Type I errors Problem set A

Find the probability of making a Type I error in each test.

1. (a) X ~ N(μ, 5.2²), n = 14, 2% significance level; H0: μ = 120, H1: μ > 120.
2. (b) X ~ N(μ, 0.8²), n = 25, 8% significance level; H0: μ = 3.5, H1: μ ≠ 3.5.

X is normally distributed with standard deviation σ. A sample of size n is used to test H0: μ = μ0. Find the probability of making a Type I error for each critical region.

1. (a) σ = 4.5, n = 9, H0: μ = 52, H1: μ < 52; critical region \(\bar X < 50\).
2. (b) σ = 2, n = 16, H0: μ = 10, H1: μ ≠ 10; critical region \(\bar X < 9.2\) or \(\bar X > 10.8\).

X ~ B(n, p). Find the probability of making a Type I error for each test.

1. (a) n = 15, H0: p = 0.6, H1: p < 0.6; critical region X ≤ 5.
2. (b) n = 10, H0: p = 0.3, H1: p > 0.3; critical region X ≥ 6.

X ~ Po(λ). Find the probability of making a Type I error for each test.

1. (a) H0: λ = 5.5, H1: λ < 5.5; critical region X ≤ 2.
2. (b) H0: λ = 4.1, H1: λ > 4.1; critical region X ≥ 9.

A factory says that 8% of its bolts are faulty. A customer thinks that the true proportion is higher. He tests a random sample of 40 bolts at the 5% significance level.

1. (a) Find the critical region for the test.
2. (b) Find the probability of making a Type I error.

The number of customers who arrive at a kiosk in 10 minutes is modelled by a Poisson distribution with mean 11. The owner thinks that a new sign has increased this mean. She will count the arrivals in one 10-minute period and test, at the 5% significance level.

1. (a) Find the critical region for the test.
2. (b) Find the probability of making a Type I error.

A bakery makes rolls with a mean mass of 80 g and a standard deviation of 6 g. An inspector takes a random sample of 16 rolls and rejects the batch if the sample mean is below 77.5 g. Assume that masses are normally distributed. Find the probability that a batch with mean mass 80 g is wrongly rejected.

A dairy states that the mean fat content of its milk is 3.5%. An inspector tests, at the 5% significance level, whether the mean fat content is different from 3.5%. She uses the mean of a random sample of 20 cartons.

1. (a) Describe a Type I error in this context.
2. (b) Find the probability of making a Type I error.

A drug is said to work for 30% of patients. A doctor thinks that it works for fewer, and tests this at the 5% significance level with a random sample of 25 patients.

1. (a) Find the critical region for the test.
2. (b) Find the probability of making a Type I error.
3. (c) Explain why this probability is not exactly 0.05.

A teacher says that the mean score on a test is 70, with standard deviation 12. A researcher decides to reject this claim in favour of a higher mean if the sample mean is above 73. Assume that scores are normally distributed.

1. (a) Find the probability of a Type I error when the sample size is 25.
2. (b) Find the probability of a Type I error when the sample size is 100.
3. (c) Explain the change.