Basic Statistics

176 REGRESSION AND CORRELATION

Most computer programs furnish only limited or no confidence intervals for linear
regression. They typically give the standard error of the regression line, the slope
coefficient, and the intercept. The variance of X can be used to compute c(X -x)’.
From these quantities the desired confidence intervals can easily be computed. Stata
does provide confidence intervals.

12.2.8 Tests of Hypotheses for Regression Line from a

Single Sample

The computation of tests of hypotheses for the population intercept and the slope are
common options in computer programs. In Chapter 8 we presented the test to decide
whether a population mean was a particular value. A similar test can be made for the
slope of a regression line. The test statistic is

b - Po

se(b)

t=-

Here, PO is the hypothesized population slope and the computed t is compared with
the t values in Table A.3 with n - 2 d.f. The se(b) is given in Section 12.2.7. For
example, if we wish to test the null hypothesis that the population slope, PO = .20,
for the 10 males and with o = .05, our test statistic would be

.29 - .20

t= = 1.45

.062

From Table A.3 for n - 2 = 8 d.f., the tabled value of t is 2.306 and we would be
unable to reject the null hypothesis of a population slope equal to .20.
Similarly, for a test that the population intercept Q takes on some particular value,
QO, we have the test statistic
a - Qo
t=-

For example, to test the null hypothesis that the population intercept a0 = 100 mmHg,

for the 10 males at the 5% level we have

se(a)

80.74 - 100
t= = -1.62
11.87
and we would not be able to reject the null hypothesis since for n - 2 = 8 d.f., we
need a t value greater in magnitude than the tabled value of 2.306 in Table A.3 to
reject at the o = .05 significance level.
Statistical programs test the null hypotheses that the population slope PO = 0 and
that the population intercept a0 = 0. If the test that the population slope PO = 0 is
not rejected, we have no proof that Y changes with different values of X. It may be
that provides just as good an estimate of the mean of the Y distribution for different
values of X as does Y. Tests of other hypothesized values must be done by hand.
However, most programs furnish the standard errors so that these other tests can be
made with very little effort. The test that the population intercept oo = 0 is a test that
the line goes through the origin (0,O). There is usually no reason for making this test.