Paired t-test

7 In order to improve their mathematics results 10 students attended an intensive Summer School course. Each student took a test at the start of the course and a similar test at the end of the course. The table shows the scores achieved in each test. It is desired to test whether there has been an increase in the population mean score.

1. Explain why a two-sample \(t\)-test would not be appropriate.
2. Stating any necessary assumptions, carry out a suitable \(t\)-test at the \(\frac { 1 } { 2 } \%\) significance level.
3. The Summer School director claims that after taking the course the population mean score increases by more than 5 . Is there sufficient evidence for this claim?

7 A factory manager wished to compare two methods of assembling a new component, to determine which method could be carried out more quickly, on average, by the workforce. A random sample of 12 workers was taken, and each worker tried out each of the methods of assembly. The times taken, in seconds, are shown in the table.

1. (a) Carry out an appropriate \(t\)-test, using a \(2 \%\) significance level, to test whether there is any difference in the times for the two methods of assembly.
   (b) State an assumption needed in carrying out this test.
   (c) Calculate a \(95 \%\) confidence interval for the population mean time difference for the two methods of assembly.
2. Instead of using the same 12 workers to try both methods, the factory manager could have used two independent random samples of workers, allocating Method 1 to the members of one sample and Method 2 to the members of the other sample.
   (a) State one disadvantage of a procedure based on two independent random samples.
   (b) State any assumptions that would need to be made to carry out a \(t\)-test based on two independent random samples.

3 The management of a large chain of shops aims to reduce the level of absenteeism among its workforce by means of an incentive bonus scheme. In order to evaluate the effectiveness of the scheme, the management measures the percentage of working days lost before and after its introduction for each of a random sample of 11 shops. The results are shown below.

1. The management decides to carry out a \(t\) test to investigate whether there has been a reduction in absenteeism.
   1. State clearly the hypotheses that should be used together with any necessary assumptions.
   2. Carry out the test using a \(5 \%\) significance level.
2. Find a 95\% confidence interval for the true mean percentage of days lost after the introduction of the incentive scheme and state any assumption needed. The management has set a target that the mean percentage should be 3.5. Do you think this has been achieved? Explain your answer.

3 A tutor gave an assessment to 6 randomly chosen new eleven-year-old students. After each student had received 30 hours of tuition, they were given a second assessment. The scores are shown in the table.

1. Show that, at the \(5 \%\) significance level, there is insufficient evidence that students' scores are higher, on average, after tuition than before tuition. State a necessary assumption.
2. Disappointed by this result, the tutor looked again at the first assessment. She discovered that the first assessment was too easy, in fact being a test for ten-year-olds, not eleven-year-olds. She decided to reduce each score for the first assessment by a constant integer \(k\). Find the least value of \(k\) for which there is evidence at the \(5 \%\) significance level that the students' scores have, on average, improved.

3 Employees at a particular company have been working seven hours each day, from 9 am to 4 pm. To try to reduce absence, the company decides to introduce 'flexi-time' and allow employees to work their seven hours each day at any time between 7 am and 9 pm. For a random sample of 10 employees, the numbers of hours of absence in the year before and the year after the introduction of flexi-time are given in the following table. Test, at the \(10\%\) significance level, whether the population mean number of hours of absence has decreased following the introduction of flexi-time, stating any assumption that you make.

6 The deterioration of a certain drug over time was investigated as follows. The drug strength was measured in each of a random sample of 8 bottles containing the drug. These were stored for two years and the strengths were then re-measured. The original and final strengths, in suitable units, are shown in the following table.

1. Stating any required assumption, test at the \(5 \%\) significance level whether the mean strength has decreased by more than 0.2 over the two years.
2. Calculate a 95\% confidence interval for the mean reduction in strength over the two years.

2 A new running track has been developed and part of the testing procedure involves 7 randomly chosen athletes. They each run 100 m on both the old and new tracks.
The results are as follows. The population mean times on the old and new tracks are denoted by \(\mu _ { \mathrm { O } }\) seconds and \(\mu _ { \mathrm { N } }\) seconds respectively. Stating any necessary assumption, carry out a suitable \(t\)-test of the null hypothesis \(\mu _ { \mathrm { O } } - \mu _ { \mathrm { N } } = 0\) against the alternative hypothesis \(\mu _ { \mathrm { O } } - \mu _ { \mathrm { N } } > 0\). Use a \(2 \frac { 1 } { 2 } \%\) significance level .

2 Four pairs of randomly chosen twins were each given identical puzzles to solve. The times taken (in seconds) are shown in the following table. Stating any necessary assumption, test at the \(10 \%\) significance level whether there is a difference between the population mean times of first-born and second-born twins.

3 Cholesterol is a lipid (fat) which is manufactured by the liver from the fatty foods that we eat. It plays a vital part in allowing the body to function normally. However, when high levels of cholesterol are present in the blood there is a risk of arterial disease. Among the factors believed to assist with achieving and maintaining low cholesterol levels are weight loss and exercise. A doctor wishes to test the effectiveness of exercise in lowering cholesterol levels. For a random sample of 12 of her patients, she measures their cholesterol levels before and after they have followed a programme of exercise. The measurements obtained are as follows.

1. A \(t\) test is to be used in order to see if, on average, the exercise programme seems to be effective in lowering cholesterol levels. State the distributional assumption necessary for the test, and carry out the test using a \(1 \%\) significance level.
2. A second random sample of 12 patients gives a \(95 \%\) confidence interval of \(( - 0.5380,1.4046 )\) for the true mean reduction (before - after) in cholesterol level. Find the mean and standard deviation for this sample. How might the doctor interpret this interval in relation to the exercise programme?

1 Technologists at a company that manufactures paint are trying to develop a new type of gloss paint with a shorter drying time than the current product. In order to test whether the drying time has been reduced, the technologists paint a square metre of each of the new and old paints on each of 10 different surfaces. The lengths of time, in hours, that each square metre takes to dry are as follows.

1. Explain why a paired sample is used in this context.
2. The mean reduction in drying time is to be investigated. Why might a \(t\) test be appropriate in this context and what assumption needs to be made?
3. Using a significance level of \(5 \%\), carry out a test to see if there appears to be any reduction in mean drying time.
4. Find a 95\% confidence interval for the true mean reduction in drying time.

10 Carpal Tunnel syndrome is a condition which affects a person's ability to grip with their hands. Researchers tested a treatment for this syndrome which was applied to 8 randomly chosen patients. A pre-treatment and a post-treatment test of grip was given to each patient, with the following results, measured in kg. Stating any required assumption, test, at the \(1 \%\) significance level, whether the mean grip of people with the syndrome increases after undergoing the treatment. It is given that there is evidence at the \(10 \%\) significance level that the mean grip increases by more than \(w \mathrm {~kg}\). Find an inequality for \(w\).

5. Seven pipes of equal length are selected at random. Each pipe is cut in half. One piece of each pipe is coated with protective paint and the other is left uncoated. All of the pieces of pipe are buried to the same depth in various soils for 6 months. The table gives the percentage area of the pieces of pipe in the various soils that are subject to corrosion.

1. Stating your hypotheses clearly and using a \(5 \%\) significance level, carry out a paired \(t\)-test to assess whether or not there is a difference between the mean percentage of corrosion on the coated pipes and the mean percentage of corrosion on the uncoated pipes.
2. 1. State an assumption that has been made in order to carry out this test.
   2. Comment on the validity of this assumption.
3. State what difference would be made to the conclusion in part (a) if the test had been to determine whether or not the percentage of corrosion on the uncoated pipes was higher than the mean percentage of corrosion on the coated pipes. Justify your answer.

4. A doctor believes that the span of a person's dominant hand is greater than that of the weaker hand. To test this theory, the doctor measures the spans of the dominant and weaker hands of a random sample of 8 people. He subtracts the span of the weaker hand from that of the dominant hand. The spans, in mm , are summarised in the table below. Test, at the 5\% significance level, the doctor's belief.
(9)

4. A farmer set up a trial to assess the effect of two different diets on the increase in the weight of his lambs. He randomly selected 20 lambs. Ten of the lambs were given \(\operatorname { diet } A\) and the other 10 lambs were given diet \(B\). The gain in weight, in kg , of each lamb over the period of the trial was recorded.

1. State why a paired \(t\)-test is not suitable for use with these data.
2. Suggest an alternative method for selecting the sample which would make the use of a paired \(t\)-test valid.
3. Suggest two other factors that the farmer might consider when selecting the sample. The following paired data were collected.

   Diet \(A\)	5	6	7	4.6	6.1	5.7	6.2	7.4	5	3
   Diet \(B\)	7	7.2	8	6.4	5.1	7.9	8.2	6.2	6.1	5.8

4. Using a paired \(t\)-test, at the \(5 \%\) significance level, test whether or not there is evidence of a difference in the weight gained by the lambs using \(\operatorname { diet } A\) compared with those using \(\operatorname { diet } B\).
5. State, giving a reason, which diet you would recommend the farmer to use for his lambs.
   (Total 13 marks)

3. As part of an investigation into the effectiveness of solar heating, a pair of houses was identified where the mean weekly fuel consumption was the same. One of the houses was then fitted with solar heating and the other was not. Following the fitting of the solar heating, a random sample of 9 weeks was taken and the table below shows the weekly fuel consumption for each house. Units of fuel used per week

1. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is evidence that the solar heating reduces the mean weekly fuel consumption.
   (8)
2. State an assumption about weekly fuel consumption that is required to carry out this test.

1. A medical student is investigating two methods of taking a person's blood pressure. He takes a random sample of 10 people and measures their blood pressure using an arm cuff and a finger monitor. The table below shows the blood pressure for each person, measured by each method.

1. 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.
   (8)
2. State an assumption about the underlying distribution of measured blood pressure required for this test.
   (1)

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.

2. An emission-control device is tested to see if it reduces \(\mathrm { CO } _ { 2 }\) emissions from cars. The emissions from 6 randomly selected cars are measured with and without the device. The results are as follows.

1. State an assumption that needs to be made in order to carry out a \(t\)-test in this case.
2. State why a paired \(t\)-test is suitable for use with these data.
3. Using a \(5 \%\) level of significance, test whether or not there is evidence that the device reduces \(\mathrm { CO } _ { 2 }\) emissions from cars.
4. Explain, in context, what a type II error would be in this case.

3. Manuel is planning to buy a new machine to squeeze oranges in his cafe and he has two models, at the same price, on trial. The manufacturers of machine \(B\) claim that their machine produces more juice from an orange than machine \(A\). To test this claim Manuel takes a random sample of 8 oranges, cuts them in half and puts one half in machine \(A\) and the other half in machine \(B\). The amount of juice, in ml , produced by each machine is given in the table 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 \(A\).

1. In a trial for a new cough medicine, a random sample of 8 healthy patients were given steadily increasing doses of a pepper extract until they started coughing. The level of pepper that triggered the coughing was recorded. Each patient completed the trial after taking a standard cough medicine and, at a later time, after taking the new medicine. The results are given in the table below.

1. Using a suitable test, at the \(5 \%\) level of significance, state whether or not, on the basis of this trial, you would recommend using the new medicine. State your hypotheses clearly.
2. State an assumption needed to carry out this 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.