Paired t-test

1 A manager is investigating the times taken by employees to complete a particular task as a result of the introduction of new technology. He claims that the mean time taken to complete the task is reduced by more than 0.4 minutes. He chooses a random sample of 10 employees. The times taken, in minutes, before and after the introduction of the new technology are recorded in the table.

1. Test at the 10\% significance level whether the manager's claim is justified.
2. State an assumption that is necessary for this test to be valid.

6 Jade is a swimming instructor at a sports college. She claims that, as a result of an intensive training course, the mean time taken by students to swim 50 metres has reduced by more than 1 second. She chooses a random sample of 10 students. The times taken, in seconds, before and after the training course are recorded in the table.

1. Test, at the 10\% significance level, whether Jade's claim is justified.
2. State an assumption that is necessary for this test to be valid.

4 Members of the Sprints athletics club have been taking part in an intense training scheme, aimed at reducing their times taken to run 400 m . For a random sample of 9 athletes from the club, the times taken, in seconds, before and after the training scheme are given in the following table. The organiser of the training scheme claims that on average an athlete's time will be reduced by at least 0.3 seconds. Test at the 10\% significance level whether the organiser's claim is justified, stating any assumption that you make.

4 Manet has developed a new training course to help athletes improve their time taken to run 800 m . Manet claims that his course will decrease an athlete's time by more than 2 s on average. For a random sample of 10 athletes the times taken, in seconds, before and after the course are given in the following table. Use a \(t\)-test, at the \(5 \%\) significance level, to test whether Manet's claim is justified, stating any assumption that you make.

3 Scientists are studying the effects of exercise on LDL blood cholesterol levels. Over a three-month period, a large group of people exercised for 20 minutes each day. For a randomly chosen sample of 10 of these people, the LDL blood cholesterol levels were measured at the beginning and the end of the three-month period. The results, measured in suitable units, are as follows.

1. Test, at the \(2.5 \%\) significance level, whether there is evidence that the population mean LDL blood cholesterol level has reduced by more than 2 units after the three-month period.
2. State any assumption that you have made in part (a). \includegraphics[max width=\textwidth, alt={}, center]{44829994-2ef0-488d-aa3b-99fb0e36d733-06_399_1383_269_324} As shown in the diagram, the continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} m x & 0 \leqslant x \leqslant 2 \\ \frac { k } { x ^ { 2 } } + c & 2 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(m , k\) and \(c\) are constants.

3 Scientists are studying the effects of exercise on LDL blood cholesterol levels. Over a three-month period, a large group of people exercised for 20 minutes each day. For a randomly chosen sample of 10 of these people, the LDL blood cholesterol levels were measured at the beginning and the end of the three-month period. The results, measured in suitable units, are as follows.

1. Test, at the \(2.5 \%\) significance level, whether there is evidence that the population mean LDL blood cholesterol level has reduced by more than 2 units after the three-month period.
2. State any assumption that you have made in part (a). \includegraphics[max width=\textwidth, alt={}, center]{b6635fbc-3c9d-4f93-b51a-b1cbd71ddbb1-06_399_1383_269_324} As shown in the diagram, the continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} m x & 0 \leqslant x \leqslant 2 \\ \frac { k } { x ^ { 2 } } + c & 2 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(m , k\) and \(c\) are constants.

5 Of two brands of lawnmower, \(A\) and \(B\), brand \(A\) was claimed to take less time, on average, than brand \(B\) to mow similar stretches of lawn. In order to test this claim, 9 randomly selected gardeners were each given the task of mowing two regions of lawn, one with each brand of mower. All the regions had the same size and shape and had grass of the same height. The times taken, in seconds, are given in the table.

1. Test the claim using a paired-sample \(t\)-test at the \(5 \%\) significance level. State a distributional assumption required for the test to be valid.
2. Give a reason why a paired-sample \(t\)-test should be used, rather than a 2 -sample \(t\)-test, in this case.

3 Ten randomly chosen athletes were coached for a 200 m event. For each athlete, the times taken to run 200 m before and after coaching were measured. The sample mean times before and after coaching were 23.43 seconds and 22.84 seconds respectively. For each athlete the difference, \(d\) seconds, in the times before and after coaching was calculated and an unbiased estimate of the population variance of \(d\) was found to be 0.548 . Stating any required assumption, test at the \(5 \%\) significance level whether the population mean time for the 200 m run decreased after coaching.

3 A nurse was asked to measure the blood pressure of 12 patients using an aneroid device. The nurse's readings were immediately checked using an accurate electronic device. The differences, \(x\), given by \(x =\) (aneroid reading - electronic reading), in appropriate units, are shown below. $$\begin{array} { c c c c c c c c c c c } - 1.3 & 4.7 & - 0.9 & 3.8 & - 1.5 & 4.0 & - 1.9 & 4.4 & - 0.8 & 5.5 & - 2.9 \end{array} 4.1$$ Stating any assumption you need to make, test, at the \(10 \%\) significance level, whether readings with an aneroid device, on average, overestimate patients’ blood pressure.

7 A tutor gives a randomly selected group of 8 students an English Literature test, and after a term's further teaching, she gives the group a similar test. The marks for the two tests are given in the table.

1. Stating a necessary condition, show by carrying out a suitable \(t\)-test, at the \(1 \%\) significance level, that the marks do not give evidence of an improvement.
2. The tutor later found that she had marked the second test too severely, and she decided to add a constant amount \(k\) to each mark. Find the least integer value of \(k\) for which the increased marks would give evidence of improvement at the \(1 \%\) significance level.

4 A group of students were tested in geography before and after a fieldwork course. The marks of 10 randomly selected students are shown in the table.

1. Use a suitable \(t\)-test, at the \(5 \%\) level of significance, to test whether the students' performance has improved.
2. State the necessary assumption in applying the test.

3 Pathology departments in hospitals routinely analyse blood specimens. Ideally the analysis should be done while the specimens are fresh to avoid any deterioration, but this is not always possible. A researcher decides to study the effect of freezing specimens for later analysis by measuring the concentrations of a particular hormone before and after freezing. He collects and divides a sample of 15 specimens. One half of each specimen is analysed immediately, the other half is frozen and analysed a month later. The concentrations of the particular hormone (in suitable units) are as follows. A \(t\) test is to be used in order to see if, on average, there is a reduction in hormone concentration as a result of being frozen.

1. Explain why a paired test is appropriate in this situation.
2. State the hypotheses that should be used, together with any necessary assumptions.
3. Carry out the test using a \(1 \%\) significance level.
4. A \(p \%\) confidence interval for the true mean reduction in hormone concentration is found to be ( \(0.4869,2.8131\) ). Determine the value of \(p\).

2 A company supplying cattle feed to dairy farmers claims that its new brand of feed will increase average milk yields by 10 litres per cow per week. A farmer thinks the increase will be less than this and decides to carry out a statistical investigation using a paired \(t\) test. A random sample of 10 dairy cows are given the new feed and then their milk yields are compared with their yields when on the old feed. The yields, in litres per week, for the 10 cows are as follows.

1. Why is it sensible to use a paired test?
2. State the condition necessary for a paired \(t\) test.
3. Assuming the condition stated in part (ii) is met, carry out the test, using a significance level of \(5 \%\), to see whether it appears that the company's claim is justified.
4. Find a 95\% confidence interval for the mean increase in the milk yield using the new feed.

4 An insurance company is investigating a new system designed to reduce the average time taken to process claim forms. The company has decided to use 10 experienced employees to process claims using the old system and the new system. Two procedures for comparing the systems are proposed.
Procedure \(A\) There are two sets of claim forms, set 1 and set 2. Each contains the same number of forms. Each employee processes set 1 on the old system and set 2 on the new system. The times taken are compared. Procedure \(B\) There is just one set of claim forms which each employee processes firstly on the old system and then on the new system. The times taken are compared.

1. State one weakness of each of these procedures. In fact a third procedure which avoids these two weaknesses is adopted. In this procedure each employee is given a randomly selected set of claim forms. Each set contains the same number of forms. The employees each process their set of claim forms on both systems. The times taken, in minutes, are shown in the table.

   Employee	1	2	3	4	5	6	7	8	9	10
   Old system	40.5	42.9	52.8	51.7	77.2	66.7	65.2	49.2	55.6	58.3
   New system	39.2	40.7	50.6	50.7	71.4	70.5	71.1	47.7	52.1	55.5

2. Carry out a paired \(t\) test at the \(5 \%\) level of significance to investigate whether the mean length of time taken to process a set of forms has reduced using the new system.
3. State fully the usual conditions for a paired \(t\) test.
4. Construct a \(99 \%\) confidence interval for the mean reduction in time taken to process a set of forms using the new system.

8 Part of a research study of identical twins who had been separated at birth involved a random sample of 9 pairs, in which one twin had been raised by the natural parents and the other by adoptive parents. The IQ scores of these twins were measured, with the following results. It may be assumed that the difference in IQ scores has a normal distribution. The mean IQ scores of separated twins raised by natural parents and by adoptive parents are denoted by \(\mu _ { N }\) and \(\mu _ { A }\) respectively. Obtain a \(90 \%\) confidence interval for \(\mu _ { N } - \mu _ { A }\). One of the researchers claimed that there was no evidence of a difference between the two population means. State, giving a reason, whether the confidence interval supports this claim.

8 A company decides that its employees should follow an exercise programme for 30 minutes each day, with the aim that they lose weight and increase productivity. The weights, in kg , of a random sample of 8 employees at the start of the programme and after following the programme for 6 weeks are shown in the table. Assuming that loss in weight is normally distributed, find a 95\% confidence interval for the mean loss in weight of the company's employees. Test at the \(5 \%\) significance level whether, after the exercise programme, there is a reduction of more than 2.5 kg in the population mean weight.

7 Each of a random sample of 6 cyclists from a cycling club is timed over two different 10 km courses. Their times, in minutes, are recorded in the following table. Assuming that differences in time over the two courses are normally distributed, test at the \(10 \%\) significance level whether the mean times over the two courses are different.

10 During the summer months, all members of a large swimming club take part in intensive training. The times taken to swim 50 metres at the beginning of the summer and at the end of the summer are recorded for each member of the club. The time taken, in seconds, at the beginning of the summer is denoted by \(x\) and the time taken at the end of the summer is denoted by \(y\). For a random sample of 9 members the results are shown in the following table. The swimming coach believes that, on average, the time taken by a swimmer to swim 50 metres will decrease by more than one second as a result of the intensive training.

1. Stating suitable hypotheses and assuming a normal distribution, test the coach's belief at the \(10 \%\) significance level.
2. Find a 95\% confidence interval for the population mean time taken to swim 50 metres after the intensive training, assuming a normal distribution.

8 A large number of runners are attending a summer training camp. A random sample of 6 runners is chosen and their times to run 1500 m at the beginning of the camp and at the end of the camp are recorded. Their times, in minutes, are shown in the following table. The organiser of the training camp claims that a runner's time will improve by more than 0.05 minutes between the beginning and end of the camp. Assuming that differences in time over the two runs are normally distributed, test at the \(10 \%\) significance level whether the organiser's claim is justified. [8]

1. As part of an investigation, a random sample of 10 people had their heart rate, in beats per minute, measured whilst standing up and whilst lying down. The results are summarised below.

1. State one assumption that needs to be made in order to carry out a paired \(t\)-test.
2. Test, at the \(5 \%\) level of significance, whether or not there is any evidence that standing up increases people's mean heart rate by more than 5 beats per minute. State your hypotheses clearly.

1. Students studying for their Mathematics GCSE are assessed by two examination papers. A teacher believes that on average the score on paper I is more than 1 mark higher than the score on paper II. To test this belief the scores of 8 randomly selected students are recorded. The results are given in the table below.

Assuming that the scores are normally distributed and stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence to support the teacher's belief.

1. The Sales Manager of a large chain of convenience stores is studying the sale of lottery tickets in her stores. She randomly selects 8 of her stores. From these stores she collects data for the total sales of lottery tickets in the previous January and July. The data are shown below

1. Use a paired \(t\)-test to determine whether or not there is evidence, at the \(5 \%\) level of significance, that the mean sales of lottery tickets in this chain's stores are higher in July than in January. You should state your hypotheses and show your working clearly.
2. State what assumption the Sales Manager needs to make about the sales of lottery tickets in her stores for the test in part (a) to be valid.

1. A new diet has been designed. Its designers claim that following the diet for a month will result in a mean weight loss of more than 2 kg . In a trial, a random sample of 10 people followed the new diet for a month. Their weights, in kg, before starting the diet and their weights after following the diet for a month were recorded. The results are given in the table below.

1. Using a suitable \(t\)-test, at the \(5 \%\) level of significance, state whether or not the trial supports the designers’ claim. State your hypotheses and show your working clearly.
2. State an assumption necessary for the test in part (a).

4. A coach believes that the average score in the final round of a golf tournament is more than one point below the average score in the first round. To test this belief, the scores of 8 randomly selected players are recorded. The results are given in the table below.

1. 1. State why a paired \(t\)-test is suitable for use with these data.
   2. State an assumption that needs to be made in order to carry out a paired \(t\)-test in this case.
2. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the coach's belief. Show your working clearly.
3. Explain, in the context of the coach's belief, what a Type II error would be in this case.

7. Two methods of extracting juice from an orange are to be compared. Eight oranges are halved. One half of each orange is chosen at random and allocated to Method \(A\) and the other half is allocated to Method \(B\). The amounts of juice extracted, in ml , are given in the table. The lengths of components produced by the machines can be assumed to follow normal distributions.

1. Use a two tail test to show, at the \(10 \%\) significance level, that the variances of the lengths of components produced by each machine can be assumed to be equal.
2. Showing your working clearly, find a \(95 \%\) confidence interval for \(\mu _ { B } - \mu _ { A }\), where \(\mu _ { A }\) and \(\mu _ { B }\) are the mean lengths of the populations of components produced by machine \(A\) and machine \(B\) respectively. There are serious consequences for the production at the factory if the difference in mean lengths of the components produced by the two machines is more than 0.7 cm .
3. State, giving your reason, whether or not the factory manager should be concerned.
   5. Rolls of cloth delivered to a factory contain defects at an average rate of \(\lambda\) per metre. A quality assurance manager selects a random sample of 15 metres of cloth from each delivery to test whether or not there is evidence that \(\lambda > 0.3\). The criterion that the manager uses for rejecting the hypothesis that \(\lambda = 0.3\) is that there are 9 or more defects in the sample.
4. Find the size of the test. Table 1 gives some values, to 2 decimal places, of the power function of this test. \begin{table}[h]
5. Use a paired \(t\)-test to determine, at the \(10 \%\) level of significance, whether or not there is a difference in the mean blood pressure measured using the two methods. State your hypotheses clearly.
6. State an assumption about the underlying distribution of measured blood pressure required for this test.
   2. The value of orders, in \(\pounds\), made to a firm over the internet has distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). A random sample of \(n\) orders is taken and \(\bar { X }\) denotes the sample mean.
7. Write down the mean and variance of \(\bar { X }\) in terms of \(\mu\) and \(\sigma ^ { 2 }\). A second sample of \(m\) orders is taken and \(\bar { Y }\) denotes the mean of this sample.
   An estimator of the population mean is given by $$U = \frac { n \bar { X } + m \bar { Y } } { n + m }$$
8. Show that \(U\) is an unbiased estimator for \(\mu\).
9. Show that the variance of \(U\) is \(\frac { \sigma ^ { 2 } } { n + m }\).
10. State which of \(\bar { X }\) or \(U\) is a better estimator for \(\mu\). Give a reason for your answer.
    3. The lengths, \(x \mathrm {~mm}\), of the forewings of a random sample of male and female adult butterflies are measured. The following statistics are obtained from the data.
11. Stating your hypotheses clearly, and using a \(10 \%\) level of significance, test whether or not there is evidence of a difference between the variances of the marks of the two groups.
12. State clearly an assumption you have made to enable you to carry out the test in part (a).
13. Use a two tailed test, with a \(5 \%\) level of significance, to determine if the playing of music during the test has made any difference in the mean marks of the two groups. State your hypotheses clearly.
14. Write down what you can conclude about the effect of music on a student's performance during the test.
    3. 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:
15. State an assumption that needs to be made in order to carry out a \(t\)-test in this case.
16. State why a paired \(t\)-test is suitable for use with these data.
17. Using a \(5 \%\) level of significance, test whether or not there is evidence that the device reduces \(\mathrm { CO } _ { 2 }\) emissions from cars.
18. Explain, in context, what a type II error would be in this case.
    3. Define, in terms of \(\mathrm { H } _ { 0 }\) and/or \(\mathrm { H } _ { 1 }\),
19. the size of a hypothesis test,
20. the power of a hypothesis test. The probability of getting a head when a coin is tossed is denoted by \(p\). This coin is tossed 12 times in order to test the hypotheses \(\mathrm { H } _ { 0 } : p = 0.5\) against \(\mathrm { H } _ { 1 } : p \neq 0.5\), using a \(5 \%\) level of significance.
21. Find the largest critical region for this test, such that the probability in each tail is less than \(2.5 \%\).
22. Given that \(p = 0.4\)
    1. find the probability of a type II error when using this test,
    2. find the power of this test.
23. Suggest two ways in which the power of the test can be increased.
    4. A farmer set up a trial to assess whether adding water to dry feed increases the milk yield of his cows. He randomly selected 22 cows. Thirteen of the cows were given dry feed and the other 9 cows were given the feed with water added. The milk yields, in litres per day, were recorded with the following results. You may assume that the times taken to complete the task by the students are two independent random samples from normal distributions.
24. Stating your hypotheses clearly, test, at the \(10 \%\) level of significance, whether or not the variances of the times taken to complete the task with and without background music are equal.
25. Find a \(99 \%\) confidence interval for the difference in the mean times taken to complete the task with and without background music. Experiments like this are often performed using the same people in each group.
26. Explain why this would not be appropriate in this case.
    2. As part of an investigation, a random sample of 10 people had their heart rate, in beats per minute, measured whilst standing up and whilst lying down. The results are summarized below. Stating your hypotheses clearly, test, at the \(10 \%\) level of significance, whether or not the mean amount of juice produced by machine \(B\) is more than the mean amount produced by machine \(A\).
    4. A proportion \(p\) of letters sent by a company are incorrectly addressed and if \(p\) is thought to be greater than 0.05 then action is taken. Using \(\mathrm { H } _ { 0 } : p = 0.05\) and \(\mathrm { H } _ { 1 } : p > 0.05\), a manager from the company takes a random sample of 40 letters and rejects \(\mathrm { H } _ { 0 }\) if the number of incorrectly addressed letters is more than 3 .
27. Find the size of this test.
28. Find the probability of a Type II error in the case where \(p\) is in fact 0.10 . Table 1 below gives some values, to 2 decimal places, of the power function of this test. The student decides to carry out a paired \(t\)-test to investigate whether, on average, the blood pressure of a person when sitting down is more than their blood pressure after standing up.
29. State clearly the hypotheses that should be used and any necessary assumption that needs to be made.
30. Carry out the test at the \(1 \%\) level of significance.
    2. A biologist investigating the shell size of turtles takes random samples of adult female and adult make turtles and records the length, \(x \mathrm {~cm}\), of the shell. The results are summarised below. Assuming that the scores are normally distributed and stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence to support the teacher's belief.
    (8)
    6. A machine fills bottles with water. The amount of water in each bottle is normally distributed. To check the machine is working properly, a random sample of 12 bottles is selected and the amount of water, in ml , in each bottle is recorded. Unbiased estimates for the mean and variance are $$\mu = 502 \quad s ^ { 2 } = 5.6$$ Stating your hypotheses clearly, test at the \(1 \%\) level of significance
31. whether or not the mean amount of water in a bottle is more than 500 ml ,
32. whether or not the standard deviation of the amount of water in a bottle is less than 3 ml .
    7. A machine produces bricks. The lengths, \(x \mathrm {~mm}\), of the bricks are distributed \(\mathrm { N } \left( \mu , 2 ^ { 2 } \right)\). At the start of each week a random sample of \(n\) bricks is taken to check the machine is working correctly.
    A test is then carried out at the \(1 \%\) level of significance with $$\mathrm { H } _ { 0 } : \mu = 202 \quad \text { and } \quad \mathrm { H } _ { 1 } : \mu < 202$$
33. Find, in terms of \(n\), the critical region of the test. The probability of a type II error, when \(\mu = 200\), is less than 0.05 .
34. Find the minimum value of \(n\).