One-sample t-tests Problem set B One question at a time

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.

   • Let μ grams be the population mean mass of cereal in a box.
   • H0: μ = 450
   • H1: μ < 450 (the 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.

   • Let μ m/s be the population mean sprint speed.
   • H0: μ = 8.4
   • H1: μ > 8.4 (the mean has risen)

3. (c) A gardener wants to know whether the mean height of her plants has changed from 35 cm.

   • Let μ cm be the population mean height of a plant.
   • H0: μ = 35
   • H1: μ ≠ 35 (changed means up or down)

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.

• H0: μ = 125, H1: μ > 125
• From the calculator: t = 3.37, p = 0.00598
• 0.00598 < 0.05, so reject H0.

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.

• H0: μ = 34, H1: μ < 34
• From the calculator: t = −3.34, p = 0.00780
• 0.00780 < 0.1, so reject H0.

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.

• H0: μ = 78.8, H1: μ ≠ 78.8
• From the calculator: t = −2.20, p = 0.0638
• 0.0638 > 0.02, so do not reject H0.

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.

• H0: μ = 63, H1: μ ≠ 63
• s = √12.25 = 3.50
• From the calculator: t = −2.57, p = 0.0167
• 0.0167 < 0.05, so reject H0.

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.

   • Let μ be the population mean honey yield per hive, in kilograms.
   • H0: μ = 6
   • H1: μ > 6

2. (b) Find the t-statistic and the p-value.

   • From the calculator: t = 1.84, p = 0.0519

3. (c) State the conclusion of the test, in context.

   • 0.0519 < 0.1, so reject H0.
   • There is sufficient evidence, at the 10% significance level, that the mean honey yield per hive has increased.

4. (d) State an assumption needed for the test.

   • The honey yield per hive is normally distributed.

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.

   • H0: μ = 8, H1: μ < 8
   • From the calculator, with s = 0.6: t = −3.27, p = 0.00170

2. (b) State the conclusion in context.

   • 0.00170 < 0.01, so reject H0.
   • There is sufficient evidence, at the 1% significance level, that the mean journey time is lower than 8.0 minutes.

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.

   • 0.0240 < 0.05, so reject H0.
   • There is sufficient evidence, at the 5% significance level, that the mean time spent on homework has decreased.

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

   • 0.0240 > 0.01, so do not reject H0.
   • There is insufficient evidence, at the 1% significance level, that the mean time spent on homework has decreased.

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.

   • H0: μ = 250, H1: μ ≠ 250
   • From the calculator: t = −1.34, p = 0.214
   • 0.214 > 0.05, so do not reject H0.
   • There is insufficient evidence, at the 5% significance level, that the mean mass is not 250 g.

2. (b) State the assumption about the population that this test needs.

   • The masses of packets are normally distributed.

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.

   • Let μ be the population mean journey time, in minutes.
   • H0: μ = 4, H1: μ > 4

2. (b) Find the t-statistic and the p-value.

   • s = √0.81 = 0.9
   • From the calculator: t = 2.12, p = 0.0244

3. (c) State the conclusion in context.

   • 0.0244 < 0.05, so reject H0.
   • There is sufficient evidence, at the 5% significance level, that the mean journey time has increased.