Introduction to Linear Regression Analysis. Douglas C. Montgomery

(2.66),

which we recognize from Eq. (2.47) as the coefficient of determination. That is, the coefficient of determination R² is just the square of the correlation coefficient between y and x.

      While regression and correlation are closely related, regression is a more powerful tool in many situations. Correlation is only a measure of association and is of little use in prediction. However, regression methods are useful in developing quantitative relationships between variables, which can be used in prediction.

      It is often useful to test the hypothesis that the correlation coefficient equals zero, that is,

      (2.67)

      The appropriate test statistic for this hypothesis is

      (2.68)

      which follows the t distribution with n − 2 degrees of freedom if H₀: ρ = 0 is true. Therefore, we would reject the null hypothesis if |t₀| > t_{α/2, n−2}. This test is equivalent to the t test for H₀: β₁ = 0 given in Section 2.3. This equivalence follows directly from Eq. (2.66).

      The test procedure for the hypotheses

      (2.69)

      where ρ₀ ≠ 0 is somewhat more complicated. For moderately large samples (e.g., n ≥ 25) the statistic

(2.70)

      is approximately normally distributed with mean

and variance

      Therefore, to test the hypothesis H₀: ρ = ρ₀, we may compute the statistic

      (2.71)

      and reject H₀: ρ = ρ₀ if |Z₀| > Z_α/2.

It is also possible to construct a 100(1 − α) percent CI for ρ using the transformation (2.70). The 100(1 − α) percent CI is

(2.72)

      where tanh u = (eu − e^−u)/(eu + e^−u).

      Example 2.9 The Delivery Time Data

      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