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

A batch test uses $n=30$ items to test the null hypothesis $H_0: p=0.20$ against the alternative hypothesis $H_1: p=0.30$. The critical region is defined as $X > 6$, where $X$ is the number of successful items in the batch.

Given that the true value of $p$ is $0.30$, find the probability of a Type II error.

Use a normal approximation (with continuity correction) for $X \sim \text{Bin}(300, 0.02)$ to test $H_0: p = 0.02$ against $H_1: p > 0.02$ at a significance level of $\alpha = 0.01$.

Find the critical value $c$ (the largest integer value for which the null hypothesis is not rejected) and compute the Type II error probability $\beta$ when the true proportion is $p = 0.03$.

A binomial test uses $n = 60$ items for $H_0 : p = 0.10$ with critical region $X \ge 12$. Compute the Type II error probability when $p = 0.15$.

A random variable $X$ follows a binomial distribution $X \sim \text{Bin}(200, 0.05)$. We test the null hypothesis $H_0: p = 0.05$ against the alternative hypothesis $H_1: p > 0.05$ at a significance level of $\alpha = 0.05$. The decision rule is to reject $H_0$ if $X > c$, where $c$ is an integer.

Using a normal approximation with a continuity correction, determine the value of $c$ such that the probability of a Type I error is as close as possible to, but does not exceed, $0.05$.

Hence, calculate the probability of a Type II error, $\beta$, for the case where the true proportion is $p = 0.07$.

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 $\alpha = P(X \ge k \mid \lambda=3) \le 0.05$. Then compute the type II error probability $\beta = P(X < k \mid \lambda=4)$.

The number of faults in a component follows a Poisson distribution with parameter $\lambda$. A statistical test is conducted with $H_0: \lambda = 2$ and $H_1: \lambda = 3$. The critical region for the test is defined as $X \ge 4$.

If the true rate is $\lambda = 3$, calculate the probability of a Type II error.

A Binomial distribution $\text{Bin}(500, p)$ is approximated by a Poisson distribution for testing the null hypothesis $H_0: p = 0.02$ (where $\lambda = 10$) against the alternative hypothesis $H_1: p = 0.05$ (where $\lambda = 25$).

The critical region for the test is defined as $X \ge 15$. Calculate the probability of a Type II error, $\beta$, given that $\lambda = 25$ under the alternative hypothesis.

Fault counts follow a Poisson distribution, $X \sim \text{Pois}(\lambda)$. A test of the null hypothesis $H_0 : \lambda = 4$ against the alternative hypothesis $H_1 : \lambda = 6$ is conducted. The null hypothesis is rejected when $X \ge 8$.

Calculate the probability of a Type II error, $\beta$, for this test.

In a large production run of $n = 100$ items, a hypothesis test compares $H_0: p = 0.05$ to $H_1: p = 0.08$ with critical region $X \ge 10$. Find the probability of a Type II error when $p = 0.08$.

In a lot of 50 items, a test is set up with $H_0: p = 0.05$ versus $H_1: p = 0.10$. The critical region is $X \ge 4$.

Compute the probability of a Type II error when the true defective rate is $0.10$.

A quality control engineer tests $H_0: p = 0.10$ against $H_1: p = 0.20$ using a sample of $n = 20$ items. The critical region for this test is $X \ge 5$, where $X \sim \text{B}(20, p)$.

Given that the true defective probability is $0.20$, calculate the probability of a Type II error.