s is an estimator of σ, the standard deviation of the residuals. Thus, there are actually three parameters
to worry about in regression: α, β , and σ , which are estimated by a ,b , ands , respectively.
The final statistic we need to do inference for regression is the standard error of the slope of the
regression line given by the following equation. You will not need to use this formula on the exam:
In summary, inference for regression depends upon estimating μ (^) y = α + β (^) x with = a + bx . For each
x , the response values of y are independent and follow a normal distribution, each distribution having the
same standard deviation. Inference for regression depends on the following statistics:
• a , the estimate of the y intercept, α , of μ (^) y
• b , the estimate of the slope, β , of μ (^) y
• s , the standard error of the residuals
• s (^) b , the standard error of the slope of the regression line
In the section that follows, we explore inference for the slope of a regression line in terms of a
significance test and a confidence interval for the slope.
Inference for the Slope of a Regression Line
Inference for regression consists of either a significance test or a confidence interval for the slope of a
regression line. The null hypothesis in a significance test is usually H 0 : β = 0, although it is possible to
test H 0 : β = β 0 . Our interest is the extent to which a least-squares regression line is a good model for
the data. That is, the significance test is a test of a linear model for the data.
We note that in theory we could test whether the slope of the regression line is equal to any specific
value. However, the usual test is whether the slope of the regression line is zero or not. If the slope of the
line is zero, then there is no linear relationship between the x and y variables (remember: ; if r =
O, then b = 0).
The alternative hypothesis is often two sided (i.e., H (^) A : β ≠ 0). We could do a one-sided test if we
believed that the data were positively or negatively related.
Significance Test for the Slope of a Regression Line
The basic details of a significance test for the slope of a regression line are given in the following table: