Inference for Regression Rapid Review for AP Statistics

Review the following concepts if necessary:

• Simple Linear Regression for AP Statistics
• Inference for the Slope of a Regression Line for AP Statistics
• Confidence Interval for the Slope of a Regression Line for AP Statistics
• Inference for Regression Using Technology for AP Statistics

Rapid Review

1. The regression equation for predicting grade point average from number of hours studied is determined to be GPA = 1.95 + 0.05(Hours). Interpret the slope of the regression line.

Answer: For each additional hour studied, the GPA is predicted to increase by 0.05 points.

2. Which of the following is not a necessary condition for doing inference for the slope of a regression line?
   1. For each given value of the independent variable, the response variable is normally distributed.
   2. The values of the predictor and response variables are independent.
   3. For each given value of the independent variable, the distribution of the response variable has the same standard deviation.
   4. 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.

3. True–False: Significance tests for the slope of a regression line are always based on the hypothesis H₀: β = 0 versus the alternative HA: 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 non-zero value so 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.

4. Consider the following Minitab printout:



1. What is the slope of the regression line?
2. What is the standard error of the residuals?
3. What is the standard error of the slope?
4. Do the data indicate a predictive linear relationship between x and y?

Answer:

1. 0.634
2. 9.282
3. 0.07039
4. 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.
5. 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.

6. In the printout from question #4, we were given the regression equation 282 + 0.634x. The t-test for H₀: β = 0 yielded a P-value of 0.000. What is the conclusion you would arrive at based on these data?

Answer: Because P is very small, we would reject the null hypothesis that the slope of the regression line is 0. We have strong evidence of a predictive linear relationship between x and y.

7. Suppose the computer output for regression reports P = 0.036. What is the P-value for H_A: β > 0 (assuming the test was in the correct direction for the data)?

Answer: 0.018. Computer output for regression assumes the alternative is two sided (H_A: β ≠ 0). Hence the P-value reported assumes the finding could have been in either tail of the t-distribution. The correct P-value for the one-sided test is one-half of this value.