Inference for the Slope of a Regression Line for AP Statistics

Practice problems for these concepts can be found at:

• Inference for Regression Multiple Choice Practice Problems for AP Statistics
• Inference for Regression Free Response Practice Problems for AP Statistics
• Inference for Regression Review Problems for AP Statistics
• Inference for Regression Rapid Review for AP Statistics

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 H₀: β = 0, although it is possible to test H₀: β = β₀. 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 = 0, then b = 0).

The alternative hypothesis is often two sided (i.e., H_A: β ≠ 0). We can 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:

example: The data in the following table give the top 15 states in terms of per pupil expenditure in 1985 and the average teacher salary in the state for that year.

Test the hypothesis, at the 0.01 level of significance, that there is no straight-line relationship between per pupil expenditure and teacher salary. Assume that the conditions necessary for inference for linear regression are present.

solution:

A significance test that the slope of a regression line equals zero is closely related to a test that there is no correlation between the variables. That is, if r is the population correlation coefficient, then the test statistic for H₀: β = 0 is equal to the test statistic for H₀: ρ = 0. You aren't required to know it for the AP exam, but the t-test statistic for H₀: ρ = 0, where r is the sample correlation coefficient, is

Practice problems for these concepts can be found at:

• Inference for Regression Multiple Choice Practice Problems for AP Statistics
• Inference for Regression Free Response Practice Problems for AP Statistics
• Inference for Regression Review Problems for AP Statistics
• Inference for Regression Rapid Review for AP Statistics