5.07d Paired vs two-sample: selection

6 A biologist is studying the effect of nutrients on the heights to which plants grow. A random sample of 24 similar young plants is divided into two equal groups \(A\) and \(B\). The plants in group \(A\) are fed with nutrients and water and the plants in group \(B\) are given only water. After four weeks, the height, in cm, of each plant is measured and the results are as follows. The biologist decides to carry out a test at the \(5 \%\) significance level to test whether the nutrients have resulted in an increase in growth.

1. She carries out a Wilcoxon rank-sum test. Give a reason why this is an appropriate choice of test.
2. Carry out the Wilcoxon rank-sum test for these results.
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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.

1 A college uses two assessments, \(X\) and \(Y\), when interviewing applicants for research posts at the college. These assessments have been used for a large number of applicants this year. The scores for a random sample of 9 applicants who took assessment \(X\) are as follows. $$\begin{array} { l l l l l l l l l } 21.4 & 24.6 & 25.3 & 22.7 & 20.8 & 21.5 & 22.9 & 21.3 & 22.3 \end{array}$$ The scores for a random sample of 10 applicants who took assessment \(Y\) are as follows. $$\begin{array} { l l l l l l l l l l } 20.9 & 23.5 & 24.8 & 21.9 & 23.4 & 24.0 & 23.8 & 24.1 & 25.1 & 25.8 \end{array}$$ The interviewer believes that the population median score from assessment \(X\) is lower than the population median score from assessment \(Y\). Carry out a Wilcoxon rank-sum test, at the \(1 \%\) significance level, to test whether the interviewer's belief is supported by the data. \includegraphics[max width=\textwidth, alt={}, center]{b5ff998a-fcb6-4a1b-ae86-ec66b0dccc3c-02_2714_37_143_2008}

6 The blood cholesterol levels, measured in suitable units, of a random sample of 11 women and a random sample of 12 men are shown below. Carry out a Wilcoxon rank-sum test, at the \(5 \%\) significance level, to test whether, on average, there is a difference in cholesterol levels between women and men.
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4 Applicants for a particular college take a written test when they attend for interview. There are two different written tests, \(A\) and \(B\), and each applicant takes one or the other. The interviewer wants to determine whether the medians of the distribution of marks obtained in the two tests are equal. The marks obtained by a random sample of 8 applicants who took test \(A\) and a random sample of 8 applicants who took test \(B\) are as follows.

1. Carry out a Wilcoxon rank-sum test at the \(5 \%\) significance level to determine whether there is a difference in the population median marks obtained in the two tests.
   The interviewer considers using the given information to carry out a paired sample \(t\)-test to determine whether there is a difference in the population means for the two tests.
2. Give two reasons why it is not appropriate to use this test.

6 The manager of a technology company \(A\) claims that his employees earn more per year than the employees at technology company \(B\). The amounts earned per year, in hundreds of dollars, by a random sample of 12 employees from company \(A\) and an independent random sample of 12 employees from company \(B\) are shown below.

1. Carry out a Wilcoxon rank-sum test at the \(5 \%\) significance level to test whether the manager's claim is supported by the data.
2. Explain whether a paired sample \(t\)-test would be appropriate to test the manager's claim if earnings are normally distributed.
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5 A company is deciding which of two machines, \(X\) and \(Y\), can make a certain type of electrical component more quickly. The times taken, in minutes, to make one component of this type are recorded for a random sample of 8 components made by machine \(X\) and a random sample of 9 components made by machine \(Y\). These times are as follows. The manager claims that on average the time taken by machine \(X\) to make one component is less than that taken by machine \(Y\).

1. Carry out a Wilcoxon rank-sum test at the \(5 \%\) significance level to test whether the manager's claim is supported by the data.
2. Assuming that the times taken to produce the components by the two machines are normally distributed with equal variances, carry out a \(t\)-test at the \(5 \%\) significance level to test whether the manager's claim is supported by the data.
   \section*{Question 5(c) is printed on the next page.}
3. In general, would you expect the conclusions from the tests in parts (a) and (b) to be the same? Give a reason for your answer.
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4

1. An amateur weather forecaster has been keeping records of air pressure, measured in atmospheres. She takes the measurement at the same time every day using a barometer situated in her garden. A random sample of 100 of her observations is summarised in the table below. The corresponding expected frequencies for a Normal distribution, with its two parameters estimated by sample statistics, are also shown in the table.

   Pressure ( \(a\) atmospheres)	Observed frequency	Frequency as given by Normal model
   \(a \leqslant 0.98\)	4	1.45
   \(0.98 < a \leqslant 0.99\)	6	5.23
   \(0.99 < a \leqslant 1.00\)	9	13.98
   \(1.00 < a \leqslant 1.01\)	15	23.91
   \(1.01 < a \leqslant 1.02\)	37	26.15
   \(1.02 < a \leqslant 1.03\)	21	18.29
   \(1.03 < a\)	8	10.99

   Carry out a test at the \(5 \%\) level of significance of the goodness of fit of the Normal model. State your conclusion carefully and comment on your findings.
2. The forecaster buys a new digital barometer that can be linked to her computer for easier recording of observations. She decides that she wishes to compare the readings of the new barometer with those of the old one. For a random sample of 10 days, the readings (in atmospheres) of the two barometers are shown below.

   Day	A	B	C	D	E	F	G	H	I	J
   Old	0.992	1.005	1.001	1.011	1.026	0.980	1.020	1.025	1.042	1.009
   New	0.985	1.003	1.002	1.014	1.022	0.988	1.030	1.016	1.047	1.025

   Use an appropriate Wilcoxon test to examine at the \(10 \%\) level of significance whether there is any reason to suppose that, on the whole, readings on the old and new barometers do not agree.

4 William takes a bus regularly on the same journey, sometimes in the morning and sometimes in the afternoon. He wishes to compare morning and afternoon journey times. He records the journey times on 7 randomly chosen mornings and 8 randomly chosen afternoons. The results, each correct to the nearest minute, are as follows, where M denotes a morning time and A denotes an afternoon time. William wishes to test for a difference between the average times of morning and afternoon journeys.

1. Given that \(s _ { M } ^ { 2 } = 16.5\) and \(s _ { A } ^ { 2 } = 64.5\), with the usual notation, explain why a \(t\)-test is not appropriate in this case.
2. William chooses a non-parametric test at the \(5 \%\) significance level. Carry out the test, stating the rejection region.

5 A test was carried out to compare the breaking strengths of two brands of elastic band, \(A\) and \(B\), of the same size. Random samples of 6 were selected from each brand and the breaking strengths were measured. The results, in suitable units and arranged in ascending order for each brand, are as follows.

1. Give one advantage that a non-parametric test might have over a parametric test in this context.
2. Carry out a suitable Wilcoxon test at the \(5 \%\) significance level of whether there is a difference between the average breaking strengths of the two brands.
3. An extra elastic band of brand \(B\) was tested and found to have a breaking strength exceeding all of the other 12 bands. Determine whether this information alters the conclusion of your test.

3 The human resources department of a large company is investigating two methods, A and B, for training employees to carry out a certain complicated and intricate task.

1. Two separate random samples of employees who have not previously performed the task are taken. The first sample is of size 10 ; each of the employees in it is trained by method A. The second sample is of size 12; each of the employees in it is trained by method B. After completing the training, the time for each employee to carry out the task is measured, in controlled conditions. The times are as follows, in minutes.

   Employees trained by method A:	35.2	47.8	25.8	38.0	53.6	31.0	33.9
   	35.4	21.6	42.5
   Employees trained by method B:	43.0	57.5	68.6	20.9	31.4	44.9	62.8
   	27.6	41.8	46.1	39.8	61.6

   Stating appropriate assumptions concerning the underlying populations, use a \(t\) test at the \(5 \%\) significance level to examine whether either training method is better in respect of leading, on the whole, to a lower time to carry out the task.
2. A further trial of method B is carried out to see if the performance of experienced and skilled workers can be improved by re-training them. A random sample of 8 such workers is taken. The times in minutes, under controlled conditions, for each worker to carry out the task before and after re-training are as follows.

   Worker	\(W _ { 1 }\)	\(W _ { 2 }\)	\(W _ { 3 }\)	\(W _ { 4 }\)	\(W _ { 5 }\)	\(W _ { 6 }\)	\(W _ { 7 }\)	\(W _ { 8 }\)
   Time before	32.6	28.5	22.9	27.6	34.9	28.8	34.2	31.3
   Time after	26.2	24.1	19.0	28.6	29.3	20.0	36.0	19.2

   Stating an appropriate assumption, use a \(t\) test at the \(5 \%\) significance level to examine whether the re-training appears, on the whole, to lead to a lower time to carry out the task.
3. Explain how the test procedure in part (ii) is enhanced by designing it as a paired comparison.

3 At an agricultural research station, trials are being made of two fertilisers, A and B, to see whether they differ in their effects on the yield of a crop. Preliminary investigations have established that the underlying variances of the distributions of yields using the two fertilisers may be assumed equal. Scientific analysis of the fertilisers has suggested that fertiliser A may be inferior in that it leads, on the whole, to lower yield. A statistical analysis is being carried out to investigate this. The crop is grown in carefully controlled conditions in 14 experimental plots, 6 with fertiliser A and 8 with fertiliser B. The yields, in kg per plot, are as follows, arranged in ascending order for each fertiliser.

1. Carry out a Wilcoxon rank sum test at the \(5 \%\) significance level to examine appropriate hypotheses.
2. Carry out a \(t\) test at the \(5 \%\) significance level to examine appropriate hypotheses.
3. Goodness of fit tests based on more extensive data sets from other trials with these fertilisers have failed to reject hypotheses of underlying Normal distributions. Discuss the relative merits of the analyses in parts (i) and (ii).

3 At an agricultural research station, trials are being carried out to compare a standard variety of tomato with one that has been genetically modified (GM). The trials are concerned with the mean weight of the tomatoes and also with the aesthetic appearance of the tomatoes.

1. 1. Tomatoes of the standard and GM varieties are grown under similar conditions. The tomatoes are weighed and the data are summarised as follows.

      Variety	Sample size	Sum of weights \(( \mathrm { g } )\)	Sum of squares of weights \(\left( \mathrm { g } ^ { 2 } \right)\)
      Standard	30	3218.3	349257
      GM	26	2954.1	338691

      Carry out a test, using the Normal distribution, to investigate whether there is evidence, at the 5\% level of significance, that the two varieties of tomato differ in mean weight. State one assumption required for this test to be valid.
   2. The data in part (i) could have been used to carry out a test for the equality of means based on the \(t\) distribution. State two additional assumptions required for this test to be valid. Discuss briefly which test would be preferable in this case.
2. In order to judge whether, on the whole, GM tomatoes have a better aesthetic appearance than standard tomatoes, a trial is carried out as follows. 10 of each variety are chosen and consumer panel is asked to arrange the 20 tomatoes in order according to their appearance.
   1. State two important features of the way in which this trial should be designed. Comment briefly on how reliable the evidence from the trial is likely to be.
   2. The order in which the consumer panel arranges the tomatoes is as follows. The tomato with best appearance is listed first. \(G\) and \(S\) denote GM and standard tomatoes respectively. $$\begin{array} { c c c c c c c c c c c c c c c c c c c c } G & G & G & S & G & G & G & S & G & S & S & S & G & G & S & G & S & S & S & S \end{array}$$ Carry out an appropriate test at the \(1 \%\) level of significance.

3 A large department in a university wished to compare the standards of literacy and numeracy of its students. A random sample of 24 students was taken and sub-divided, randomly, into two groups of 12 . The students in one group took a literacy assessment (scores denoted by \(x\) ); the students in the other group took a numeracy assessment (scores denoted by \(y\) ). The two assessments were designed to give the same distributions of scores when taken by random samples from the general population. The scores obtained by the students on the two assessments are shown in the table. $$\sum x = 598 \quad \sum x ^ { 2 } = 31196 \quad \sum y = 707 \quad \sum y ^ { 2 } = 43543$$

1. Carry out an appropriate \(t\) test, at the \(5 \%\) level of significance, to compare the standards of literacy and numeracy.
2. State the distributional assumptions required for the \(t\) test to be valid. Name the test that you would use if the assumptions required for the \(t\) test are thought not to hold. State the hypotheses for this new test. Explain, in general terms, which of the two tests is more powerful, and why. A statistician at the university looked at the data and commented that a paired sample design would have been better.
3. Explain how a paired sample design would be applied in this context, and how the data would be analysed. Explain also why it would be better than the design used.

2 Low density lipoprotein (LDL) cholesterol is known as 'bad' cholesterol.
15 randomly chosen patients, each with an LDL level of 190 mg per decilitre of blood, were given one of two treatments, chosen at random. After twelve weeks their LDL levels, in mg per decilitre, were as follows. Use a Wilcoxon rank sum test, at the \(5 \%\) level of significance, to test whether the LDL levels of patients given treatment \(B\) are lower than the LDL levels of patients given treatment \(A\).

4 The heights of eleven randomly selected primary school children are measured. The results, in metres, are

1. Use a Wilcoxon rank-sum test, at the \(1 \%\) significance level, to test whether primary school girls are taller than primary school boys.
2. It is decided to repeat the test, using larger random samples. The heights of twenty girls and eighteen boys are measured. Find the greatest value of the test statistic \(W\) which will result in the conclusion that there is evidence, at the \(1 \%\) level of significance, that primary school girls are taller than primary school boys.

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.

2 A company wishes to buy a new lathe for making chair legs. Two models of lathe, 'Allegro' and 'Vivace', were trialled. The company asked 12 randomly selected employees to make a particular type of chair leg on each machine. The times, in seconds, for each employee are shown in the table. The company wishes to test whether there is any difference in average times for the two machines.

1. State the circumstances under which a non-parametric test should be used.
2. Use two different non-parametric tests and show that they lead to different conclusions at the 5\% significance level.
3. State, with a reason, which conclusion is to be preferred.

6 In a two-tail Wilcoxon rank-sum test, the sample sizes are 13 and 15. The sum of the ranks for the sample of size 13 is 135 . Carry out the test at the \(5 \%\) level of significance.

2 The distances from home to work, in km , of 8 men and 5 women were recorded and are given below. The workers were chosen at random. Carry out a Wilcoxon rank-sum test at the \(5 \%\) significance level to test whether there is a significant difference between the distances from home to work between men and women.

8 A university course was taught by two different professors. Students could choose whether to attend the lectures given by Professor \(Q\) or the lectures given by Professor \(R\). At the end of the course all the students took the same examination. The examination marks of a random sample of 30 students taught by Professor \(Q\) and a random sample of 24 students taught by Professor \(R\) were ranked. The sum of the ranks of the students taught by Professor \(Q\) was 726 . Test at the 5\% significance level whether there is a difference in the ranks of the students taught by the two professors.

7 In a school opinion poll a random sample of 8 pupils were asked to rate school lunches on a scale of 0 to 20 . The results were as follows. \(\begin{array} { l l l l l l l l } 0 & 1 & 2 & 3 & 4 & 10 & 11 & 13 \end{array}\) After a new menu was introduced, the test was repeated with a different random sample of 8 pupils. The results were as follows. \(\begin{array} { l l l l l l l l } 7 & 8 & 9 & 14 & 15 & 17 & 19 & 20 \end{array}\)

1. Carry out an appropriate Wilcoxon test at the \(5 \%\) significance level to test whether pupils' opinions of school lunches have changed. A statistics student tells the organisers of the opinion poll that it would have been better to have asked the same 8 pupils both times.
2. Explain why the statistics student's suggestion would produce a better test.
3. State which test should be used if the student's suggestion is followed.
4. You are given that there are 12870 ways in which 8 different integers can be chosen from the integers 1 to 16 inclusive. Estimate the number of ways of selecting 8 different digits between 1 and 16 inclusive that have a sum less than or equal to the critical value used in the test in part (a).

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.

7. A psychologist gives a test to students from two different schools, \(A\) and \(B\). A group of 9 students is randomly selected from school \(A\) and given instructions on how to do the test.
A group of 7 students is randomly selected from school \(B\) and given the test without the instructions. The table shows the time taken, to the nearest second, to complete the test by the two groups. Stating your hypotheses clearly,

1. test at the \(10 \%\) significance level, whether or not the variance of the times taken to complete the test by students from school \(A\) is the same as the variance of the times taken to complete the test by students from school \(B\). (You may assume that times taken for each school are normally distributed.)
2. test at the \(5 \%\) significance level, whether or not the mean time taken to complete the test by students from school \(A\) is greater than the mean time taken to complete the test by students from school \(B\).
3. Why does the result to part (a) enable you to carry out the test in part (b)?
4. Give one factor that has not been taken into account in your analysis.

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 \(A\).