Question Type 9: Finding the critical region under a poisson on binomial distribution Exercises

In a test with $X\sim\mathrm{Bin}(60,0.05)$, the critical region is chosen as $\{X\ge8\}$. Calculate the actual Type I error probability.

For a Poisson test $X\sim\mathrm{Pois}(10)$, the critical region is $\{X\le3\}$ for testing $H_0\colon \lambda=10$ versus $H_1\colon \lambda<10$. Compute the Type I error probability.

In a Poisson test with $X\sim\mathrm{Pois}(4)$, the critical region is $\{X\ge8\}$. Calculate the actual Type I error probability at $H_0\colon \lambda=4$.

A binomial test uses $X\sim\mathrm{Bin}(60,0.05)$ with critical region $\{X\ge7\}$. Calculate the Type I error probability under $H_0\colon p=0.05$.

A call center receives on average 10 calls per hour. Let $Y\sim\mathrm{Pois}(10)$. For testing $H_0\colon \lambda=10$ versus $H_1\colon \lambda<10$ at $\alpha=0.05$, determine the lower-tail critical region.

A factory claims that the defect rate is at most 5%. A sample of 60 items is tested. Let $X\sim\mathrm{Bin}(60,0.05)$. For testing $H_0\colon p=0.05$ against $H_1\colon p>0.05$ at $\alpha=0.05$, determine the critical region.

An email server historically receives on average 4 emails per hour. Let $X\sim\mathrm{Pois}(4)$ be the count in one hour. For testing $H_0\colon \lambda=4$ versus $H_1\colon \lambda>4$ at $\alpha=0.05$, determine the critical region.

Using the same scenario ($X\sim\mathrm{Bin}(60,0.05)$) for testing $H_0\colon p=0.05$ versus $H_1\colon p>0.05$ at $\alpha=0.01$, find the critical region.

In a binomial test with $X\sim\mathrm{Bin}(60,p)$, $H_0\colon p=0.05$ versus $H_1\colon p>0.05$, the critical region is $\{X\ge8\}$. Calculate the Type II error probability if the true $p=0.10$.

In a Poisson test $X\sim\mathrm{Pois}(4)$, $H_0\colon \lambda=4$ versus $H_1\colon \lambda>4$, the critical region is $\{X\ge8\}$. Calculate the Type II error probability if the true $\lambda=6$.

For a Poisson test $X\sim\mathrm{Pois}(10)$, $H_0\colon \lambda=10$ versus $H_1\colon \lambda<10$, the critical region is $\{X\le3\}$. Calculate the Type II error probability if the true $\lambda=7$.

In a binomial test $X\sim\mathrm{Bin}(60,p)$, $H_0\colon p=0.05$ versus $H_1\colon p>0.05$, the critical region is $\{X\ge9\}$. Calculate the Type II error probability if the true $p=0.08$.