Question

Management of a soft-drink bottling company has the business objective of developing a method for allocating delivery costs to customers.  Although one cost clearly relates to travel time within a particular route, another variable cost reflects the times required to unload the cases of soft drink at the delivery point.  To begin, management decided to develop a regression model to predict delivery time based on the number of cases delivered.  A sample of 20 deliveries within a territory was selected.  The delivery times and the number of cases for the customers is found below:

Data

Customer	Number of Cases	Delivery Time (Minutes)
1	52	32.1
2	64	34.8
3	73	36.2
4	85	37.8
5	95	37.8
6	103	39.7
7	116	38.5
8	121	41.9
9	143	44.2
10	157	47.1
11	161	43.0
12	184	49.4
13	202	57.2
14	218	56.8
15	243	60.6
16	254	61.2
17	267	58.2
18	275	63.2
19	287	65.6
20	298	67.3

Please use Excel to answer the following questions.  

1. Develop an estimated regression equation showing how the delivery time is related to the number of cases delivered.  Express your equation using proper notation.

2. Test to see whether delivery time and number of cases delivered are significantly related.  Use a 0.05 level of significance for the test and include all 6 steps of the test.

3. Predict the delivery time for 150 cases of soft drink.

4. Should you use the model to predict the delivery time for a customer who is receiving 500 cases of soft drink?  Why or why not?

5. Include a plot of the residuals in your analysis. Is there any evidence of an obvious pattern in the residuals?

6. Construct a 95% confidence interval estimate of the mean delivery time for 150 cases of soft drink and a 95% prediction interval of the delivery time for a single delivery of 150 cases of soft drink.  Be sure each interval is clearly labeled in your output.

Answer 1

a)

The regression equation is
Delivery Time = 24.825 + 0.140 Number of Cases

b)

Hypothesis: H0: B1=0

H1: B1 does not equal to zero.

Predictor	Coef	SE Coef	T	P
Constant	24.825	1.054	23.56	0.000
Number of Cases	0.140	0.005626	24.91	0.000

S = 1.98595 R-Sq = 97.2% R-Sq(adj) = 97.0%

Reject the null hypothesis.

Comment: The number of the case has a significant effect on the Delivery Time at the 0.05 significance level.

c) The predicted delivery time for 150 cases of soft drink is

Delivery Time = 24.825 + 0.140*150=45.842.

d)

The model can use to predict the delivery time for a customer who is receiving 500 cases of soft drink because of the relation between these variables is linear.

e)

Comment: The normal probability plot of residual has deviated from the straight line. Hence, the residual may not follow a normal distribution. The histogram of the residual also does not show a dumb-bell shape. Hence, the assumption of the model is violated.

f) The  95% confidence interval estimate of the mean delivery time for 150 cases of soft drink is (44.880, 46.804) and the  95% prediction interval of the delivery time for a single delivery of 150 cases of soft drink is (41.560, 50.124).