8.3 Main Idea Behind Hypothesis Tests for μ

The main idea of a hypothesis test is to use the data as evidence to disprove the null $H_0$ and thus prove that the alternative $H_a$ is true. The idea behind a hypothesis test for the population mean is as follows:

Collect from the population a simple random sample: $x_1,x_2,\dots ,x_n$ and calculate the sample mean $\overline{x}=\frac{x_1+x_2+\cdots +x_n}{n}$. Our “evidence” stems from the discrepancy between the point estimate $\overline{x}$ and the hypothesized population mean ${\mu }_0$.

Reject the null hypothesis $H_0$ if the sample mean $\overline{x}$ does not support the null $H_0$. That is, we should reject $H_0$ if $\overline{x}$ is too extreme. The word “extreme” means contradicting the null $H_0$ in favour of the alternative $H_a$.

In order to quantify how the data (our evidence) contradict the null hypothesis, we first assume the null hypothesis $H_0$ is true and calculate the chance of observing a sample mean at least as extreme as the observed $\overline{x}$. Reject the null $H_0$ if the chance is small; otherwise, fail to reject $H_0$. Recall that for a normal population or a large sample size, the sample mean $\overline{X}\sim N(\mu ,\frac{\sigma }{\sqrt{n}})⟹Z=\frac{\overline{X}-\mu }{\sigma /\sqrt{n}}\sim N(0,1)$ and $t=\frac{\overline{X-\mu }}{s/\sqrt{n}}\sim t$ distribution with $df=n-1.$ We call the variables $Z=\frac{\overline{X}-\mu }{\sigma /\sqrt{n}}$ or $t=\frac{\overline{X-\mu }}{s/\sqrt{n}}$ the test statistics. We should reject the null hypothesis $H_0$ if the observed test statistic $z_o=\frac{\overline{x}-{\mu }_0}{\sigma /\sqrt{n}}$ or $t=\frac{\overline{x}-{\mu }_0}{s/\sqrt{n}}$ is too extreme.