One-sample t-tests Problem set A 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 manufacturer says its bulbs last a mean of 1200 hours. A customer thinks they last less.

   • Let μ hours be the population mean life of a bulb.
   • H0: μ = 1200
   • H1: μ < 1200 (the customer thinks the mean is less)

2. (b) A teacher thinks a new method has raised the mean test mark, which was 62.

   • Let μ be the population mean test mark.
   • H0: μ = 62
   • H1: μ > 62 (the mean has risen)

3. (c) A farmer wants to know if the mean milk yield per cow has changed from 24.0 litres a day.

   • Let μ litres be the population mean daily milk yield per cow.
   • H0: μ = 24.0
   • H1: μ ≠ 24.0 (changed means up or down)

A café sold a mean of 50 cakes a day. After a promotion, the cakes sold on 7 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 = 50, H1: μ > 50
• From the calculator: t = 2.22, p = 0.0340
• 0.0340 < 0.05, so reject H0.

A game's mean completion time was 91 seconds. After an update, a player completes it 6 times, with the times shown. Test at the 10% significance level whether the mean time has decreased: state H0 and H1, then find t and p and say whether H0 is rejected.

• H0: μ = 91, H1: μ < 91
• From the calculator: t = −2.71, p = 0.0211
• 0.0211 < 0.1, so reject H0.

A scientist's standard sample of water has a mean mass of 12.0 g. A new method gives the 8 masses shown. Test at the 2% significance level whether the mean mass is different from 12.0 g: state H0 and H1, then find t and p and say whether H0 is rejected.

• H0: μ = 12, H1: μ ≠ 12
• From the calculator: t = 2.17, p = 0.0661
• 0.0661 > 0.02, so do not reject H0.

The time to assemble a chair has a mean of 25.0 minutes. After new training, 20 randomly chosen assemblies have a mean of 23.4 minutes, and the estimate of the population variance is 9.0. Test, at the 5% significance level, whether the mean time has changed. Give t and p, and say whether H0 is rejected.

• H0: μ = 25, H1: μ ≠ 25
• s = √9 = 3.00
• From the calculator: t = −2.39, p = 0.0276
• 0.0276 < 0.05, so reject H0.

A household used a mean of 200 litres of water a week in past years. After a hot summer it thinks the mean weekly use has increased. The use in 8 randomly chosen weeks, in litres, is shown. The test is at the 10% significance level.

1. (a) State the null and alternative hypotheses.

   • Let μ be the population mean weekly water use, in litres.
   • H0: μ = 200
   • H1: μ > 200

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

   • From the calculator: t = 3.89, p = 0.00299

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

   • 0.00299 < 0.1, so reject H0.
   • There is sufficient evidence, at the 10% significance level, that the mean weekly water use has increased.

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

   • The weekly water use is normally distributed.

The battery life of a phone model is normally distributed with a mean of 50 hours. A random sample of 30 phones of a new version has a mean life of 48.9 hours and the estimate of the population standard deviation is 1.8 hours. Test, at the 1% significance level, whether the new version has a lower mean life.

1. (a) State the hypotheses, then find the t-statistic and the p-value.

   • H0: μ = 50, H1: μ < 50
   • From the calculator, with s = 1.8: t = −3.35, p = 0.00114

2. (b) State the conclusion in context.

   • 0.00114 < 0.01, so reject H0.
   • There is sufficient evidence, at the 1% significance level, that the mean battery life of the new version is lower than 50 hours.

A test of whether the mean height of a plant has increased gives a p-value of 0.036.

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

   • 0.0360 < 0.05, so reject H0.
   • There is sufficient evidence, at the 5% significance level, that the mean height of the plants has increased.

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

   • 0.0360 > 0.01, so do not reject H0.
   • There is insufficient evidence, at the 1% significance level, that the mean height of the plants has increased.

A machine fills bottles with a stated 500 mL. The volumes, in mL, of 10 randomly chosen bottles are shown. A technician thinks the mean volume is not 500 mL.

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 = 500, H1: μ ≠ 500
   • From the calculator: t = −1.19, p = 0.264
   • 0.264 > 0.05, so do not reject H0.
   • There is insufficient evidence, at the 5% significance level, that the mean volume is not 500 mL.

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

   • The volumes of bottles are normally distributed.

The marks in a school's mathematics test are normally distributed with a mean of 100. After a revision course, a random sample of 15 students has a mean mark of 104.2 and the estimate of the population variance is 36. Test, at the 5% significance level, whether the course has increased the mean mark.

1. (a) State the hypotheses.

   • Let μ be the population mean mark.
   • H0: μ = 100, H1: μ > 100

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

   • s = √36 = 6
   • From the calculator: t = 2.71, p = 0.00844

3. (c) State the conclusion in context.

   • 0.00844 < 0.05, so reject H0.
   • There is sufficient evidence, at the 5% significance level, that the mean mark has increased.