Paired t-test

1. A random sample of 8 people were given a new drug designed to help people sleep.

In a two-week period the drug was given for one week and a placebo (a tablet that contained no drug) was given for one week. In the first week 4 people, selected at random, were given the drug and the other 4 people were given the placebo. Those who were given the drug in the first week were given the placebo in the second week. Those who were given the placebo in the first week were given the drug in the second week. The mean numbers of hours of sleep per night for each of the people are shown in the table.

1. State one assumption that needs to be made in order to carry out a paired \(t\)-test.
2. Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not the drug increases the mean number of hours of sleep per night by more than 10 minutes. State the critical value for this test.

1. A random sample of 8 students is selected from a school database.

Each student's reaction time is measured at the start of the school day and again at the end of the school day. The reaction times, in milliseconds, are recorded below.

1. State one assumption that needs to be made in order to carry out a paired \(t\)-test.
   (1) The random variable \(R\) is the reaction time at the start of the school day minus the reaction time at the end of the school day. The mean of \(R\) is \(\mu\). John uses a paired \(t\)-test to test the hypotheses $$\mathrm { H } _ { 0 } : \mu = m \quad \mathrm { H } _ { 1 } : \mu \neq m$$ Given that \(\mathrm { H } _ { 0 }\) is rejected at the 5\% level of significance but accepted at the 1\% level of significance,
2. find the ranges of possible values for \(m\).

2. An engineer decided to investigate whether or not the strength of rope was affected by water. A random sample of 9 pieces of rope was taken and each piece was cut in half. One half of each piece was soaked in water over night, and then each piece of rope was tested to find its strength. The results, in coded units, are given in the table below Assuming that the strength of rope follows a normal distribution, test whether or not there is any difference between the mean strengths of dry and wet rope. State your hypotheses clearly and use a \(1 \%\) level of significance.
(8 marks)

1. A doctor believes that a four-week exercise programme can reduce the resting heart rate of her patients. She takes a random sample of 7 patients and records their resting heart rate before the exercise programme and again after the exercise programme.

1. Using a \(5 \%\) level of significance, carry out an appropriate test of the doctor's belief. You should state your hypotheses, test statistic and critical value.
2. State the assumption made about the resting heart rates that was required to carry out the test.

1. A psychologist claims to have developed a technique to improve a person's memory.

A random sample of 8 people are each given the same list of words to memorise and recall. Each person then receives memory training from the psychologist. After the training, each person is given the same list of new words to memorise and recall. The table shows the percentage of words recalled by each person before and after the training.

1. State why a paired \(t\)-test is suitable for these data.
2. State an assumption that needs to be made in order to carry out a paired \(t\)-test in this case.
3. Test, at the \(5 \%\) level of significance, whether or not there is evidence of an increase in the percentage of words recalled after receiving the psychologist's training. State your hypotheses, test statistic and critical value used for this test.

1. A researcher set up a trial to assess the effect that a food supplement has on the increase in weight of Herdwick lambs. The researcher randomly selected 8 sets of twin lambs. One of each set of twins was given the food supplement and the other had no food supplement. The gain in weight, in kg, of each lamb over the period of the trial was recorded.

1. State why a two sample \(t\)-test is not suitable for use with these data.
2. Suggest 2 other factors about the lambs that the researcher may need to control when selecting the sample.
3. State one assumption, in context, that needs to be made for a paired \(t\)-test to be valid. For a pair of twin lambs, the random variable \(W\) represents the weight gain of the lamb given the food supplement minus the weight gain of the lamb not given the food supplement.
4. Using the data in the table, calculate a \(98 \%\) confidence interval for the mean of \(W\) Show your working clearly. The researcher believes that the mean of \(W\) is greater than 200 g
5. Stating your hypotheses clearly, use your confidence interval to explain whether or not there is evidence to support the researcher's belief.

1. The weights, in grams, of mice are normally distributed. A biologist takes a random sample of 10 mice. She weighs each mouse and records its weight.

The ten mice are then fed on a special diet. They are weighed again after two weeks.
Their weights in grams are as follows: Stating your hypotheses clearly, and using a \(1 \%\) level of significance, test whether or not the diet causes an increase in the mean weight of the mice.

1. The random variable \(X\) has an \(F\) distribution with 10 and 12 degrees of freedom. Find \(a\) and \(b\) such that \(\mathrm { P } ( a < X < b ) = 0.90\).
   (3)
2. A chemist has developed a fuel additive and claims that it reduces the fuel consumption of cars. To test this claim, 8 randomly selected cars were each filled with 20 litres of fuel and driven around a race circuit. Each car was tested twice, once with the additive and once without. The distances, in miles, that each car travelled before running out of fuel are given in the table below.

Assuming that the distances travelled follow a normal distribution and stating your hypotheses clearly test, at the \(10 \%\) level of significance, whether or not there is evidence to support the chemist's claim.
(8)