Type I errors Problem set B

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

1. (a) X ~ N(μ, 2.9²), n = 10, 1% significance level; H0: μ = 64, H1: μ < 64.
2. (b) X ~ N(μ, 4.1²), n = 18, 6% significance level; H0: μ = 22, H1: μ ≠ 22.

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 = 1.8, n = 6, H0: μ = 7.4, H1: μ < 7.4; critical region \(\bar X < 6.5\).
2. (b) σ = 15, n = 25, H0: μ = 200, H1: μ ≠ 200; critical region \(\bar X < 194\) or \(\bar X > 206\).

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

1. (a) n = 18, H0: p = 0.45, H1: p < 0.45; critical region X ≤ 4.
2. (b) n = 14, H0: p = 0.15, H1: p > 0.15; critical region X ≥ 5.

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

1. (a) H0: λ = 7.3, H1: λ < 7.3; critical region X ≤ 3.
2. (b) H0: λ = 2.6, H1: λ > 2.6; critical region X ≥ 7.

A call centre says that 12% of its calls are not resolved. A manager thinks that the true proportion is higher. She tests a random sample of 30 calls 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 emails that a teacher receives in an hour is modelled by a Poisson distribution with mean 14. She thinks that a new school notice has reduced this mean. She will count the emails in one hour 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 supplier says that its phone batteries last 40 hours on average, with a standard deviation of 3.2 hours. A tester takes a random sample of 10 batteries and concludes that the mean life is greater than 40 hours if the sample mean is above 41.5 hours. Assume that lives are normally distributed. Find the probability that the tester wrongly makes this conclusion.

A teacher states that the mean time for her class to finish a quiz is 25 minutes. A colleague tests, at the 1% significance level, whether the mean time has changed. He uses the mean of a random sample of 15 students.

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

A farmer says that 10% of her eggs are cracked. The buyer thinks that the true proportion is higher, and tests this at the 5% significance level with a random sample of 30 eggs.

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 cereal company says that the mean mass of its boxes is 500 g, with standard deviation 20 g. An inspector decides to reject this claim in favour of a lower mean if the sample mean is below 494 g. Assume that masses are normally distributed.

1. (a) Find the probability of a Type I error when the sample size is 16.
2. (b) Find the probability of a Type I error when the sample size is 64.
3. (c) Explain the change.