Hypothesis tests for a mean: mixed practice Problem set A

A park ranger records the mass, in kg, of random samples of trout from two lakes. Assume the masses are normally distributed with equal variance. She tests, at the 5% significance level, whether the mean masses in the two lakes are different. Lake P (kg): 1.82, 2.05, 1.67, 2.21, 1.94, 1.78, 2.1 Lake Q (kg): 2.14, 2.32, 1.99, 2.45, 2.27, 2.08

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

A café says that its takeaway coffees contain a mean of 250 ml. A customer measures a random sample of 12 coffees. The mean is 244.8 ml and an unbiased estimate of the population variance is 64.0 ml². He thinks the mean volume is less than 250 ml. Test his 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 mass, in kg, of 8 athletes is recorded at the start and at the end of a training season.

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

The resting heart rate of the members of a gym is normally distributed with mean 72 beats per minute (bpm) and standard deviation 6.4 bpm. A new fitness programme is claimed to lower the mean. A random sample of 25 members follows the programme. Assume the standard deviation is unchanged. Test the claim at the 1% 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 69.4 bpm. State the conclusion.

A machine fills bottles with a mean of 500 ml and standard deviation 4.2 ml. A supervisor takes a random sample of 36 bottles to test, at the 5% significance level, whether the mean volume has changed. Assume the volumes 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 498.3 ml. State the conclusion.

A psychologist tests whether listening to music reduces the reaction time of students. Nine students' reaction times, in milliseconds, are measured without and with music. 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 = with music − without music.
3. (c) Find the p-value.
4. (d) State the conclusion in context.

A teacher wants to compare the achievement of her class in two tests.

1. (a) She compares the marks of the same 15 students in an algebra test and a geometry test. Should she use a paired test or a two-sample test? Give a reason.
2. (b) She also compares the mean heights of 12 boys and 14 girls from the class. 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 coach splits a squad into two groups for sprint training. After a season, 9 swimmers in group 1 have a mean 50 m time of 52.6 s (sample variance 4.4) and 11 swimmers in group 2 have a mean time of 49.1 s (sample variance 3.9). The coach thinks the mean time of group 1 is more than 2 seconds greater than that of group 2. Assume the times are normally distributed with equal variance.

1. (a) Find unbiased estimates of the two population variances.
2. (b) Test the claim at the 5% 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.

Marks in a trial exam are normally distributed. For exam board X the standard deviation is known to be 9 and for exam board Y it is known to be 11. A random sample of 40 students sits the X paper and has a mean mark of 71.8. A different random sample of 50 students sits the Y paper and has a mean mark of 65.2. A tutor believes that the mean mark with board X is more than 4 marks greater than with board Y. Test her belief at the 5% 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 farmer thinks a new feed raises the daily milk yield of cows. The yield of one cow, in litres, is normally distributed with standard deviation 1.2, both before and after the change, and the yields are independent. The yields of 10 cows are measured before and after the change. Let D = yield after − yield before. Test at the 5% 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 0.9 litres. State the conclusion.

Customers' waiting times at a service desk are skewed, with mean 8.0 minutes and standard deviation 5.2 minutes. A new booking system is introduced, and the manager believes it reduces the mean waiting time. A random sample of 64 customers has a mean waiting time of 6.9 minutes. Assume the standard deviation is unchanged. Test at the 5% significance level.

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

A farm shop says its jars of jam contain a mean of 115 g. A customer thinks they contain more. The masses of a random sample of 9 jars are recorded. Assume the masses are normally distributed. Masses (g): 118, 124, 109, 131, 122, 127, 115, 129, 120

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 customer says: 'The p-value is small, so the jars are definitely heavier.' Comment on this statement.