z-intervals, t-intervals and choosing between them Problem set A

The lengths of a type of fish are normally distributed with \(\sigma=3.1\) cm. A random sample of 22 fish has mean length 16.8 cm. Find a 95% confidence interval for the mean length.

The times (seconds) of eight runners are 48, 52, 45, 57, 50, 54, 49, 53. The times are normal with known standard deviation 3.5 s. Find a 90% confidence interval for the mean time.

The heights, in metres, of eight trees are 7.3, 8.1, 6.8, 7.9, 8.4, 7.0, 7.6, 8.2. They are normally distributed with unknown variance. Find a 95% confidence interval for the mean height.

Test scores are normally distributed. A random sample of 20 scores has mean 71.4 and unbiased estimate of the population variance 22.3. Find a 90% confidence interval for the mean score.

A random sample of 18 jars has mean mass 212 g and standard deviation 14.8 g, calculated by dividing by n. Assume the masses are normal. Find \(s_{n-1}\), then a 95% confidence interval for the mean mass.

A 90% confidence interval for the mean daily number of visitors to a museum is \(23.4<\mu<28.2\), in hundreds. It was found from a random sample of 20 days from a normal population, using \(s_{n-1}\).

1. (a) Write down the sample mean.
2. (b) Write down the margin of error.
3. (c) Find \(s_{n-1}\).

A factory makes rods that should be 1.50 m long. A random sample of 16 rods has mean 1.52 m and standard deviation 0.09 m, calculated by dividing by n. Assume the lengths are normal.

1. (a) Find \(s_{n-1}\).
2. (b) Find a 95% confidence interval for the mean length.
3. (c) The factory will adjust the machine if the mean length is not 1.50 m. Does the interval show that it needs adjusting?
4. (d) An inspector says the rods are longer than 1.45 m on average. Does the interval support this?

State whether to use a z-interval or a t-interval for the mean in each case, and justify your choice.

1. (a) Heights of players are normal with a known standard deviation of 7 cm. A sample of 15 players is taken.
2. (b) Times are normal, but the standard deviation is not known. A sample of 15 times has \(s_{n-1}=4.2\) s.

The times spent on homework in a school have standard deviation 2.4 hours, but the distribution is skewed. A random sample of 90 students has mean 8.6 hours.

1. (a) Find a 95% confidence interval for the mean time.
2. (b) State, with a reason, whether the interval is valid when the population is not known to be normal.

The masses of sacks are normal with \(\sigma=6.5\) kg. A random sample of 16 sacks has mean mass 52.4 kg.

1. (a) Find a 80% confidence interval.
2. (b) Find a 99% confidence interval.
3. (c) Which is wider? Explain why.

The times, in minutes, that 60 commuters spend travelling are in the table. Assume the times are normal.

1. (a) Find an approximate 95% confidence interval for the mean.
2. (b) Explain why the interval is approximate.

Eight athletes' maximum lift (kg) was measured before and after a training programme. Before: 68, 72, 65, 80, 75, 70, 66, 78. After: 71, 70, 69, 84, 79, 76, 66, 83. Assume the differences are normally distributed.

1. (a) Find the difference for each athlete (after − before).
2. (b) Find a 90% confidence interval for the mean difference.
3. (c) Does the interval suggest that the programme increases the lift? Explain.

The scores on a reaction test are normally distributed with \(\sigma=15\). A random sample of 36 people has mean score 102.

1. (a) Find a 95% confidence interval for the mean score.
2. (b) A second random sample of 144 people also has mean score 102. Find the 95% confidence interval from this sample.
3. (c) Compare the two widths and explain the change.

A population is normal with \(\sigma=4.5\). Find the smallest sample size for which the 95% z-interval for the mean has width at most 2.