IB Mathematics Applications & Interpretation 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 $X\ge 8$.

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 $H_1:p\ne 0.1$ 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.