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 H0:σ2=σ20 vs. H1:σ2>σ20 in a normal population N(0,σ2). The power curve represents the probability of rejecting H0:σ2=σ20, as a function of σ, from a sample of size n from N(0,σ2). The dashed vertical line is the value of σ0=1 and the dotted horizontal line is the significance level α=0.10. The power increases as n increases and as σ increases, as the evidence against H0:σ2=σ20 is stronger in those cases. Observe how a one-sided test has no power when σ2<σ20, since it is designed to “face” H1:σ2>σ20.

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.