Exercises for Question Type 9: Finding the critical region under a poisson on binomial distribution

Question 1

A call center receives on average 10 calls per hour. Let Y \sim Pois (10) . For testing H_{0} : λ = 10 versus H_{1} : λ < 10 at α = 0.05, determine the lower-tail critical region.

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Question 2

For a Poisson test X \sim Po (10), the critical region is {X \leq 3} for testing H_{0} : λ = 10 versus H_{1} : λ < 10 . Calculate the Type I error probability.

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Question 3

Using the scenario where X \sim B (60, 0.05), find the critical region for a hypothesis test of H_{0} : p = 0.05 versus H_{1} : p > 0.05 at the α = 0.01 significance level.

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Question 4

A binomial test uses X \sim Bin (60, 0.05) with critical region {X \geq 7} . Calculate the Type I error probability under H_{0} : p = 0.05 .

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Question 5

In a Poisson test X \sim Pois (4), H_{0} : λ = 4 versus H_{1} : λ > 4, the critical region is {X \geq 8} . Calculate the Type II error probability if the true λ = 6 .

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Question 6

A factory claims that the defect rate is at most 5%. A sample of 60 items is tested. Let X \sim Bin (60, 0.05) . For testing H_{0} : p = 0.05 against H_{1} : p > 0.05 at the 5% significance level, determine the critical region.

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Question 7

For a Poisson test X \sim Pois (10), H_{0} : λ = 10 versus H_{1} : λ < 10, the critical region is {X \leq 3} . Calculate the Type II error probability if the true λ = 7 .

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Question 8

An email server historically receives on average 4 emails per hour. Let X \sim Pois (4) be the number of emails received in one hour. For testing H_{0} : λ = 4 versus H_{1} : λ > 4 at a significance level of α = 0.05, determine the critical region.

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Question 9

Probability and Statistics

In a binomial test with X \sim Bin (60, p), H_{0} : p = 0.05 versus H_{1} : p > 0.05, the critical region is {X \geq 8} . Calculate the Type II error probability if the true p = 0.10 .

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Question 10

In a binomial test, the Type II error occurs when the null hypothesis H_{0} is not rejected, even though the alternative hypothesis H_{1} is true. This probability is typically denoted by β .

In a binomial test X \sim Bin (60, p), H_{0} : p = 0.05 versus H_{1} : p > 0.05, the critical region is defined as {X \geq 9} . Calculate the Type II error probability if the true value of p is 0.08 .

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Question 11

In a test with X \sim Bin (60, 0.05), the critical region is chosen as {X \geq 8} . Calculate the actual Type I error probability.

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Question 12

In a Poisson test with X \sim Pois (4), the critical region is defined as {X \geq 7} . Calculate the actual Type I error probability for the hypothesis H_{0} : λ = 4 .

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IB Mathematics Applications & Interpretation Question Type 9: Finding the Critical Region Under a Poisson on Binomial Distribution Exercises