Type I errors Problem set B Worked solutions

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.

   • The distribution of the sample mean is continuous, so P(Type I error) is the significance level.
   • P(Type I error) = 0.01

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

   • The distribution of the sample mean is continuous, so P(Type I error) is the significance level.
   • P(Type I error) = 0.06

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 \(\bar X < 6.5\).

   • Under H0, X̄ ~ N(7.4, 1.8²/6), standard deviation 0.735
   • P(Type I error) = P(X̄ < 6.5) = 0.110

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

   • Under H0, X̄ ~ N(200, 15²/25), standard deviation 3.00
   • P(Type I error) = P(X̄ < 194) + P(X̄ > 206)
   • = 0.0228 + 0.0228 = 0.0455

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.

   • Under H0, X ~ B(18, 0.45)
   • From the calculator: P(Type I error) = P(X ≤ 4) = 0.0411

2. (b) n = 14, H0: p = 0.15, H1: p > 0.15; critical region X ≥ 5.

   • Under H0, X ~ B(14, 0.15)
   • P(Type I error) = P(X ≥ 5) = 1 − P(X ≤ 4)
   • From the calculator: 1 − 0.95326 = 0.0467

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.

   • Under H0, X ~ Po(7.3)
   • From the calculator: P(Type I error) = P(X ≤ 3) = 0.0674

2. (b) H0: λ = 2.6, H1: λ > 2.6; critical region X ≥ 7.

   • Under H0, X ~ Po(2.6)
   • P(Type I error) = P(X ≥ 7) = 1 − P(X ≤ 6)
   • From the calculator: 1 − 0.98283 = 0.0172

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.

   • H0: p = 0.12, H1: p > 0.12. Under H0, X ~ B(30, 0.12).
   • From the calculator: P(X ≥ 7) = 0.0606 > 0.05 and P(X ≥ 8) = 0.0221 ≤ 0.05
   • Critical region: X ≥ 8

2. (b) Find the probability of making a Type I error.

   • P(Type I error) = P(X ≥ 8) = 0.0221
   • This is less than 5%, because X takes whole values only.

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.

   • H0: λ = 14, H1: λ < 14. Under H0, X ~ Po(14).
   • From the calculator: P(X ≤ 7) = 0.0316 ≤ 0.05 and P(X ≤ 8) = 0.0621 > 0.05
   • Critical region: X ≤ 7

2. (b) Find the probability of making a Type I error.

   • P(Type I error) = P(X ≤ 7) = 0.0316
   • This is less than 5%.

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.

• This is a Type I error: H0 (mean 40) is true but is rejected.
• Under H0, X̄ ~ N(40, 3.2²/10), standard deviation 1.01
• P(X̄ > 41.5) = 0.0691

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.

   • A Type I error is rejecting H0 when H0 is true.
   • H0: the mean time is 25 minutes. Rejecting it when it is true means wrongly deciding that the mean has changed.

2. (b) Find the probability of making a Type I error.

   • The sample mean has a continuous distribution, so P(Type I error) is the significance level.
   • P(Type I error) = 0.01

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.

   • H0: p = 0.1, H1: p > 0.1. Under H0, X ~ B(30, 0.1).
   • From the calculator: P(X ≥ 6) = 0.0732 > 0.05 and P(X ≥ 7) = 0.0258 ≤ 0.05
   • Critical region: X ≥ 7

2. (b) Find the probability of making a Type I error.

   • P(Type I error) = P(X ≥ 7) = 0.0258

3. (c) Explain why this probability is not exactly 0.05.

   • The next value would give 0.0732, which is above 0.05.
   • So the actual probability 0.0258 is below the nominal level of 5%.

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.

   • Under H0, X̄ ~ N(500, 20²/16), standard deviation 5.00
   • P(Type I error) = P(X̄ < 494) = 0.115

2. (b) Find the probability of a Type I error when the sample size is 64.

   • Under H0, X̄ ~ N(500, 20²/64), standard deviation 2.50
   • P(Type I error) = P(X̄ < 494) = 0.00820

3. (c) Explain the change.

   • A larger sample makes the sample mean less spread out around the H0 mean.
   • The same critical value is further away in standard deviations, so the probability falls from 0.115 to 0.00820.