Wilcoxon rank-sum test (Mann-Whitney U test)

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.

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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2 Of 9 randomly chosen students attending a lecture, 4 were found to be smokers and 5 were nonsmokers. During the lecture their pulse-rates were measured, with the following results in beats per minute. It may be assumed that these two groups of students were random samples from the student populations of smokers and non-smokers. Using a suitable Wilcoxon test at the \(10 \%\) significance level, test whether there is a difference in the median pulse-rates of the two populations.

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 Because of the large number of students enrolled for a university geography course and the limited accommodation in the lecture theatre, the department provides a filmed lecture. Students are randomly assigned to two groups, one to attend the lecture theatre and the other the film. At the end of term the two groups are given the same examination. The geography professor wishes to test whether there is a difference in the performance of the two groups and selects the marks of two random samples of students, 6 from the group attending the lecture theatre and 7 from the group attending the films. The marks for the two samples, ordered for convenience, are shown below.

1. Stating a necessary assumption, carry out a suitable non-parametric test, at the \(10 \%\) significance level, for a difference between the median marks of the two groups.
2. State conditions under which a two-sample \(t\)-test could have been used.
3. Assuming that the tests in parts (i) and (ii) are both valid, state with a reason which test would be preferable.

4 The effect of water salinity on the growth of a type of grass was studied by a biologist. A random sample of 22 seedlings was divided into two groups \(A\) and \(B\), each of size 11 .
Group \(A\) was treated with water of \(0 \%\) salinity and group \(B\) was treated with water of \(0.5 \%\) salinity. After three weeks the height (in cm) of each seedling was measured with the following results, which are ordered for convenience. Jeffery was asked to test whether the two treatments resulted, on average, in a difference in growth. He chose the Wilcoxon rank sum test.

1. Justify Jeffery's choice of test.
2. Carry out the test at the \(5 \%\) significance level.

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).

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.

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.

5 An historian has reason to believe that the average age at which men got married in the seventeenth century was higher in urban areas compared to rural areas. The historian collected data from a random sample of 8 men in an urban area and a random sample of 6 men in a rural area, all of whom were married in the seventeenth century. The results were as follows, given in the form years/months.

1. Use an appropriate non-parametric method to test at the \(5 \%\) significance level whether the average age at marriage of men is higher in urban areas than in rural areas.
2. When checking the data, the historian found that the age of one of the men, Mr X, which had been recorded as 28/11, had been wrongly recorded. When corrected, the result of the test in part (a) was unchanged. Determine the youngest age that Mr X could have been, given that it was not the same, in years and months, as that of any of the other men in the sample.

3 Jo can use either of two different routes, A or B, for her journey to school. She believes that route A has shorter journey times. She measures how long her journey takes for 17 journeys by route A and 12 journeys by route B . She ranks the 29 journeys in increasing order of time taken, and she finds that the sum of the ranks of the journeys by route B is 219 .

1. Test at the \(10 \%\) significance level whether route A has shorter journey times than route B .
2. State an assumption about the 29 journeys which is necessary for the conclusion of the test to be valid.

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).

7. Nathan believes that shearers from Wales can shear more sheep, on average, in a given time than shearers from New Zealand. He takes a random sample of 8 shearers from Wales and 7 shearers from New Zealand. The numbers below indicate how many sheep were sheared in 45 minutes by the 15 shearers. Use a Mann-Whitney U test at the \(1 \%\) significance level to test whether Nathan is correct. You must state your hypotheses clearly and state the critical region.

6. A triathlon race organiser wishes to know whether competitors who are members of a triathlon club race more frequently than competitors who are not members of a triathlon club. Six competitors from a triathlon club and six competitors who are not members of a triathlon club are selected at random. The table below shows the number of triathlon races they each entered last year.

1. Use a Mann-Whitney U test at a significance level as close to \(5 \%\) as possible to carry out the race organiser's investigation.
2. Briefly explain why a Wilcoxon signed-rank test is not appropriate in this case.

6. Alana is a PhD student researching language acquisition. She gives one group of randomly selected participants, Group A, 4 minutes to memorise 40 words that are similar in meaning. She gives a different, randomly selected group of participants, Group B, 4 minutes to memorise 40 words that are different in meaning. Alana believes that the students in Group B will do better than the students in Group A. The following results are the number of words recalled on testing the students from the two groups. Conduct a Mann-Whitney U test at a significance level as close as possible to \(5 \%\) to test Alana's belief.

3. A motoring organisation wishes to determine whether or not the petrol consumption of two different car models \(A\) and \(B\) are the same. A trial is therefore carried out in which 6 cars of each model are given 10 litres of petrol and driven at a predetermined speed around a track until the petrol is used up. The distances travelled, in miles, are shown below It is proposed to use a test with significance level \(5 \%\) based on the Mann-Whitney statistic \(U\).

1. State suitable hypotheses.
2. Find the critical region for the test.
3. Determine the value of \(U\) for the above data and state your conclusion in context. You must justify your answer.

8 In an experiment to investigate the effect of background music in carrying out work, ten students were each given a task. Five of the students did the task in silence and the other five did the task with background music. The scores on the tasks were as follows.

1. Use a Wilcoxon rank-sum test to test at the 10\% level whether the presence of background music affects scores.
2. A statistician suggests that the experiment is redesigned so that each student takes one task in silence and another task with background music. The differences in the test scores would then be analysed using a paired-sample method. State an advantage in redesigning the experiment in this way.

3. A statistics teacher wants to investigate whether students from the north of a county and students from the south of the same county feel similarly stressed about examinations. The teacher carries out a psychometric test on 10 randomly selected students to give a score between 0 (low stress) and 100 (high stress) to measure their stress levels before a set of examinations. The results are shown in the table below.

1. State one reason why a Mann-Whitney test is appropriate.
2. Conduct a Mann-Whitney test at a significance level as close to \(5 \%\) as possible. State your conclusion clearly.
3. How could this investigation be improved?

5 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.
[0pt] [10] Total Marks for Question Set 3: 38 \section*{Mark scheme} \section*{Marking Instructions} a An element of professional judgement is required in the marking of any written paper. Remember that the mark scheme is designed to assist in marking incorrect solutions. Correct solutions leading to correct answers are awarded full marks but work must not always be judged on the answer alone, and answers that are given in the question, especially, must be validly obtained; key steps in the working must always be looked at and anything unfamiliar must be investigated thoroughly. Correct but unfamiliar or unexpected methods are often signalled by a correct result following an apparently incorrect method. Such work must be carefully assessed.
b The following types of marks are available. \section*{M} A suitable method has been selected and applied in a manner which shows that the method is essentially understood. Method marks are not usually lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. In some cases the nature of the errors allowed for the award of an M mark may be specified.
A method mark may usually be implied by a correct answer unless the question includes the DR statement, the command words "Determine" or "Show that", or some other indication that the method must be given explicitly. \section*{A} Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated Method mark is earned (or implied). Therefore M0 A1 cannot ever be awarded. \section*{B} Mark for a correct result or statement independent of Method marks. \section*{E} A given result is to be established or a result has to be explained. This usually requires more working or explanation than the establishment of an unknown result. Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored. Sometimes this is reinforced in the mark scheme by the abbreviation isw. However, this would not apply to a case where a candidate passes through the correct answer as part of a wrong argument.
c When a part of a question has two or more 'method' steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. (The notation 'dep*' is used to indicate that a particular mark is dependent on an earlier, asterisked, mark in the scheme.) Of course, in practice it may happen that when a candidate has once gone wrong in a part of a question, the work from there on is worthless so that no more marks can sensibly be given. On the other hand, when two or more steps are successfully run together by the candidate, the earlier marks are implied and full credit must be given.
d The abbreviation FT implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A and B marks are given for correct work only - differences in notation are of course permitted. A (accuracy) marks are not given for answers obtained from incorrect working. When A or B marks are awarded for work at an intermediate stage of a solution, there may be various alternatives that are equally acceptable. In such cases, what is acceptable will be detailed in the mark scheme. Sometimes the answer to one part of a question is used in a later part of the same question. In this case, A marks will often be 'follow through'.
e We are usually quite flexible about the accuracy to which the final answer is expressed; over-specification is usually only penalised where the scheme explicitly says so.

• When a value is given in the paper only accept an answer correct to at least as many significant figures as the given value.
• When a value is not given in the paper accept any answer that agrees with the correct value to \(\mathbf { 3 ~ s } . \mathbf { f }\). unless a different level of accuracy has been asked for in the question, or the mark scheme specifies an acceptable range.

Follow through should be used so that only one mark in any question is lost for each distinct accuracy error.
Candidates using a value of \(9.80,9.81\) or 10 for \(g\) should usually be penalised for any final accuracy marks which do not agree to the value found with 9.8 which is given in the rubric.
f Rules for replaced work and multiple attempts:

• If one attempt is clearly indicated as the one to mark, or only one is left uncrossed out, then mark that attempt and ignore the others.
• If more than one attempt is left not crossed out, then mark the last attempt unless it only repeats part of the first attempt or is substantially less complete.
• if a candidate crosses out all of their attempts, the assessor should attempt to mark the crossed out answer(s) as above and award marks appropriately.

For a genuine misreading (of numbers or symbols) which is such that the object and the difficulty of the question remain unaltered, mark according to the scheme but following through from the candidate's data. A penalty is then applied; 1 mark is generally appropriate, though this may differ for some units. This is achieved by withholding one A or B mark in the question. Marks designated as cao may be awarded as long as there are no other errors.
If a candidate corrects the misread in a later part, do not continue to follow through. Note that a miscopy of the candidate's own working is not a misread but an accuracy error.
h If a calculator is used, some answers may be obtained with little or no working visible. Allow full marks for correct answers, provided that there is nothing in the wording of the question specifying that analytical methods are required such as the bold "In this question you must show detailed reasoning", or the command words "Show" or "Determine". Where an answer is wrong but there is some evidence of method, allow appropriate method marks. Wrong answers with no supporting method score zero. \begin{table}[h] \end{table}