Question Type 11: Calculating the probability of a type II error for a critical region under poisson on binomial distribution Exercises

A quality control engineer tests $H_0 egthinspace: olimits p=0.10$ against $H_1 egthinspace: olimits p=0.20$ using a sample of $n=20$ items. The critical region is $X\ge5$, where $X\\sim\\mathrm{Bin}(20,p)$. If the true defective probability is $0.20$, calculate the probability of a type II error.

In a lot of 50 items, a test is set up with $H_0 egthinspace: olimits p=0.05$ versus $H_1 egthinspace: olimits p=0.10$. The critical region is $X\ge4$. Compute the type II error probability when the true defective rate is $0.10$.

A batch test uses $n=30$ items to test $H_0 egthinspace: olimits p=0.20$ against $H_1 egthinspace: olimits p=0.30$. The critical region is $X\le6$ (i.e. accept $H_1$ only if $X>6$). If the true $p=0.30$, find the probability of a type II error.

The number of faults in a component follows a Poisson distribution. A test uses $H_0 egthinspace: olimits\lambda=2$ versus $H_1 egthinspace: olimits\lambda=3$. The critical region is $X\ge4$. If the true rate is $\lambda=3$, compute the type II error probability.

Approximate $\mathrm{Bin}(500,0.02)$ by Poisson for testing $H_0\!:p=0.02$ ($\lambda=10$) vs $H_1\!:p=0.05$ ($\lambda=25$). The critical region is $X\ge15$. Compute $\beta$ at $\lambda=25$.

A binomial test uses $n=60$ items for $H_0\!:p=0.10$ with critical region $X\ge12$. Compute the type II error probability when $p=0.15$.

In a large production run of $n=100$ items, a hypothesis test compares $H_0 egthinspace: olimits p=0.05$ to $H_1 egthinspace: olimits p=0.08$ with critical region $X\ge10$. Find the type II error probability when $p=0.08$.

Using a normal approximation, let $X\sim\mathrm{Bin}(200,0.05)$. We test $H_0\!:p=0.05$ at $\alpha=0.05$ (upper tail) and reject if $X>c$. Determine $c$ (with continuity correction) and then compute $\beta$ for $p=0.07$.

Fault counts follow $\mathrm{Pois}(\lambda)$. A test of $H_0\!:\lambda=4$ vs $H_1\!:\lambda=6$ rejects when $X\ge8$. Calculate the type II error probability under $\lambda=6$.

Determine the smallest integer $k$ such that for testing $H_0\!:\lambda=3$ vs $H_1\!:\lambda=4$ in a Poisson model the type I error $\alpha=P(X\ge k\mid\lambda=3)\le0.05$. Then compute the type II error probability $\beta=P(X<k\mid\lambda=4)$.

Use a normal approximation (with continuity correction) for $X\sim\mathrm{Bin}(300,0.02)$ to test $H_0\!:p=0.02$ at $\alpha=0.01$ (upper tail). Find the critical value $c$ and compute the type II error probability when $p=0.03$.