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Example 2.9 The Delivery Time Data

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Consider the soft drink delivery time data introduced in Chapter 1. The 25 observations on delivery time y and delivery volume x are listed in Table 2.11. The scatter diagram shown in Figure 1.1 indicates a strong linear relationship between delivery time and delivery volume. The Minitab output for the simple linear regression model is in Table 2.12.

The sample correlation coefficient between delivery time y and delivery volume x is


TABLE 2.11 Data Example 2.9

Observation Delivery Time, y Number of Cases, x
1 16.68 7
2 11.50 3
3 12.03 3
4 14.88 4
5 13.75 6
6 18.11 7
7 8.00 2
8 17.83 7
9 79.24 30
10 21.50 5
11 40.33 16
12 21.00 10
13 13.50 4
14 19.75 6
15 24.00 9
16 29.00 10
17 15.35 6
18 19.00 7
19 9.50 3
20 35.10 17
21 17.90 10
22 52.32 26
23 18.75 9
24 19.83 8
25 10.75 4

TABLE 2.12 MlNITAB Output for Soft Drink Delivery Time Data

Regression Analysis: Time versus Cases
The regression equation is
Time = 3.32 + 2.18 Cases
Predictor Coef SE Coef T P
Constant 3.321 1.371 2.42 0.024
Cases 2.1762 0.1240 17.55 0.000
S = 4.18140 R- Sq= 93.0% R- Sq(adj) = 92.7%
Analysis of Variance
Source DF SS MS F P
Regression 1 5382.4 5382.4 307.85 0.000
Residual Error 23 402.1 17.5
Total 24 5784.5

If we assume that delivery time and delivery volume are jointly normally distributed, we may test the hypotheses


using the test statistic


Since t0.025,23 = 2.069, we reject H0 and conclude that the correlation coefficient ρ ≠ 0. Note from the Minitab output in Table 2.12 that this is identical to the t-test statistic for H0: β1 = 0. Finally, we may construct an approximate 95% CI on ρ from (2.72). Since arctanh r = arctanh 0.9646 = 2.0082, Eq. (2.72) becomes


which reduces to


Although we know that delivery time and delivery volume are highly correlated, this information is of little use in predicting, for example, delivery time as a function of the number of cases of product delivered. This would require a regression model. The straight-line fit (shown graphically in Figure 1.1b) relating delivery time to delivery volume is


Further analysis would be required to determine if this equation is an adequate fit to the data and if it is likely to be a successful predictor.

Introduction to Linear Regression Analysis

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