Hypothesis tests for a mean: mixed practice Problem set B

A teacher times how long, in seconds, students take to solve a puzzle. One random group had hints and a different random group had none. Assume the times are normally distributed with equal variance. He tests, at the 10% significance level, whether the mean times of the two groups are different. With hints (s): 48, 55, 62, 41, 57, 50, 66, 53 No hints (s): 61, 72, 58, 69, 75, 64, 70

1. (a) State suitable hypotheses.
2. (b) Use your calculator to find the p-value.
3. (c) State the conclusion in context.

A bus company says that the journey to school takes a mean of 35 minutes. A parent times a random sample of 15 journeys. The mean is 37.9 minutes and an unbiased estimate of the population variance is 28.8 min². She thinks the mean journey time is more than 35 minutes. Test her belief at the 5% significance level.

1. (a) Explain why a t-test, not a z-test, is used.
2. (b) Write down hypotheses, defining μ.
3. (c) Find the p-value.
4. (d) State the conclusion in context.

The number of sit-ups that 8 pupils can do in one minute is counted before and after a month of practice.

1. (a) Find the difference, after − before, for each pupil.
2. (b) Use your calculator to find the mean and the standard deviation s of the differences.

The drying time of a load of washing in a tumble dryer is normally distributed with mean 55 minutes and standard deviation 7 minutes. After a repair the owner thinks the mean drying time has increased. She times a random sample of 30 loads and assumes the standard deviation is unchanged. Test her belief at the 5% significance level.

1. (a) Write down suitable hypotheses.
2. (b) Find the distribution of the sample mean \(\bar X\) when H0 is true.
3. (c) Find the critical region.
4. (d) The sample mean is 57.6 minutes. State the conclusion.

A saw cuts rods with a mean length of 120 cm and standard deviation 0.9 cm. An engineer takes a random sample of 25 rods to test, at the 1% significance level, whether the mean length has changed. Assume the lengths are normally distributed and the standard deviation is unchanged.

1. (a) Write down the distribution of \(\bar X\) when H0 is true.
2. (b) Find the critical region.
3. (c) The sample mean is 120.5 cm. State the conclusion.

A coach tests whether a ten-minute walk raises the resting pulse rate of pupils. Eight pupils' pulse rates, in beats per minute, are measured before and after the walk. Test at the 5% significance level.

1. (a) Explain why a paired test is suitable.
2. (b) Write down hypotheses in terms of μD, where D = after − before.
3. (c) Find the p-value.
4. (d) State the conclusion in context.

A sports scientist is comparing training methods.

1. (a) She measures the jump height of the same 10 athletes before and after a month of training. Should she use a paired test or a two-sample test? Give a reason.
2. (b) She also compares the mean jump heights of 9 athletes who trained with weights and 11 different athletes who trained with bands. Should she use a paired test or a two-sample test? Give a reason.
3. (c) State the assumption needed for a paired t-test.
4. (d) State two assumptions needed for a pooled two-sample t-test.
5. (e) State one advantage of a paired design over two independent samples.

A school compares two teaching methods. Students taught with the old method (12 students) have a mean mark of 71.5 (sample variance 16.8) and students taught with the new method (10 students) have a mean mark of 66.2 (sample variance 14.4). A teacher thinks the mean mark with the old method is more than 3 marks greater. Assume the marks are normally distributed with equal variance.

1. (a) Find unbiased estimates of the two population variances.
2. (b) Test the claim at the 10% significance level. Give the hypotheses, the pooled estimate of the variance, the test statistic and the p-value.
3. (c) State the conclusion in context.

Daily sales, in items, are normally distributed. At shop M the standard deviation is known to be 14 and at shop N it is known to be 10. Over a random sample of 35 days, shop M sells a mean of 253 items. Over a different random sample of 45 days, shop N sells a mean of 232 items. The owner believes that the mean daily sales at shop M are more than 15 greater than at shop N. Test his belief at the 10% significance level.

1. (a) Explain why a two-sample z-test can be used.
2. (b) Find the distribution of the difference between the two sample means when H0 is true.
3. (c) Find the p-value.
4. (d) State the conclusion in context.

A canteen manager thinks a new layout shortens the time students queue for lunch. The queue time of one student, in minutes, is normally distributed with standard deviation 0.8, both before and after the change, and the times are independent. The queue times of 12 students are measured before and after the change. Let D = queue time before − queue time after. Test at the 10% significance level.

1. (a) Write down suitable hypotheses in terms of μD.
2. (b) Find Var(D).
3. (c) Find the distribution of \(\bar D\) when H0 is true.
4. (d) Find the critical region.
5. (e) The sample mean difference is 0.6 minutes. State the conclusion.

The daily rainfall in a valley is skewed, with mean 3.4 mm and standard deviation 2.9 mm. A scientist thinks the mean daily rainfall has increased over the last 50 days, which have a mean of 4.2 mm. Assume the standard deviation is unchanged and treat the 50 days as a random sample. Test at the 5% significance level.

1. (a) Explain why a test based on the normal distribution can be used although the distribution of daily rainfall is not normal.
2. (b) Find the p-value.
3. (c) State the conclusion in context.

A gym says its fitness classes last a mean of 50 minutes. A member thinks they are shorter. The lengths of a random sample of 9 classes are recorded. Assume the lengths are normally distributed. Times (min): 46, 52, 44, 49, 55, 47, 51, 43, 50

1. (a) Use your calculator to find the mean and the sample standard deviation s.
2. (b) Find the p-value of a one-sample t-test of the claim.
3. (c) The member says: 'The p-value is large, so the classes last exactly 50 minutes.' Comment on this statement.