Inference for Regression Using Technology 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
If you had to do them from the raw data, the computations involved in doing inference for the slope of a regression line would be daunting.
For example, how would you like to compute
Fortunately, you probably will never have to do this by hand, but instead can rely on computer output you are given, or you will be able to use your calculator to do the computations.
Consider the following data that were gathered by counting the number of cricket chirps in 15 seconds and noting the temperature.
We want to use technology to test the hypothesis that the slope of the regression line is 0 and to construct a confidence interval for the true slope of the regression line.
First let us look at the Minitab regression output for this data.
You should be able to read most of this table, but you are not responsible for all of it. You see the following table entries:
- The regression equation, Temp = 44.0 + 0.993 Number, is the Least Squares Regression Line (LSRL) for predicting temperature from the number of cricket chirps.
- Under "Predictor" are the y-intercept and explanatory variable of the regression equation, called "Constant" and "Number" in this example.
- Under "Coef " are the values of the "Constant" (which equals the y-intercept, the a in = a + bx; here, a = 44.013) and the slope of the regression line (which is the coefficient of "Number" in this example, the b in = a + bx; here, b = 0.99340).
- For the purposes of this course, we are not concerned with the "Stdev," "t-ratio," or "p" for "Constant" (the last three entries in the first line of printout—only the "44.013" is meaningful for us).
- "Stdev" of "Number" is the standard error of the slope (what we have called sb, the variability of the estimates of the slope of the regression line, which equals here sb = 0.06523); "t-ratio" is the value of the t-test statistic (t = , df = n – 2; here, t = = 15.23); and P is the P-value associated with the test statistic assuming a two-sided test (here, P = 0.000; if you were doing a one-sided test, you would need to divide the given P-value by 2).
- s is the standard error of the residuals (which is the variability of the vertical distances of the y-values from the regression line; ; (here, s = 1.538.)
- "R-sq" is the coefficient of determination (or, r2; here R-sq = 95.9% 95.9% of the variation in temperature that is explained by the regression on the number of chirps in 15 seconds; note that, here, r = = 0.979—it's positive since b = 0.9934 is positive). You don't need to worry about "R-sq(adj)."
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