Chapter 7 Appendix

7.1 Unbounded sample size & statistical significance – sketch of proof

Let the sample mean, \(\hat{\mu}\), be the parameter estimate for our mean parameter \(\mu\) and the null hypothesis of the t-test be \(H_0\): \(/mu = 0\). The test statistic is given by \(\hat{\mu} / (\hat{\sigma} / \sqrt{n})\). Remember that the p-value is determined by the test statistic and the t-distribution with \((n – 2)\) degrees of freedom in this case. By the Central Limit Theorem, \(\sqrt{n}*(\hat{\mu}-\mu) \rightarrow N(0,\sigma^2)\) as \(n \rightarrow \infty\), or written differently as \(\hat{\mu} \rightarrow \mu + \frac{\sigma}{\sqrt{n}}N(0,1)\). Implicitly, \(\hat{\mu} = \mu + O(\frac{1}{\sqrt{n}})\) and \(\hat{sigma} = \sigma + O(\frac{1}{\sqrt{n}})\). By substitution,

\[\frac{\hat{\mu}}{\hat{\sigma}/\sqrt{n}} = \sqrt{n}\frac{\hat{\mu}}{\hat{\sigma}}\] \[= \sqrt{n}*\frac{\mu + O(n^{-1/2})}{\sigma + O(n^{-1/2})}\] \[= \sqrt{n}[\frac{\mu}{\sigma}+O(n^{-1/2})]\] \[= \sqrt{n}*\frac{\mu}{\sigma}+O(1)\]

Therefore, the test statistic will approach positive or negative infinity at a constant rate of \(\sqrt{n}\) the mean parameter does not exactly equal 0. As \(n\) approaches infinity and, therefore, the degrees of freedom \((n – 2)\) approach infinity, the t-distribution converges to a standard normal distribution. So, for any nonzero mean parameter, the p-value will always approach 0 as \(n\) goes to infinity.

Similar arguments can be made for differences in means, regression coefficients or linear combinations of regression coefficients. For each parameter type, the p-value is similarly determined by the size of the parameter and the precision with which it is estimated, as well as the sample size. In each case, the p-value will converge stochastically to zero as \(n\) goes to infinity (unless the true parameter value is exactly equal to the null value).