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

Question 1

Skill question

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.

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Question 2

Skill question

The question covers hypothesis testing for a binomial proportion using a normal approximation with a continuity correction, and the calculation of a Type II error probability.

Use a normal approximation (with continuity correction) for $X \sim Bin (300, 0.02)$ to test $H_{0} : p = 0.02$ against $H_{1} : p > 0.02$ at a significance level of $α = 0.01$ .

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

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Question 3

Skill question

A binomial test uses $n = 60$ items for $H_{0} : p = 0.10$ with critical region $X \geq 12$ . Compute the Type II error probability when $p = 0.15$ .

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Question 4

Skill question

A random variable $X$ follows a binomial distribution $X \sim Bin (200, 0.05)$ . We test the null hypothesis $H_{0} : p = 0.05$ against the alternative hypothesis $H_{1} : p > 0.05$ at a significance level of $α = 0.05$ . 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, $β$ , for the case where the true proportion is $p = 0.07$ .

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Question 5

Skill question

Determine the smallest integer $k$ such that for testing $H_{0} : λ = 3$ vs $H_{1} : λ = 4$ in a Poisson model the type I error $α = P (X \geq k ∣ λ = 3) \leq 0.05$ . Then compute the type II error probability $β = P (X < k ∣ λ = 4)$ .

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Question 6

Skill question

The number of faults in a component follows a Poisson distribution with parameter $λ$ . A statistical test is conducted with $H_{0} : λ = 2$ and $H_{1} : λ = 3$ . The critical region for the test is defined as $X \geq 4$ .

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

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Question 7

Skill question

The question concerns hypothesis testing with a Poisson distribution as an approximation for a Binomial distribution. It requires the calculation of Type II error probability $(β)$ for a given critical region and alternative hypothesis parameter.

A Binomial distribution $Bin (500, p)$ is approximated by a Poisson distribution for testing the null hypothesis $H_{0} : p = 0.02$ (where $λ = 10$ ) against the alternative hypothesis $H_{1} : p = 0.05$ (where $λ = 25$ ).

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

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Question 8

Skill question

The question assesses the ability to calculate the probability of a Type II error for a hypothesis test involving a Poisson distribution.

Fault counts follow a Poisson distribution, $X \sim Pois (λ)$ . A test of the null hypothesis $H_{0} : λ = 4$ against the alternative hypothesis $H_{1} : λ = 6$ is conducted. The null hypothesis is rejected when $X \geq 8$ .

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

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Question 9

Skill question

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 $X \geq 10$ . Find the probability of a Type II error when $p = 0.08$ .

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Question 10

Skill question

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 $X \geq 4$ .

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

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Question 11

Skill question

The probability of a Type II error is calculated for a binomial hypothesis test given a specific true population parameter.

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 $X \geq 5$ , where $X \sim B (20, p)$ .

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

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IB Mathematics Applications & Interpretation Question Type 11: Calculating the Probability of a Type II Error for a Critical Region Under Poisson on Binomial Distribution Exercises