5.07b Sign test: and Wilcoxon signed-rank

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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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 The times, in milliseconds, taken by a computer to perform a certain task were recorded on 10 randomly chosen occasions. The times were as follows. $$\begin{array} { l l l l l l l l l l } 6.44 & 6.16 & 5.62 & 5.82 & 6.51 & 6.62 & 6.19 & 6.42 & 6.34 & 6.28 \end{array}$$ It is claimed that the median time to complete the task is 6.4 milliseconds.

1. Carry out a Wilcoxon signed-rank test at the \(5 \%\) significance level to test this claim.
2. State an underlying assumption that is made when using a Wilcoxon signed-rank test.

6 A teacher at a large college gave a mathematical puzzle to all the students. The median time taken by a random sample of 24 students to complete the puzzle was 18.0 minutes. The students were then given practice in solving puzzles. Two weeks later, the students were given another mathematical puzzle of the same type as the first. The times, in minutes, taken by the random sample of 24 students to complete this puzzle are as follows. The teacher claims that the practice has not made any difference to the average time taken to complete a puzzle of this type. Carry out a Wilcoxon signed-rank test, at the 10\% significance level, to test whether there is sufficient evidence to reject the teacher's claim.
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5 A manager claims that the lengths of the rubber tubes that his company produces have a median of 5.50 cm . The lengths, in cm , of a random sample of 11 tubes produced by this company are as follows. It is required to test at the \(10 \%\) significance level the null hypothesis that the population median length is 5.50 cm against the alternative hypothesis that the population median length is not equal to 5.50 cm . Show that both a sign test and a Wilcoxon signed-rank test give the same conclusion and state this conclusion.

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 number of students are taking a Physics course. They are assessed by a practical examination and a written examination. The marks out of 100 obtained by a random sample of 15 students in each of the examinations are as follows. Use a sign test, at the \(10 \%\) significance level, to test whether, on average, the practical examination marks are higher than the written examination marks.

3 A factory produces metal discs. The manager claims that the diameters of these discs have a median of 22.0 mm . The diameters, in mm , of a random sample of 12 discs produced by this factory are as follows. $$\begin{array} { l l l l l l l l l l l l } 22.4 & 20.9 & 22.8 & 21.5 & 23.2 & 22.9 & 23.9 & 21.7 & 19.8 & 23.6 & 22.6 & 23.0 \end{array}$$

1. Carry out a Wilcoxon signed-rank test, at the \(10 \%\) significance level, to test whether there is any evidence against the manager's claim.
2. State an assumption that is necessary for this test to be valid.

2 Metal rods produced by a certain factory are claimed to have a median breaking strength of 200 tonnes. For a random sample of 9 rods, the breaking strengths, measured in tonnes, were as follows. $$\begin{array} { l l l l l l l l l } 210 & 186 & 188 & 208 & 184 & 191 & 215 & 198 & 196 \end{array}$$ A scientist believes that the median breaking strength of metal rods produced by this factory is less than 200 tonnes.

1. Use a Wilcoxon signed-rank test, at the \(5 \%\) significance level, to test whether there is evidence to support the scientist's belief.
2. Give a reason why a Wilcoxon signed-rank test is preferable to a sign test, when both are valid.

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.

6 A school is conducting an experiment to see whether the distance that children can throw a ball increases in hot weather. On a cold day, all the children at the school were asked to throw a ball as far as possible. The distances thrown were measured and recorded. The median distance thrown by a random sample of 25 of the children was 22.0 m . The children were asked to throw the ball again on a hot day. The distances thrown by the same 25 children were measured and recorded and these distances, in m , are shown below. The teacher claims that on average the distances thrown will be further when it is hot.
Carry out a Wilcoxon signed-rank test, at the 5\% significance level, to test whether the data supports the teacher's claim.
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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).

6 A sports college keeps records of the times taken by students to run one lap of a running track. The population median time taken is 51.0 seconds. After a month of intensive training, a random sample of 22 new students run one lap of the track, giving times, in seconds, as follows. It is claimed that the intensive training has led to a decrease in the median time taken to run one lap of the track. Carry out a Wilcoxon signed-rank test, at the \(5 \%\) significance level, to test whether there is sufficient evidence to support the claim. \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-13_2726_35_97_20}
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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).

2 The manager of a large country estate is preparing to plant an area of woodland. He orders a large number of saplings (young trees) from a nursery. He selects a random sample of 12 of the saplings and measures their heights, which are as follows (in metres). $$\begin{array} { l l l l l l l l l l l l } 0.63 & 0.62 & 0.58 & 0.56 & 0.59 & 0.62 & 0.64 & 0.58 & 0.55 & 0.61 & 0.56 & 0.52 \end{array}$$

1. The manager requires that the mean height of saplings at planting is at least 0.6 metres. Carry out the usual \(t\) test to examine this, using a \(5 \%\) significance level. State your hypotheses and conclusion carefully. What assumption is needed for the test to be valid?
2. Find a \(95 \%\) confidence interval for the true mean height of saplings. Explain carefully what is meant by a \(95 \%\) confidence interval.
3. Suppose the assumption needed in part (i) cannot be justified. Identify an alternative test that the manager could carry out in order to check that the saplings meet his requirements, and state the null hypothesis for this test.

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 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.

4 A machine produces plastic strip in a continuous process. Occasionally there is a flaw at some point along the strip. The length of strip (in hundreds of metres) between successive flaws is modelled by a continuous random variable \(X\) with probability density function \(\mathrm { f } ( x ) = \frac { 18 } { ( 3 + x ) ^ { 3 } }\) for \(x > 0\). The table below gives the frequencies for 100 randomly chosen observations of \(X\). It also gives the probabilities for the class intervals using the model.

1. Examine the fit of this model to the data at the \(5 \%\) level of significance. You are given that the median length between successive flaws is 124 metres. At a later date the following random sample of ten lengths (in metres) between flaws is obtained. $$\begin{array} { l l l l l l l l l l } 239 & 77 & 179 & 221 & 100 & 312 & 52 & 129 & 236 & 42 \end{array}$$
2. Test at the \(10 \%\) level of significance whether the median length may still be assumed to be 124 metres.

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 The levels of impurity in a particular alloy were measured using a random sample of 20 specimens. The results, in suitable units, were as follows.

1. Use the sign test, at the \(5 \%\) significance level, to decide if there is evidence that the population median level of impurity is greater than 2.70 .
2. State what other test might have been used, and give one advantage and one disadvantage this other test has over the sign test.

2 Part of Helen's psychology dissertation involved the reaction times to a certain stimulus. She measured the reaction times of 30 randomly selected students, in seconds correct to 2 decimal places. The results are shown in the following stem-and-leaf diagram. Key: 18 | 3 means 1.83 seconds Helen wishes to test whether the population median time exceeds 1.80 seconds.

1. Give a reason why the Wilcoxon signed-rank test should not be used.
2. Carry out a suitable non-parametric test at the \(5 \%\) significance level.

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.

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.