One-sample t-tests Problem set B

State the null and alternative hypotheses for a one-sample t-test in each situation. Define μ.

1. (a) A cereal box says it holds a mean of 450 g. An inspector thinks the mean is less.
2. (b) A coach thinks a new training plan has increased the mean sprint speed, which was 8.4 m/s.
3. (c) A gardener wants to know whether the mean height of her plants has changed from 35 cm.

A swimming pool had a mean of 125 visitors a day. After a new slide is built, the visitors on 8 randomly chosen days are shown. Test at the 5% significance level whether the mean has increased: state H0 and H1, then find t and p and say whether H0 is rejected.

A cyclist's mean time on a hill climb was 34 minutes. After a new bike, she records the 7 times shown. Test at the 10% significance level whether her mean time has decreased: state H0 and H1, then find t and p and say whether H0 is rejected.

A pharmacy's standard tablet has a mean mass of 78.8 mg. A new batch gives the 8 masses shown. Test at the 2% significance level whether the mean mass is different from 78.8 mg: state H0 and H1, then find t and p and say whether H0 is rejected.

The time to complete a puzzle has a mean of 63.0 seconds. After a hint is added, 25 randomly chosen attempts have a mean of 61.2 seconds, and the estimate of the population variance is 12.25. Test, at the 5% significance level, whether the mean time has changed. Give t and p, and say whether H0 is rejected.

A beekeeper's hives used to give a mean of 6.0 kg of honey. He thinks a new feeding plan has increased the mean. The honey from 9 randomly chosen hives, in kilograms, is shown. The test is at the 10% significance level.

1. (a) State the null and alternative hypotheses.
2. (b) Find the t-statistic and the p-value.
3. (c) State the conclusion of the test, in context.
4. (d) State an assumption needed for the test.

The mean time for a tram journey is normally distributed with a mean of 8.0 minutes. After a new timetable, a random sample of 24 journeys has a mean time of 7.6 minutes and the estimate of the population standard deviation is 0.6 minutes. Test, at the 1% significance level, whether the mean time is lower.

1. (a) State the hypotheses, then find the t-statistic and the p-value.
2. (b) State the conclusion in context.

A test of whether the mean time spent on homework has decreased gives a p-value of 0.024.

1. (a) State the conclusion at the 5% significance level, in context.
2. (b) State the conclusion at the 1% significance level, in context.

A packet of rice is labelled 250 g. The masses, in grams, of 10 randomly chosen packets are shown. A trading standards officer thinks the mean mass is not 250 g.

1. (a) Carry out the test at the 5% significance level. Give the t-statistic, the p-value and the conclusion in context.
2. (b) State the assumption about the population that this test needs.

The time a bus takes between two stops is normally distributed. Last year the mean was 4.0 minutes. In a random sample of 18 journeys this year the mean is 4.45 minutes and the estimate of the population variance is 0.81. Test, at the 5% significance level, whether the mean time has increased.

1. (a) State the hypotheses.
2. (b) Find the t-statistic and the p-value.
3. (c) State the conclusion in context.