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

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

For a Poisson test $X \sim \text{Po}(10)$, the critical region is $\{X \le 3\}$ for testing $H_0: \lambda = 10$ versus $H_1: \lambda < 10$. Calculate the Type I error probability.

Using the scenario where $X \sim \text{B}(60, 0.05)$, find the critical region for a hypothesis test of $H_0: p = 0.05$ versus $H_1: p > 0.05$ at the $\alpha = 0.01$ significance level.

A binomial test uses $X \sim \text{Bin}(60, 0.05)$ with critical region $\{X \ge 7\}$. Calculate the Type I error probability under $H_0: p = 0.05$.

In a Poisson test $X \sim \text{Pois}(4)$, $H_0: \lambda = 4$ versus $H_1: \lambda > 4$, the critical region is $\{X \ge 8\}$. Calculate the Type II error probability if the true $\lambda = 6$.

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

For a Poisson test $X \sim \text{Pois}(10)$, $H_0: \lambda = 10$ versus $H_1: \lambda < 10$, the critical region is $\{X \le 3\}$. Calculate the Type II error probability if the true $\lambda = 7$.

An email server historically receives on average 4 emails per hour. Let $X \sim \text{Pois}(4)$ be the number of emails received in one hour. For testing $H_0: \lambda = 4$ versus $H_1: \lambda > 4$ at a significance level of $\alpha = 0.05$, determine the critical region.

In a binomial test with $X \sim \text{Bin}(60, p)$, $H_0: p = 0.05$ versus $H_1: p > 0.05$, the critical region is $\{X \ge 8\}$. Calculate the Type II error probability if the true $p = 0.10$.

In a binomial test $X \sim \mathrm{Bin}(60, p)$, $H_0: p = 0.05$ versus $H_1: p > 0.05$, the critical region is defined as $\{X \ge 9\}$. Calculate the Type II error probability if the true value of $p$ is $0.08$.

In a test with $X \sim \text{Bin}(60, 0.05)$, the critical region is chosen as $\{X \ge 8\}$. Calculate the actual Type I error probability.

In a Poisson test with $X \sim \text{Pois}(4)$, the critical region is defined as $\{X \ge 7\}$. Calculate the actual Type I error probability for the hypothesis $H_0: \lambda = 4$.