Statistics Minus the Maths

To conclude this chapter, let’s briefly revisit the very first type of hypothesis test we encountered in this textbook (though we did not call it that at the time): testing for the significance of a regression slope coefficient (Section 3.5). Now that we know more about hypothesis testing, let’s fill in some of the details of how to calculate the p-values we rely upon to determine the significance of these coefficients.

The appropriate type of significance test in the case of the regression coefficients we have learned about is a t test. Recall the general formula for a t test:

As applied to the case of the slope in a simple regression, the statistic is the sample value of the slope coefficient (\(\hat{\beta}\)). Generally, the hypothesized value is 0, meaning that we want to test a null hypothesis of no relationship between the independent and dependent variables.

Just as when we generated a confidence interval for the slope coefficient in Subsection 6.3.3, the degrees of freedom for this t test is n-2. We also use the same calculation for the estimated standard error as when calculating a confidence interval, so refer back to Chapter 6 (specifically Appendix II) if you would like to review how we calculate it.

With the data example we used when learning precise confidence interval calculations (Subsection 6.3.3), we had a sample slope coefficient (\(\hat{\beta}\)) of 0.425, a standard error (\(s_{\beta}\)) of 0.305, and a sample size of 5. Given these numbers and a hypothesized value of 0:

With these values of \(t\) and \(df\text{,}\) the p value for a two-tailed t test is 0.26. Therefore, the slope is not significantly different from 0 under this example.