c. For each given value of the independent variable, the distribution of the response variable has the
same standard deviation.
d. The mean response values lie on a line.
Answer: (b) is not a condition for doing inference for the slope of a regression line. In fact, we are
trying to find out the degree to which they are not independent.
True–False: Significance tests for the slope of a regression line are always based on the hypothesis H
0 : β^ = 0 versus the alternative H^ A : β^ ≠ 0.
Answer: False. While the stated null and alternative may be the usual hypotheses in a test about the
slope of the regression line, it is possible to test that the slope has some particular nonzero value or
that the alternative can be one sided (H (^) A : B > 0 or H (^) A : β < 0). Note that most computer programs
will test only the two-sided alternative by default. The TI-83/84 will test either a one- or two-sided
alternative.
Consider the following Minitab printout:
a. What is the slope of the regression line?
b. What is the standard error of the residuals?
c. What is the standard error of the slope?
d. Do the data indicate a predictive linear relationship between x and y ?
Answer:
a. 0.634
b. 9.282
c. 0.07039
d. Yes, the t -test statistic = 9.00 P -value = .000. That is, the probability is close to zero of
getting a slope of 0.634 if, in fact, the true slope was zero.
A t -test for the slope of a regression line is to be conducted at the 0.02 level of significance based on
18 data values. As usual, the test is two sided. What is the upper critical value for this test (that is,
find the minimum positive value of t * for which a finding would be considered significant)?
Answer: There are 18 – 2 = 16 degrees of freedom. Since the alternative is two sided, the rejection
region has 0.01 in each tail. Using Table B, we find the value at the intersection of the df = 16 row and the
0.01 column: t = 2.583. If you have a TI-84 with the invT function, invT(0.99,16)=2.583 . This is, of
course, the same value of t you would use to construct a 98% confidence interval for the slope of the
regression line.