Wilcoxon matched-pairs signed-rank test

5 Georgio has designed two new uniforms \(X\) and \(Y\) for the employees of an airline company. A random sample of 11 employees are each asked to assess each of the two uniforms for practicality and appearance, and to give a total score out of 100. The scores are given in the table.

1. Give a reason why a Wilcoxon signed-rank test may be more appropriate than a \(t\)-test for investigating whether there is any evidence of a preference for one of the uniforms.
2. Carry out a Wilcoxon matched-pairs signed-rank test at the \(10 \%\) significance level.

2 A company is developing a new flavour of chocolate by varying the quantities of the ingredients. A random selection of 9 flavours of chocolate are judged by two tasters who each give marks out of 100 to each flavour of chocolate. Carry out a Wilcoxon matched-pairs signed-rank test at the \(10 \%\) significance level to investigate whether, on average, there is a difference between marks awarded by the two tasters.

4 A random sample of 13 technology companies is chosen and the numbers of employees in 2018 and in 2022 are recorded. A researcher claims that there has been an increase in the median number of employees at technology companies between 2018 and 2022.

1. Carry out a Wilcoxon matched-pairs signed-rank test, at the \(5 \%\) significance level, to test whether the data supports this claim.
   The researcher notices that the figures for company \(G\) have been recorded incorrectly. In fact, the number of employees in 2018 was 32 and the number of employees in 2022 was 35.
2. Explain, with numerical justification, whether or not the conclusion of the test in part (a) remains the same.

3 A large number of students took two test papers in mathematics. The teacher believes that the marks obtained in Paper 1 will be higher than the marks obtained in Paper 2. She chooses a random sample of 9 students and compares their marks. The marks are shown in the table.

1. Carry out a Wilcoxon matched-pairs signed-rank test, at the \(5 \%\) significance level, to test whether the data supports the teacher’s belief.
2. State an assumption that you have made in carrying out the test in part (a).

2 A large school is holding an essay competition and each student has submitted an essay. To ensure fairness, each essay is given a mark out of 100 by two different judges. The marks awarded to the essays submitted by a random sample of 12 students are shown in the following table.

1. One of the students claims that Judge 2 is awarding higher marks than Judge 1. Carry out a Wilcoxon matched-pairs signed-rank test at the \(5 \%\) significance level to test whether the data supports the student’s claim.
   It is discovered later that the marks awarded to student \(A\) have been entered incorrectly. In fact, Judge 1 awarded 65 marks and Judge 2 awarded 62 marks.
2. By considering how this change affects the test statistic, explain why the conclusion of the test carried out in part (a) remains the same.

3 A large college is holding a piano competition. Each student has played a particular piece of music and two judges have each awarded a mark out of 80 . The marks awarded to a random sample of 14 students are shown in the following table.

1. One of the students claims that on average Judge 1 is awarding higher marks than Judge 2. Carry out a Wilcoxon matched-pairs signed-rank test at the 5\% significance level to test whether the data supports the student's claim.
2. Give a reason why it is preferable to use a Wilcoxon matched-pairs signed-rank test in this situation rather than a paired sample \(t\)-test.

2 A school with a large number of students is updating its logo. Each student has designed a new logo and two teachers have each awarded a mark out of 50 for each logo. The marks awarded to a random sample of 12 students are shown in the following table. One of the students claims that Teacher 2 is awarding higher marks than Teacher 1.

1. Carry out a Wilcoxon matched-pairs signed-rank test, at the \(5 \%\) significance level, to test whether the data supports the claim. \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-04_2720_38_109_2010} \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-05_2717_29_105_22} It was later discovered that Teacher 1 had entered her mark for student \(C\) incorrectly. Her intended mark was 24 not 40 . This was corrected.
2. Determine whether this correction affects the conclusion of the test carried out in part (a).

2 A school with a large number of students is updating its logo. Each student has designed a new logo and two teachers have each awarded a mark out of 50 for each logo. The marks awarded to a random sample of 12 students are shown in the following table. One of the students claims that Teacher 2 is awarding higher marks than Teacher 1.

1. Carry out a Wilcoxon matched-pairs signed-rank test, at the \(5 \%\) significance level, to test whether the data supports the claim. \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-04_2715_38_109_2010} \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-05_2716_29_107_22} It was later discovered that Teacher 1 had entered her mark for student \(C\) incorrectly. Her intended mark was 24 not 40 . This was corrected.
2. Determine whether this correction affects the conclusion of the test carried out in part (a).

4 A company has many factories. It is concerned about incidents of trespassing and, in the hope of reducing if not eliminating these, has embarked on a programme of installing new fencing.

1. Records for a random sample of 9 factories of the numbers of trespass incidents in typical weeks before and after installation of the new fencing are as follows.

   Factory	A	B	C	D	E	F	G	H	I
   Number before installation	8	12	6	4	14	22	4	13	14
   Number after installation	6	11	0	1	18	10	11	5	4

   Use a Wilcoxon test to examine at the \(5 \%\) level of significance whether it appears that, on the whole, the number of trespass incidents per week is lower after the installation of the new fencing than before.
2. Records are also available of the costs of damage from typical trespass incidents before and after the introduction of the new fencing for a random sample of 7 factories, as follows (in £).

   Factory	T	U	V	W	X	Y	Z
   Cost before installation	1215	95	546	467	2356	236	550
   Cost after installation	1268	110	578	480	2417	318	620

   Stating carefully the required distributional assumption, provide a two-sided \(99 \%\) confidence interval based on a \(t\) distribution for the population mean difference between costs of damage before and after installation of the new fencing. Explain why this confidence interval should not be based on the Normal distribution.

2 A botanist believes that some species of plants produce more flowers at high altitudes than at low altitudes. In order to investigate this belief the botanist randomly samples 11 species of plants each of which occurs at both altitudes. The numbers of flowers on the plants are shown in the table.

1. Use the Wilcoxon signed rank test at the 5\% significance level to test the botanist's belief.
2. Explain why the Wilcoxon rank sum test should not be used for this test.

2 Two drugs, I and II, for alleviating hay fever are trialled in a hospital on each of 12 volunteer patients. Each received drug I on one day and drug II on a different day. After receiving a drug, the number of times each patient sneezed over a period of one hour was noted. The results are given in the table. The patients may be considered to be a random sample of all hay fever sufferers.
A researcher believes that patients taking drug II sneeze less than patients taking drug I.
Test this belief using the Wilcoxon signed rank test at the \(5 \%\) significance level.

1 A teacher believes that the calculator paper in a GCSE Mathematics examination was easier than the non-calculator paper. The marks of a random sample of ten students are shown in the table.

1. Use a Wilcoxon signed-rank test, at the \(5 \%\) significance level, to test the teacher's belief.
2. State the assumption necessary for this test to be applied.

4

1. At a college, two examiners are responsible for marking, independently, the students' projects. Each examiner awards a mark out of 100 to each project. There is some concern that the examiners' marks do not agree, on average. Consequently a random sample of 12 projects is selected and the marks awarded to them are compared.
   1. Describe how a random sample of projects should be chosen.
   2. The marks given for the projects in the sample are as follows.

      Project	1	2	3	4	5	6	7	8	9	10	11	12
      Examiner A	58	37	72	78	67	77	62	41	80	60	65	70
      Examiner B	73	47	74	71	78	96	54	27	97	73	60	66

      Carry out a test at the \(10 \%\) level of significance of the hypotheses \(\mathrm { H } _ { 0 } : m = 0 , \mathrm { H } _ { 1 } : m \neq 0\), where \(m\) is the population median difference.
2. A calculator has a built-in random number function which can be used to generate a list of random digits. If it functions correctly then each digit is equally likely to be generated. When it was used to generate 100 random digits, the frequencies of the digits were as follows.

   Digit	0	1	2	3	4	5	6	7	8	9
   Frequency	6	8	11	14	12	9	15	5	14	6

   Use a goodness of fit test, with a significance level of \(10 \%\), to investigate whether the random number function is generating digits with equal probability.

3 A company which employs 600 staff wishes to improve its image by introducing new uniforms for the staff to wear. The human resources manager would like to obtain the views of the staff. She decides to do this by means of a systematic sample of \(10 \%\) of the staff.

1. How should she go about obtaining such a sample, ensuring that all members of staff are equally likely to be selected? Explain whether this constitutes a simple random sample. At a later stage in the process, the choice of uniform has been reduced to two possibilities. Twelve members of staff are selected to take part in deciding which of the two uniforms to adopt. Each of the twelve assesses each uniform for comfort, appearance and practicality, giving it a total score out of 10. The scores are as follows.

   Staff member	1	2	3	4	5	6	7	8	9	10	11	12
   Uniform A	4.2	2.6	10.0	9.0	8.2	2.8	5.0	7.4	2.8	6.8	10.0	9.8
   Uniform B	5.0	5.2	1.4	2.8	2.2	6.4	7.4	7.8	6.8	1.2	3.4	7.6

   A Wilcoxon signed rank test is to be used to decide whether there is any evidence of a preference for one of the uniforms.
2. Explain why this test is appropriate in these circumstances and state the hypotheses that should be used.
3. Carry out the test at the \(5 \%\) significance level.

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.

4 A psychologist investigated the scores of pairs of twins on an aptitude test. Seven pairs of twins were chosen randomly, and the scores are given in the following table.

1. Carry out an appropriate Wilcoxon test at the \(10 \%\) significance level to investigate whether there is evidence of a difference in test scores between the elder and the younger of a pair of twins.
2. Explain the advantage in this case of a Wilcoxon test over a sign test.

5. To qualify as a music examiner, a trainee must listen to a series of performances by 8 randomly chosen students. An experienced examiner and the trainee both award scores for each of the 8 performances. In order for the trainee to qualify, there must not be a significant difference between the average scores given by the experienced examiner and the trainee.

1. Explain why the Wilcoxon signed rank test is appropriate. The scores awarded are shown below.

   Student	A	B	C	D	E	F	G	H
   Experienced Examiner	108	109	92	95	145	148	134	120
   Trainee	114	116	95	93	137	144	133	110

2. 1. Carry out an appropriate Wilcoxon signed rank test on this dataset, using a \(5 \%\) significance level.
   2. What conclusion should be reached about the suitability of the trainee to qualify?

4. Llŷr believes that he will have more social media followers by appearing on a certain Welsh television show. To investigate his belief, he collects data on 9 randomly selected contestants who have appeared on the show. Llŷr records the number of social media followers one week before and one week after the contestants appeared on the show. The data he collects are shown in the table below.

1. 1. Carry out a Wilcoxon signed-rank test on this data set, at a significance level as close to 10\% as possible.
   2. Suggest a possible course of action that Llŷr might take.
2. Give two reasons why the Wilcoxon signed-rank test is appropriate in this case.

6. A medical student is investigating two different methods, A and B , of measuring a patient's blood pressure. He believes that Method B gives, on average, a higher reading than Method A so he defines the following hypotheses. \(H _ { 0 }\) : There is on average no difference in the readings obtained using Methods A and B; \(H _ { 1 }\) : The reading obtained using Method B is on average higher than the reading obtained using Method A. He selects 10 patients at random and he measures their blood pressures using both methods. He obtains the following results.

1. Carry out an appropriate Wilcoxon signed rank test on this data set, using a 5\% significance level.
2. State what conclusion the medical student should reach, justifying your answer.

5 Alexa believes that students are equally likely to achieve the same percentage score on each of two tests, paper I and paper II. She randomly selects 8 students and gives them each paper I and paper II. The percentage scores for each paper are recorded. The following paired data are collected. Test, at the \(1 \%\) significance level, whether or not there is evidence to support Alexa's belief. State your hypotheses clearly and show your working.