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 A production line has two machines, A and B , for delivering liquid soap into bottles. Each machine is set to deliver a nominal amount of 250 ml , but it is not expected that they will work to a high level of accuracy. In particular, it is known that the ambient temperature affects the rate of flow of the liquid and leads to variation in the amounts delivered. The operators think that machine B tends to deliver a somewhat greater amount than machine A , no matter what the ambient temperature. This is being investigated by an experiment. A random sample of 10 results from the experiment is shown below. Each column of data is for a different ambient temperature.

1. Use an appropriate \(t\) test to examine, at the \(5 \%\) level of significance, whether the mean amount delivered by machine B may be taken as being greater than that delivered by machine A , stating carefully your null and alternative hypotheses and the required distributional assumption.
2. Using the data for machine A in the table above, provide a two-sided \(95 \%\) confidence interval for the mean amount delivered by this machine, stating the required distributional assumption. Explain whether you would conclude that the machine appears to be working correctly in terms of the nominal amount as set.

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

1. A tea grower is testing two types of plant for the weight of tea they produce. A trial is set up in which each type of plant is grown at each of 8 sites. The total weight, in grams, of tea leaves harvested from each plant is measured and shown below.

   Site	A	B	C	D	E	F	G	H
   Type I	225.2	268.9	303.6	244.1	230.6	202.7	242.1	247.5
   Type II	215.2	242.1	260.9	241.7	245.5	204.7	225.8	236.0

   1. The grower intends to perform a \(t\) test to examine whether there is any difference in the mean yield of the two types of plant. State the hypotheses he should use and also any necessary assumption.
   2. Carry out the test using a \(5 \%\) significance level.
2. The tea grower deals with many types of tea and employs tasters to rate them. The tasters do this by giving each tea a score out of 100. The tea grower wishes to compare the scores given by two of the tasters. Their scores for a random selection of 10 teas are as follows.

   Tea	Q	R	S	T	U	V	W	X	Y	Z
   Taster 1	69	79	85	63	81	65	85	86	89	77
   Taster 2	74	75	99	66	75	64	96	94	96	86

   Use a Wilcoxon test to examine, at the \(5 \%\) level of significance, whether it appears that, on the whole, the scores given to teas by these two tasters differ.

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.

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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?

3

1. In order to prevent and/or control the spread of infectious diseases, the Government has various vaccination programmes. One such programme requires people to receive a booster injection at the age of 18. It is felt that the proportion of people receiving this booster could be increased and a publicity campaign is undertaken for this purpose. In order to assess the effectiveness of this campaign, health authorities across the country are asked to report the percentage of 18-year-olds receiving the booster before and after the campaign. The results for a randomly chosen sample of 9 authorities are as follows.

   Authority	A	B	C	D	E	F	G	H	I
   Before	76	98	88	81	86	84	83	93	80
   After	82	97	93	77	83	95	91	95	89

   This sample is to be tested to see whether the campaign appears to have been successful in raising the percentage receiving the booster.
   1. Explain why the use of paired data is appropriate in this context.
   2. Carry out an appropriate Wilcoxon signed rank test using these data, at the \(5 \%\) significance level.
2. Benford's Law predicts the following probability distribution for the first significant digit in some large data sets.

   Digit	1	2	3	4	5	6	7	8	9
   Probability	0.301	0.176	0.125	0.097	0.079	0.067	0.058	0.051	0.046

   On one particular day, the first significant digits of the stock market prices of the shares of a random sample of 200 companies gave the following results.

   Digit	1	2	3	4	5	6	7	8	9
   Frequency	55	34	27	16	15	17	12	15	9

   Test at the \(10 \%\) level of significance whether Benford's Law provides a reasonable model in the context of share prices.

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 \(w\).

9 A national athletics coach suspects that, on average, 200-metre runners' indoor times exceed their outdoor times by more than 0.1 seconds. In order to test this, the coach randomly selects eight 200 -metre runners and records their indoor and outdoor times. The results, in seconds, are shown in the table. Stating suitable hypotheses and any necessary assumption that you make, test the coach's suspicion at the 2.5\% level of significance.

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. A medical student is investigating whether there is a difference in a person's blood pressure when sitting down and after standing up. She takes a random sample of 12 people and measures their blood pressure, in mmHg , when sitting down and after standing up.

The results are shown below. 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.

1. State clearly the hypotheses that should be used and any necessary assumption that needs to be made.
2. Carry out the test at the \(1 \%\) level of significance.
   \section*{L}

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.