Chapter 6 Hypothesis tests

$Power curve of the one-sided hypothesis test for $H_0:\sigma^2=\sigma_0^2$ vs. $H_1:\sigma^2>\sigma_0^2$ in a normal population $\mathcal{N}(0,\sigma^2).$ The power curve represents the probability of rejecting $H_0:\sigma^2=\sigma_0^2,$ as a function of $\sigma,$ from a sample of size $n$ from $\mathcal{N}(0,\sigma^2).$ The dashed vertical line is the value of $\sigma_0=1$ and the dotted horizontal line is the significance level $\alpha=0.10.$ The power increases as $n$ increases and as $\sigma$ increases, as the evidence against $H_0:\sigma^2=\sigma_0^2$ is stronger in those cases. Observe how a one-sided test has no power when $\sigma^2<\sigma_0^2,$ since it is designed to “face” $H_1:\sigma^2>\sigma_0^2$.$

Figure 6.1: Power curve of the one-sided hypothesis test for $H_0:\sigma^2=\sigma_0^2$ vs. $H_1:\sigma^2>\sigma_0^2$ in a normal population $\mathcal{N}(0,\sigma^2).$ The power curve represents the probability of rejecting $H_0:\sigma^2=\sigma_0^2,$ as a function of $\sigma,$ from a sample of size $n$ from $\mathcal{N}(0,\sigma^2).$ The dashed vertical line is the value of $\sigma_0=1$ and the dotted horizontal line is the significance level $\alpha=0.10.$ The power increases as $n$ increases and as $\sigma$ increases, as the evidence against $H_0:\sigma^2=\sigma_0^2$ is stronger in those cases. Observe how a one-sided test has no power when $\sigma^2<\sigma_0^2,$ since it is designed to “face” $H_1:\sigma^2>\sigma_0^2$.

Hypothesis tests are arguably the main tool in statistical inference to answer research questions in a precise and formal way that takes into account the uncertainty behind the data generation and measurement process. This chapter introduces the basics of hypothesis testing, the most-known tests for one and two normal populations, and the theory of likelihood ratio tests.