IB Mathematics Applications & Interpretation 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 $H_0:\lambda =4$.