83	8	635
24	10.75	4	150

a) Fit the model Y=Bo+B1X1+B2X2+E to the delivery time data, and give the least squares function.

b) Find the value of SSE that is minimized by the least squares method.

c) Estimate s, the standard deviation of the model.

d)   Conduct the ANOVA F-test for model usefulness at the a = 0.05 significance level (be sure to specify the null and alternative hypotheses).

e)   Conduct the individual t-tests for b₁ andb₂ at the a = 0.05 significance level (be sure to specify the null and alternative hypotheses).

f)    Find and interpret the coefficient of determination R², and the adjusted coefficient of determination R_a².

g)   Which model do you think would be best: a simple linear regression with X₁ as the predictor variable, a simple linear regression with X₂ as the predictor variable, or the first-order multiple regression model with both X₁ and X₂?

h)   Using the chosen model from part g), predict the delivery time when 10 cases need to be stocked, and the distance to be walked is 500 ft. Give a 95% prediction interval for this estimate.

Answer 1

We can easily fit this regression model in Excel.

Steps to do Regression analysis.

Data->data analysis–>Regression -> put x and y in input range -> click okay.

A)

As we can see that coefficinets are highlighted in yellow colour.

so the fitted model is

B)

SSE is given in ANOVA table which is highlighted in yellow colour(Residual SS)

SSE = 124.0023974

C)

Standard deviation of Model is given by square root of MSE

D)

Now we want to check the that overall model is significant.

So p value corresponding to it is given in the anova table as Significance F = 2.85856*E-14 0 < which implies that we have enough evidenct to reject H0

which suggests that the model is significant.