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

The journey times on a route are normally distributed with \(\sigma=5.2\) minutes. A random sample of 28 journeys has mean 31.4 minutes. Find a 95% confidence interval for the mean journey time.

The masses (g) of seven loaves are 210, 224, 198, 231, 217, 205, 226. The masses are normal with known standard deviation 11 g. Find a 90% confidence interval for the mean mass.

The heights, in centimetres, of seven young plants are 132, 141, 127, 138, 145, 130, 136. They are normally distributed with unknown variance. Find a 95% confidence interval for the mean height.

Reaction times (in hundredths of a second) are normal. A random sample of 25 has mean 41.2 and unbiased estimate of the population variance 9.7. Find a 90% confidence interval for the mean.

A random sample of 14 cartons has mean volume 86.4 ml and standard deviation 5.3 ml, calculated by dividing by n. Assume the volumes are normal. Find \(s_{n-1}\), then a 95% confidence interval for the mean volume.

A 95% confidence interval for the mean time, in minutes, that a customer waits at a bank is \(180.6<\mu<197.4\). It was found from a random sample of 12 customers 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 bakery's loaves should have mass 2.00 kg. A random sample of 12 loaves has mean 2.04 kg and standard deviation 0.15 kg, calculated by dividing by n. Assume the masses are normal.

1. (a) Find \(s_{n-1}\).
2. (b) Find a 95% confidence interval for the mean mass.
3. (c) The baker will change the recipe if the mean mass is not 2.00 kg. Does the interval show that it needs changing?
4. (d) A customer says the loaves are heavier than 1.90 kg 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) Reaction times are normal and the standard deviation is not known. A sample of 12 times has \(s_{n-1}=0.06\) s.
2. (b) Masses of bags are normal with a known standard deviation of 4 g. A sample of 12 bags is taken.

The distances driven by delivery drivers in a day have standard deviation 3.9 km, but the distribution is skewed. A random sample of 110 drivers has mean 14.3 km.

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

The heights of seedlings are normal with \(\sigma=0.24\) m. A random sample of 20 seedlings has mean height 1.82 m.

1. (a) Find a 90% confidence interval.
2. (b) Find a 98% confidence interval.
3. (c) Which is wider? Explain why.

The masses, in grams, of 50 eggs are in the table. Assume the masses are normal.

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

Nine players' scores in a quiz were recorded before and after a practice session. Before: 14, 18, 11, 20, 16, 13, 17, 12, 15. After: 17, 19, 15, 22, 16, 18, 20, 13, 19. Assume the differences are normally distributed.

1. (a) Find the difference for each player (after − before).
2. (b) Find a 95% confidence interval for the mean difference.
3. (c) Does the interval suggest that practice improves the score? Explain.

The lengths of bolts are normally distributed with \(\sigma=0.6\) cm. A random sample of 20 bolts has mean length 12.4 cm.

1. (a) Find a 95% confidence interval for the mean length.
2. (b) A second random sample of 80 bolts also has mean length 12.4 cm. Find the 95% confidence interval from this sample.
3. (c) Compare the two widths and explain the change.

A population is normal with \(\sigma=2.8\). Find the smallest sample size for which the 99% z-interval for the mean has width at most 1.5.