Lecture 12

Regression and Correlation Analysis Continued:

(Chapter 12)

Inference concerning the values of A and B:

Like any other sample statistic, the slope of the sample regression line b and its intercept a have sampling distributions. An estimate of the standard deviation of the sampling distribution of b is:

s_b = s_{e /Ö(S (Xi}² – x^bar)²

= s_{e /Ö(S Xi}² – nX^bar2)

which is known as the standard error of b. We often want to test if the value of B is zero or any specific value. If B=0, then X does not affect Y; i.e., there is no relationship between the dependent and the independent variable. Thus,

to test the Null hypothesis: B = 0,

Alternative hypothesis can be

i) B>0

ii) B<0, or

iii) B not equals to 0.

The test for the null hypothesis H₀ is carried out in the following way depending on the type of alternative:

Alternative hypothesis B>0:

Reject H₀ if (b/s_e) > t_a. n is the sample size and a is the level of significance. This is a one-sided test.

Alternative hypothesis B<0:

Reject H₀ if (b/s_e) < - t_a. n is the sample size and a is the level of significance. This is also a one-sided test.

In the above two cases you can alternatively calculate the absolute value of the test statistic (b/s_e), and check if the value exceeds the tabulated value t_a (see Table 6, A16). n-2 is the degrees of freedom.

Alternative hypothesis that B is that B is not 0:

Reject the null hypothesis if the absolute value of (b/s_e) is greater than t_a/2. Note that this a two-sided test.

Let us look at the examples in page 487.

Do exercises 12.14a, 12.15a, 12.16a, 12.17, 12.18a, 12.19a, 12.21, and12.23.