Type II errors Problem set A

Find the probability of a Type II error for each test of a normal mean. Give answers to 3 significant figures.

1. (a) \(X \sim N(\mu, 0.6^2)\), sample size 9. H0: μ = 12, H1: μ < 12. Critical region \(\bar X < 11.6\). True mean μ = 11.2.
2. (b) \(X \sim N(\mu, 4^2)\), sample size 16. H0: μ = 50, H1: μ ≠ 50. Critical region \(\bar X < 48\) or \(\bar X > 52.5\). True mean μ = 51.5.

Find the probability of a Type II error for each test. Give answers to 3 significant figures.

1. (a) \(X \sim B(25, p)\). H0: p = 0.4, H1: p > 0.4. Critical region \(X \ge 14\). True p = 0.6.
2. (b) \(X \sim Po(\lambda)\). H0: λ = 6, H1: λ < 6. Critical region \(X \le 2\). True λ = 3.5.

Find the probability of a Type I error in each test. Give answers to 3 significant figures.

1. (a) \(X \sim N(\mu, 2.5^2)\), sample size 10. H0: μ = 30, H1: μ ≠ 30. Critical region \(\bar X < 28.6\) or \(\bar X > 31.8\).
2. (b) \(X \sim B(18, p)\). H0: p = 0.35, H1: p < 0.35. Critical region \(X \le 3\).
3. (c) The number of events in one hour is Poisson with mean 1.5 under H0. The test counts events over 4 hours, and the critical region is \(X \ge 11\).

Answer the following.

1. (a) A coffee shop says that 40% of its customers order a flat white. The owner thinks the proportion has risen after a menu change and will test this with 30 customers. State suitable hypotheses, defining p.
2. (b) A website averages 7.2 sales a day. The designer hopes a new layout has increased the daily mean. State suitable hypotheses, defining λ.
3. (c) In the coffee shop test, describe a Type I error and a Type II error in context.

A machine makes steel rods whose length is normally distributed with standard deviation 0.5 mm. The mean length should be 14.0 mm. After a service the engineer thinks the mean length has increased. She takes a sample of 10 rods and tests H0: μ = 14.0 against H1: μ > 14.0 at the 5% significance level.

1. (a) Find the critical region. Give the critical value to 3 significant figures.
2. (b) The true mean is 14.3 mm. Find the probability of a Type II error. Use the unrounded critical value.

A packing line fills sacks of grain. The mass is normally distributed with standard deviation 3.0 kg. A manager tests H0: μ = 80.0 against H1: μ ≠ 80.0 using the mean of 8 sacks. He rejects H0 if the sample mean is below 77.4 kg or above 82.6 kg.

1. (a) The true mean mass is 78.5 kg. Find the probability of a Type II error.
2. (b) The manager now uses 30 sacks and keeps the same significance level. Explain the effect on the probability of a Type I error and on the probability of a Type II error.

A hospital screens patients for a rare condition. The test result is used to decide between H0: the patient does not have the condition, and H1: the patient has the condition.

1. (a) Describe a Type I error and a Type II error in this context.
2. (b) Which error is more important to avoid? Give a reason.

A cyclist says that she wins 50% of her sprint finishes. Her coach thinks the claim is too high and records the results of 20 sprint finishes. The coach rejects the claim if the cyclist wins 5 or fewer, using H0: p = 0.5 against H1: p < 0.5.

1. (a) Find the probability that the coach makes a Type I error.
2. (b) The cyclist's true probability of winning is 0.3. Find the probability that the coach makes a Type II error.

A café owner says that 30% of customers buy a cake. A student thinks the proportion is higher and counts the number of cake buyers among 25 customers. The student rejects the owner's claim for a critical region \(X \ge c\) with probability of a Type I error less than 0.05.

1. (a) State H0 and H1 and find the critical region that gives the smallest probability of a Type II error.
2. (b) Given that the true probability that a customer buys a cake is 0.5, find the smallest probability of a Type II error.

The number of parcels delivered to a building in one hour is Poisson with mean 9.0. After a change of courier the building manager thinks the mean has decreased. She counts the parcels in one hour and rejects H0 if the count is in a critical region \(X \le c\). The true mean is now 3.0. She wants the probability of a Type II error to be less than 0.15.

1. (a) State the hypotheses and find the critical region that gives the smallest probability of a Type I error.
2. (b) Find that smallest probability of a Type I error.

A drawer holds 15 socks, r of them black and the rest white. Ravi knows that r is either 4 or 10. He takes 3 socks without looking and tests H0: r = 4 against H1: r = 10. He rejects H0 if at least 2 of the socks are black.

1. (a) Find the probability that he makes a Type I error.
2. (b) Find the probability that he makes a Type II error.