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

Suppose $X \sim \text{Pois}(6)$ represents the count of events occurring over a 2-hour interval. A hypothesis test is performed with the null hypothesis $H_0: \lambda = 6$ and the alternative hypothesis $H_1: \lambda > 6$. The null hypothesis is rejected if $X \ge 10$.

Calculate the significance level, $\alpha$, for this test.

A sample of $n = 60$ items is taken from a population where the number of successes $X$ follows a binomial distribution $\text{B}(60, p)$. The null hypothesis $H_0: p = 0.05$ is tested against the alternative hypothesis $H_1: p > 0.05$. The null hypothesis is rejected if $X \ge 6$.

Calculate the probability of a Type I error, $\alpha$.

Let $X \sim \text{Bin}(40, 0.05)$. You decide to reject $H_0: p = 0.05$ in favour of $H_1: p > 0.05$ when $X \ge 4$. Determine the Type I error, $\alpha$.

In a quality control process, a random variable $X$ follows a binomial distribution such that $X \sim \text{B}(25, 0.2)$. A hypothesis test is performed with the null hypothesis $H_0: p = 0.2$ against the alternative hypothesis $H_1: p > 0.2$. The null hypothesis is rejected if $X \geq 8$.

Find the probability of a Type I error.

Let $X$ follow a Poisson distribution with mean $\lambda$, such that $X \sim \text{Po}(\lambda)$. A hypothesis test is conducted with $H_0: \lambda = 2$ and $H_1: \lambda > 2$. The null hypothesis is rejected when $X \ge 5$.

Calculate the probability of a type I error, $\alpha$.

For $X \sim \text{Bin}(50, 0.1)$, consider the two-sided test $H_0: p = 0.1$ vs $H_1: p \neq 0.1$ with critical region $\{X \le 3\} \cup \{X \ge 9\}$. Compute the overall type I error $\alpha$.

For a batch of $n = 20$ components, let $X \sim \text{B}(20, 0.1)$. You decide to reject $H_0: p = 0.1$ in favour of $H_1: p > 0.1$ if $X \ge 3$. Calculate the probability of a Type I error, $\alpha$.

Let $X \sim \text{Po}(10)$. The null hypothesis $H_0: \lambda = 10$ is rejected in favour of $H_1: \lambda > 10$ when $X \ge 15$.

Determine the probability of a Type I error, $\alpha$.

For $X \sim \text{Po}(4)$, consider a two-sided test of $H_0: \lambda = 4$ with the critical region $\{X \le 1\} \cup \{X \ge 7\}$. Calculate the significance level of the test, $\alpha$ (the probability of a Type I error).

In a time interval of 10 hours, the number of events follows $X \sim \text{Pois}(5)$. You wish to test $H_0: \lambda = 5$ against $H_1: \lambda < 5$ and reject $H_0$ if $X \le 2$. Find the significance level, $\alpha$.

For $X \sim B(30, 0.2)$, test $H_0: p = 0.2$ against $H_1: p < 0.2$ and reject $H_0$ if $X \le 3$. Compute the probability of a type I error, $\alpha$.

For $X \sim \text{Bin}(100, 0.1)$, test $H_0: p = 0.1$ vs $H_1: p < 0.1$ with a critical region defined by $X \le 5$. Calculate the Type I error, $\alpha$, for this test.