Type II errors Problem set B

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

1. (a) \(X \sim N(\mu, 1.8^2)\), sample size 12. H0: μ = 40, H1: μ < 40. Critical region \(\bar X < 38.9\). True mean μ = 38.0.
2. (b) \(X \sim N(\mu, 0.35^2)\), sample size 10. H0: μ = 8, H1: μ ≠ 8. Critical region \(\bar X < 7.8\) or \(\bar X > 8.25\). True mean μ = 7.7.

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

1. (a) \(X \sim B(30, p)\). H0: p = 0.5, H1: p > 0.5. Critical region \(X \ge 20\). True p = 0.7.
2. (b) \(X \sim Po(\lambda)\). H0: λ = 5, H1: λ < 5. Critical region \(X \le 1\). True λ = 2.6.

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

1. (a) \(X \sim N(\mu, 0.9^2)\), sample size 6. H0: μ = 15, H1: μ ≠ 15. Critical region \(\bar X < 14.0\) or \(\bar X > 16.3\).
2. (b) \(X \sim B(25, p)\). H0: p = 0.4, H1: p < 0.4. Critical region \(X \le 5\).
3. (c) The number of events in one day is Poisson with mean 0.8 under H0. The test counts events over 5 days, and the critical region is \(X \ge 7\).

Answer the following.

1. (a) A school says that 65% of its students walk to school. The head thinks the proportion has fallen since a new bus route opened and will test this with 40 students. State suitable hypotheses, defining p.
2. (b) An office receives a mean of 14 emails an hour. A new filter is meant to reduce the hourly mean. State suitable hypotheses, defining λ.
3. (c) In the walking test, describe a Type I error and a Type II error in context.

A dairy sells yoghurt pots whose mass is normally distributed with standard deviation 12 g. The mean mass is meant to be 250 g. A trading standards officer thinks the mean mass is too low. She takes a sample of 6 pots and tests H0: μ = 250 against H1: μ < 250 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 238.0 g. Find the probability of a Type II error. Use the unrounded critical value.

A carpenter cuts planks whose length is normally distributed with standard deviation 4.0 cm. An inspector tests H0: μ = 120.0 against H1: μ ≠ 120.0 using the mean of 9 planks. He rejects H0 if the sample mean is below 117.4 cm or above 122.6 cm.

1. (a) The true mean length is 123.0 cm. Find the probability of a Type II error.
2. (b) The inspector now uses 36 planks 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 bank uses software to decide whether a card payment is fraudulent. For each payment the software chooses between H0: the payment is genuine, and H1: the payment is fraudulent.

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 football club says that 60% of its season ticket holders attend each home game. A supporter thinks the figure is too high and counts the attendance of 16 season ticket holders at one game. She rejects the claim if 5 or fewer attend, using H0: p = 0.6 against H1: p < 0.6.

1. (a) Find the probability that the supporter makes a Type I error.
2. (b) The true probability that a season ticket holder attends is 0.35. Find the probability that the supporter makes a Type II error.

A pizza shop says that 20% of its orders include a dessert. An inspector thinks the proportion is higher and counts the orders with a dessert among 30 orders. The inspector rejects the claim for a critical region \(X \ge c\) with probability of a Type I error less than 0.04.

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 an order includes a dessert is 0.4, find the smallest probability of a Type II error.

The number of customers entering a kiosk in a ten-minute period is Poisson with mean 11.0. After road works start the owner thinks the mean has decreased. She counts the customers in one ten-minute period and rejects H0 if the count is in a critical region \(X \le c\). The true mean is now 4.0. She wants the probability of a Type II error to be less than 0.2.

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 tray holds 10 cupcakes, r of them chocolate and the rest vanilla. Nadia knows that r is either 3 or 7. She takes 4 cupcakes without looking and tests H0: r = 3 against H1: r = 7. She rejects H0 if at least 3 of the cupcakes are chocolate.

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