Testing a mean: z-test or t-test Problem set A

For each situation, write down the null and alternative hypotheses, defining \(\mu\), and say whether a z-test or a t-test is needed. The population is normally distributed in each case.

1. (a) A fruit farm says its oranges have a mean mass of 180 g. Records over many seasons give a population standard deviation of 14 g. A sample of 30 oranges has a mean mass of 175.2 g. Has the mean mass decreased?
2. (b) A café says a medium coffee has a mean volume of 250 ml. A sample of 12 coffees has a mean of 247 ml and a sample standard deviation of 5 ml. Is the mean volume different from 250 ml?
3. (c) A train company says journeys take a mean of 42 minutes. The population variance is known to be 9 min². A sample of 20 journeys has a mean of 43.6 minutes. Is the mean journey time more than 42 minutes?

A juice plant fills bottles with a mean volume of 330 ml. A manager thinks the machine has drifted and measures a sample. The population standard deviation is 4.2 ml. A random sample of 18 bottles has a mean of 331.9 ml. Assume the population is normally distributed. Test, at the 5% significance level, whether the mean volume has changed. Give the p-value and your conclusion.

A taxi firm says the mean length of a journey to the airport is 25 minutes. A rival thinks it takes longer. The sample standard deviation is 4.6 minutes. A random sample of 16 journeys has a mean of 27.5 minutes. Assume the population is normally distributed. Test, at the 5% significance level, whether the mean journey time is more than 25 minutes. Give the p-value and your conclusion.

A runner says her mean time for a 5 km run is 24.5 minutes. After a month of training she records 9 times, in minutes. The data are 24.6, 23.1, 25.4, 22.8, 24.0, 23.5, 22.9, 24.3, 23.7 Assume the population is normally distributed. Test, at the 5% significance level, whether her mean time has fallen below 24.5 minutes. Give the p-value and your conclusion.

A packing line fills bags of rice that should have a mean mass of 500 g. A sample of 7 bags is weighed, in grams. The population standard deviation is 6 g. The data are 498, 507, 501, 512, 495, 509, 503 Assume the population is normally distributed. Test, at the 5% significance level, whether the mean mass of a bag is different from 500 g. Give the p-value and your conclusion.

A teacher says the mean time students spend on homework each week is 8.0 hours. The sample variance is 2.25 hours². A random sample of 20 students has a mean of 8.8 hours. Assume the population is normally distributed. Test, at the 5% significance level, whether the mean weekly homework time is different from 8.0 hours. Give the p-value and your conclusion.

A factory makes rods with a mean length of 70 cm. The population variance is 0.81 cm². A random sample of 12 rods has a mean of 70.6 cm. Assume the population is normally distributed. Test, at the 5% significance level, whether the mean length of a rod is more than 70 cm. Give the p-value and your conclusion.

A bakery's bread rolls have a mean mass of 240 g. From years of records the population standard deviation is 5.4 g. A sample of 25 rolls has a mean of 242.4 g and a sample standard deviation of 6.1 g. The population is normally distributed. Test, at the 5% significance level, whether the mean mass of a roll has changed.

1. (a) State the hypotheses and say which test you will use.
2. (b) Find the p-value.
3. (c) Write your conclusion at the 5% significance level, and at the 1% significance level.

A fish farm says the mean length of its trout is 18.5 cm. The farm changes its feed and measures 10 trout, in centimetres: 18.4, 19.2, 17.9, 20.1, 18.8, 19.6, 20.4, 18.1, 19.0, 19.9 Assume the population is normally distributed.

1. (a) Find the sample mean and the sample standard deviation.
2. (b) Test, at the 5% significance level, whether the new feed has increased the mean length.
3. (c) Say why a t-test and not a z-test is used here.