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448 CHAPTER 9 SIMPLE LINEAR REGRESSION AND CORRELATION
An alternative formula for computing r is given by
r =
ng x i y i - 1g x i 21g y i 2
2n g x 2 i - 1g x i 2 2 2ng y 2 i - 1g y i 2 2
(9.7.2)
An advantage of this formula is that r may be computed without first computing
b. This is the desirable procedure when it is not anticipated that the regression equation
will be used.
Remember that the sample correlation coefficient, r, will always have the same
sign as the sample slope, b.
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EXAMPLE 9.7.2
Refer to Example 9.7.1. We wish to see if the sample value of r = .848 is of sufficient
magnitude to indicate that, in the population, height and Cv SEP levels are correlated.
Solution:
We conduct a hypothesis test as follows.
1. Data. See the initial discussion of Example 9.7.1.
2. Assumptions. We presume that the assumptions given in Section 9.6
are applicable.
3. Hypotheses.
H 0 : r = 0
H A : r Z 0
4. Test statistic. When r = 0, it can be shown that the appropriate test
statistic is
t = r A
n - 2
1 - r 2
(9.7.3)
5. Distribution of test statistic. When H 0 is true and the assumptions are
met, the test statistic is distributed as Student’s t distribution with n - 2
degrees of freedom.
6. Decision rule. If we let a = .05, the critical values of t in the present
example are ; 1.9754 (by interpolation). If, from our data, we compute
a value of t that is either greater than or equal to +1.9754 or less
than or equal to -1.9754, we will reject the null hypothesis.
7. Calculation of test statistic. Our calculated value of t is
t = .848 A
153
1 - .719 = 19.787
8. Statistical decision. Since the computed value of the test statistic does
exceed the critical value of t, we reject the null hypothesis.