# 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).