Confidence Interval 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

In addition to doing hypothesis tests on H₀: β = β₀, we can construct a confidence interval for the true slope of a regression line. The details follow:

example: Consider once again the earlier example on predicting teacher salary from per pupil expenditure. Construct a 95% confidence interval for the slope of the population regression line.

solution: When we were doing a test of H_o : β = 0 for that problem, we found that Salary = 12027 + 3.34 PPE. The slope of the regression line for the 15 points, and hence our estimate of β, is b = 3.34. We also had t = 6.04.

Our confidence interval is of the form b ± t*s_b. We need to find t* and s_b. For C = 0.95, df = 15 – 2 = 13, we have t* = 2.160 (from Table B; if you have a TI-84 with the invT function, use invT(0.975,13)). Now, as mentioned earlier, .

Hence, b ± t*s_b = 3.34 ± 2.160(0.5530) = (2.15, 4.53). We are 95% confident that the true slope of the regression line is between 2.15 and 4.53. Note that, since 0 is not in this interval, this finding is consistent with our earlier rejection of the hypothesis that the slope equals 0. This is another way of saying that we have statistically significant evidence of a predictive linear relationship between PPE and Salary.

(If your TI-84 has the invT function in the DISTR menu, it also has, in the STAT TESTS menu, LinRegTInt, which will return the interval (2.146, 4.538). It still doesn't tell you the value of s_b. There is a more complete explanation of how to use technology to do inference for regression in the next section.)

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