Critical regions for a test of a mean Problem set A

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

1. (a) H0: μ = 64; H1: μ < 64; 5% significance level; n = 25; population standard deviation σ = 5.
2. (b) H0: μ = 2.4; H1: μ < 2.4; 1% significance level; n = 36; population standard deviation σ = 0.3.

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

1. (a) H0: μ = 120; H1: μ > 120; 5% significance level; n = 30; population standard deviation σ = 15.
2. (b) H0: μ = 0.85; H1: μ > 0.85; 2% significance level; n = 20; population standard deviation σ = 0.1.

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

1. (a) H0: μ = 35; H1: μ ≠ 35; 5% significance level; n = 16; population standard deviation σ = 4.
2. (b) H0: μ = 210; H1: μ ≠ 210; 10% significance level; n = 40; population standard deviation σ = 12.

The volume of coffee in a takeaway cup is normally distributed with mean 340 ml and standard deviation 6 ml. After a new dispenser is fitted, a café owner takes a random sample of 25 cups to test, at the 5% significance level, whether the mean volume has decreased.

1. (a) State the hypotheses.
2. (b) Find the critical region for the sample mean.
3. (c) The sample mean volume is 337.6 ml. State the conclusion of the test.

A courier's delivery times, in minutes, are normally distributed with mean 32 and standard deviation 7. After a change of route, the mean time for a random sample of 49 deliveries is 34.6 minutes. The courier tests, at the 1% significance level, whether the mean delivery time 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.

Resistors are labelled 470 Ω. Their resistances are normally distributed with standard deviation 9 Ω. A sample of 30 resistors from a new supplier has mean 473.6 Ω. Test, at the 5% significance level, whether the mean resistance has changed.

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

A cable maker claims a mean breaking load of 800 N, with standard deviation 40 N. A buyer tests H0: μ = 800 against H1: μ < 800 at the 5% significance level.

1. (a) Find the critical value when the sample has 16 cables.
2. (b) Find the critical value when the sample has 64 cables.
3. (c) Explain why the critical value in part (b) is closer to 800 N.

A test of H0: μ = 100 against H1: μ < 100 uses a sample of 36 observations from a normal population with standard deviation 12. The critical region is \(\bar X < 96\). Find the significance level of the test.