Critical regions for a test of a mean Problem set B

Find the critical region for each test of a mean. Give critical values to 3 significant figures.

1. (a) H0: μ = 18.5; H1: μ < 18.5; 1% significance level; n = 64; population standard deviation σ = 2.4.
2. (b) H0: μ = 0.62; H1: μ < 0.62; 10% significance level; n = 25; population standard deviation σ = 0.08.

Find the critical region for each test of a mean. Give critical values to 3 significant figures.

1. (a) H0: μ = 72; H1: μ > 72; 1% significance level; n = 36; population standard deviation σ = 9.
2. (b) H0: μ = 4.2; H1: μ > 4.2; 5% significance level; n = 30; population standard deviation σ = 0.5.

Find the critical region for each test of a mean. Give critical values to 3 significant figures.

1. (a) H0: μ = 18; H1: μ ≠ 18; 10% significance level; n = 25; population standard deviation σ = 2.5.
2. (b) H0: μ = 0.95; H1: μ ≠ 0.95; 2% significance level; n = 45; population standard deviation σ = 0.06.

The journey time of a train between two cities is normally distributed with mean 54 minutes and standard deviation 5 minutes. After a timetable change, a rail company takes a random sample of 36 journeys to test, at the 5% significance level, whether the mean journey time has decreased.

1. (a) State the hypotheses.
2. (b) Find the critical region for the sample mean.
3. (c) The sample mean journey time is 52.9 minutes. State the conclusion of the test.

The mass of a tomato from a grower is normally distributed with mean 82 g and standard deviation 9 g. After a change of feed, the mean mass of a random sample of 40 tomatoes is 84.1 g. The grower tests, at the 2% significance level, whether the mean mass has increased.

1. (a) Find the critical region for the sample mean.
2. (b) Is the sample mean in the critical region? State the conclusion.

Rolls of fabric are cut to a mean length of 25.0 m. The lengths are normally distributed with standard deviation 1.5 m. A sample of 25 rolls from a new cutting machine has mean length 25.6 m. Test, at the 1% significance level, whether the mean length has changed.

1. (a) Find the critical region for the sample mean.
2. (b) State the conclusion of the test.

A school says that the mean score on a reading test is 55, with standard deviation 10. An inspector tests H0: μ = 55 against H1: μ < 55 at the 1% significance level.

1. (a) Find the critical value when the sample has 25 students.
2. (b) Find the critical value when the sample has 100 students.
3. (c) Explain why the critical value in part (b) is closer to 55.

A test of H0: μ = 50 against H1: μ < 50 uses a sample of 81 observations from a normal population with standard deviation 9. The critical region is \(\bar X < 48.5\). Find the significance level of the test.