Question Type 10: Calculating the probability of a type 1 error under a critical region for a binomial or poisson distribution Exercises

For a batch of $n=20$ components, let $X\sim\mathrm{Bin}(20,0.1)$. You decide to reject $H_0: p=0.1$ in favour of $H_1: p>0.1$ if $X\ge3$. Calculate the probability of a type I error $\alpha$.

Let $X\sim\mathrm{Pois}(2)$. You test $H_0: \lambda=2$ versus $H_1: \lambda>2$ and reject $H_0$ when $X\ge5$. Calculate the type I error $\alpha$.

Let $X\sim\mathrm{Bin}(40,0.05)$. You decide to reject $H_0: p=0.05$ for $H_1: p>0.05$ when $X\ge4$. Determine the type I error $\alpha$.

Suppose $X\sim\mathrm{Pois}(6)$, representing counts over 2 hours. Test $H_0: \lambda=6$ vs $H_1: \lambda>6$ with rejection if $X\ge10$. Calculate $\alpha$.

Out of $n=60$ items, $X\sim\mathrm{Bin}(60,0.05)$. You reject $H_0: p=0.05$ in favour of $H_1: p>0.05$ if $X\ge6$. Find the type I error probability $\alpha$.

In a time interval of 10 hours, the number of events follows $X\sim\mathrm{Pois}(5)$. You wish to test $H_0: \lambda=5$ against $H_1: \lambda<5$ and reject if $X\le2$. Find $\alpha$.

In quality control, $X\sim\mathrm{Bin}(25,0.2)$. You test $H_0: p=0.2$ vs $p>0.2$ and reject if $X\ge8$. Find the type I error probability.

Let $X\sim\mathrm{Pois}(10)$. You reject $H_0: \lambda=10$ in favour of $\lambda>10$ when $X\ge15$. Determine the type I error $\alpha$.

For $X\sim\mathrm{Bin}(100,0.1)$, test $H_0: p=0.1$ vs $H_1: p<0.1$ with critical region $X\le5$. Calculate the type I error $\alpha$.

For $X\sim\mathrm{Bin}(30,0.2)$, test $H_0: p=0.2$ vs $H_1: p<0.2$ and reject $H_0$ if $X\le3$. Compute the type I error $\alpha$.

For $X\sim\mathrm{Pois}(4)$, carry out a two‐sided test of $H_0: \lambda=4$ with critical region $\{X\le1\}\cup\{X\ge7\}$. Compute the type I error $\alpha$.

For $X\sim\mathrm{Bin}(50,0.1)$, consider the two‐sided test $H_0: p=0.1$ vs $H_1: p\neq0.1$ with critical region $\{X\le3\}\cup\{X\ge9\}$. Compute the overall type I error $\alpha$.