Fig. 3
Optimal (a) sample size, n∗, (b) type II error rate, β∗, (c) \(z_{\alpha }^{*}\) and (d) type I error rate, α∗ in order to detect an alternative θ A =$24819 against the size of the population, N, with fixed \(\mu _{0}=\$96000,\sigma _{0}^{2}=(\$49638)^{2}, \tau ^{2}=(\$363202)^{2}, c_{1}=\$5000\) and c f =$1 million for different values of; c2=0 (light grey dotted line), c2=σ0/4=$12409 (light grey dashed line), c2=σ0/2=$24819 (light grey solid line), c2=$61032 (heavy black solid line), c2=μ0=$96000 (heavy black dashed line) and c2=σ0/2+μ0=$120819 (heavy black dotted line)