Question 545:

Asked on December 10, 2008

Tags: Regression Analysis , Slope

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1. Q545_1_q545.xls

a.       The fitted regression equation is the y intercept plus the slope of the regression line. We see that Income Tax Withheld = 30.7963 + .343(Weekly Pay).
b.      There is a surprising amount of disagreement on the degrees of freedom for the significance of a slope, either (N, N-1 or N-2). This output shows us N-2, so there are 33 degrees of freedom. The critical value of the t for alpha of .05 and 33 degrees of freedom can be found using the excel function =TINV(.05,33) = 2.0345
c.       We're given the p-value of the slope as .0068 so we'd say that the slope of the regression line is significantly different than 0 since it's p-value is less than .05.
d.      The 95% Confidence intervals have a low-point at .01 and high point at .0563 meaning we'd expect the angle of the slope to fall betrween .01 and .0563 around 95% of the time if we were to repeat this sample.
e.       The F-statistic can be used to test the significance of the slope, as well as other values. The t-statistic with d degrees of freedom can be squared to produce the F-statistic on 1 and d degrees of freedom. So the t of 2.889 with 33 degrees of freedom becomes an F of 8.346 with 1 and 33 degrees of freedom. Using the excel function =FDIST(8.346, 1,33) we get p = .0068, the same as the p-value given in the output.
f.        The fit of this equation can be described by using the R2 statistic which is given as .202. We would say that weekly pay describes around 20.2% of the variation in what you pay in taxes, meaning the equation is not describing 80% of the variation. Overall, while this is low, it still explains a reasonable amount of variation and we can learn a lot about tax payments with only this one predictor. In short, the fit is reasonable.