Wilcoxon signed-rank test (single sample)

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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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.

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.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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}
If you use the following page to complete the answer to any question, the question number must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-14_2715_33_109_2012}

1

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2. Use a Wilc sig d rah test to test wh th r th a rag p ltu in le l in th rie r is mo e th iU se a \(\%\) sig fican e le 1 .
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1

1. A continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \lambda x ^ { c } , \quad 0 \leqslant x \leqslant 1 ,$$ where \(c\) is a constant and the parameter \(\lambda\) is greater than 1 .
   1. Find \(c\) in terms of \(\lambda\).
   2. Find \(\mathrm { E } ( X )\) in terms of \(\lambda\).
   3. Show that \(\operatorname { Var } ( X ) = \frac { \lambda } { ( \lambda + 2 ) ( \lambda + 1 ) ^ { 2 } }\).
2. Every day, Godfrey does a puzzle from the newspaper and records the time taken in minutes. Last year, his median time was 32 minutes. His times for a random sample of 12 puzzles this year are as follows. $$\begin{array} { l l l l l l l l l l l l } 40 & 20 & 18 & 11 & 47 & 36 & 38 & 35 & 22 & 14 & 12 & 21 \end{array}$$ Use an appropriate test, with a 5\% significance level, to examine whether Godfrey's times this year have decreased on the whole.

2

1. 1. What is stratified sampling? Why would it be used?
   2. A local authority official wishes to conduct a survey of households in the borough. He decides to select a stratified sample of 2000 households using Council Tax property bands as the strata. At the time of the survey there are 79368 households in the borough. The table shows the numbers of households in the different tax bands.

      Tax band	A - B	C - D	E - F	G - H
      Number of households	32298	33211	9739	4120

      Calculate the number of households that the official should choose from each stratum in order to obtain his sample of 2000 households so that each stratum is represented proportionally.
2. 1. What assumption needs to be made when using a Wilcoxon single sample test?
   2. As part of an investigation into trends in local authority spending, one of the categories of expenditure considered was 'Highways and the Environment'. For a random sample of 10 local authorities, the percentages of their total expenditure spent on Highways and the Environment in 1999 and then in 2009 are shown in the table.

      Local authority	A	B	C	D	E	F	G	H	I	J
      1999	9.60	8.40	8.67	9.32	9.89	9.35	7.91	8.08	9.61	8.55
      2009	8.94	8.42	7.87	8.41	10.17	10.11	8.31	9.76	9.54	9.67

      Use a Wilcoxon test, with a significance level of \(10 \%\), to determine whether there appears to be any change to the average percentage of total expenditure spent on Highways and the Environment between 1999 and 2009.

2

1. 1. Give two reasons why an investigator might need to take a sample in order to obtain information about a population.
   2. State two requirements of a sample.
   3. Discuss briefly the advantage of the sampling being random.
2. 1. Under what circumstances might one use a Wilcoxon single sample test in order to test a hypothesis about the median of a population? What distributional assumption is needed for the test?
   2. On a stretch of road leading out of the centre of a town, highways officials have been monitoring the speed of the traffic in case it has increased. Previously it was known that the median speed on this stretch was 28.7 miles per hour. For a random sample of 12 vehicles on the stretch, the following speeds were recorded. $$\begin{array} { l l l l l l l l l l l l } 32.0 & 29.1 & 26.1 & 35.2 & 34.4 & 28.6 & 32.3 & 28.5 & 27.0 & 33.3 & 28.2 & 31.9 \end{array}$$ Carry out a test, with a \(5 \%\) significance level, to see whether the speed of the traffic on this stretch of road seems to have increased on the whole.
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1 In the past, the times for workers in a factory to complete a particular task had a known median of 7.4 minutes. Following a review, managers at the factory wish to know if the median time to complete the task has been reduced.

1. A random sample of 12 times, in minutes, gives the following results. $$\begin{array} { l l l l l l l l l l l l } 6.90 & 7.23 & 6.54 & 7.62 & 7.04 & 7.33 & 6.74 & 6.45 & 7.81 & 7.71 & 7.50 & 6.32 \end{array}$$ Carry out an appropriate test using a \(5 \%\) level of significance.
2. Some time later, a much larger random sample of times gives the following results. $$n = 80 \quad \sum x = 555.20 \quad \sum x ^ { 2 } = 3863.9031$$ Find a \(95 \%\) confidence interval for the true mean time for the task. Justify your choice of which distribution to use.
3. Describe briefly one advantage and one disadvantage of having a \(99 \%\) confidence interval instead of a \(95 \%\) confidence interval.

5 In order to test whether the median salary of employees in a certain industry who had worked for three years was \(\pounds 19500\), the salaries \(x\), in thousands of pounds, of 50 randomly chosen employees were obtained.

1. The values \(| x - 19.5 |\) were calculated and ranked. No two values of \(x\) were identical and none was equal to 19.5 . The sum of the ranks corresponding to positive values of \(( x - 19.5 )\) was 867. Stating a required assumption, carry out a suitable test at the \(5 \%\) significance level.
2. If the assumption you stated in part (i) does not hold, what test could have been used?

2 The manufacturer of a painkiller, designed to relieve headaches, claims that people taking the painkiller feel relief in at most 30 minutes, on average. A random sample of eight users of the painkiller recorded the times it took for them to feel relief from their headaches. These times, in minutes, were as follows: $$\begin{array} { l l l l l l l l } 33 & 39 & 29 & 35 & 40 & 32 & 26 & 37 \end{array}$$ Use a Wilcoxon single-sample signed-rank test at the \(5 \%\) significance level to test the manufacturer's claim, stating a necessary assumption.

1 A Wilcoxon signed-rank test is carried out at the \(5 \%\) level of significance on a random sample of size 32 . The hypotheses are \(\mathrm { H } _ { 0 } : m = m _ { 0 } , \mathrm { H } _ { 1 } : m < m _ { 0 }\) where \(m\) is the population median and \(m _ { 0 }\) is a specific numerical value. The value obtained for the test statistic \(T\) is 162 . Find the outcome of the test.

4 John regularly downloads podcasts onto his mobile phone. From past experience he knows that the average time to download one 30 -minute podcast is 12.7 s . He believes that this time has recently increased. At each of 12 randomly chosen times, he downloads a 30-minute podcast. The times in seconds to download the 12 podcasts are as follows. \(\begin{array} { l l l l l l l l l l l } 12.63 & 13.24 & 11.73 & 14.91 & 13.17 & 13.53 & 12.33 & 14.27 & 11.48 & 13.51 & 13.05 \end{array} 13.83\)

1. Given that it is not known whether the times are Normally distributed, carry out a suitable test at the \(5 \%\) significance level to investigate whether the average download time has increased.
2. What assumption is required to carry out the test in part (a)?

4 An online encyclopedia claims that the average mass of an adult European hedgehog is 720 g . In an investigation to check this average figure, the masses in grams of twelve randomly chosen adult European hedgehogs are measured and shown below.

1. What assumption is required to carry out a Wilcoxon test in this situation?
2. Given that this assumption is met, carry out a 2 -tail Wilcoxon test at the \(5 \%\) level to test whether the median mass is 720 g . You should state your hypotheses and complete the table of calculations in the Printed Answer Booklet.

5 A particular alloy of bronze is specified as containing \(11.5 \%\) copper on average. A researcher takes a random sample of 14 specimens of this bronze and undertakes an analysis of each of them. The percentages of copper are found to be as follows. The researcher uses software to draw a Normal probability plot for these data and to conduct a Kolmogorov-Smirnov test for Normality. The output is shown in Fig 5.1. \begin{figure}[h] \end{figure}

1. Comment on what the Normal probability plot and the \(p\)-value of the test suggest about the data. The researcher uses software to produce a \(99 \%\) confidence interval for the mean percentage of copper in the alloy, based on the \(t\) distribution. The output from the software is shown in Fig 5.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0de8222f-7df5-4e17-ab68-0f9d84fc615d-5_1058_615_434_726} \captionsetup{labelformat=empty} \caption{Fig 5.2}
   \end{figure}
2. State the confidence interval which the software gives, in the form \(a < \mu < b\).
3. (A) State an assumption necessary for the use of the \(t\) distribution in the construction of this confidence interval.
   (B) State whether the assumption in part (iii) (A) seems reasonable.
4. Does the confidence interval suggest that the copper content is different from \(11.5 \%\), on average? Explain your answer.
5. In the output from the software shown in Fig 5.2, SE stands for 'standard error'.
   (A) Explain what a standard error is.
   (B) Show how the standard error was calculated in this case.
6. Suggest a way in which the researcher could produce a narrower confidence interval.

6 The reaction times, in milliseconds, of all adult males in a standard experiment have a symmetrical distribution with mean and median both equal to 700 and standard deviation 125. The reaction times of a random sample of 6 international athletes are measured and the results are as follows: \(\begin{array} { l l l l l l } 702 & 631 & 540 & 714 & 575 & 480 \end{array}\) It is required to test whether international athletes have a mean reaction time which is less than 700.

1. Assume first that the reaction times of international athletes have the distribution \(\mathrm { N } \left( \mu , 125 ^ { 2 } \right)\). Test at the \(5 \%\) significance level whether \(\mu < 700\).
2. Now assume only that the distribution of the data is symmetrical, but not necessarily normal.
   1. State with a reason why a Wilcoxon test is preferable to a sign test.
   2. Use an appropriate Wilcoxon test at the \(5 \%\) significance level to test whether the median reaction time of international athletes is less than 700 .
3. Explain why the significance tests in part (a) and part (b)(ii) could produce different results.

6. A zoologist knows that the median body length of adults in a species of fire-bellied toads is 4.2 cm . The zoologist thinks he has discovered a new subspecies of fire-bellied toads. If there is sufficient evidence to suggest the median body length differs from 4.2 cm , he will continue his studies to confirm whether he has discovered a new subspecies. Otherwise, he will abandon his studies on fire-bellied toads. The lengths of 10 randomly selected adult toads from the group being investigated are given below. $$\begin{array} { l l l l l l l l l l } 5 \cdot 0 & 3 \cdot 2 & 4 \cdot 9 & 4 \cdot 0 & 3 \cdot 3 & 4 \cdot 2 & 6 \cdot 1 & 4 \cdot 3 & 4 \cdot 8 & 5 \cdot 9 \end{array}$$ Carry out a suitable Wilcoxon signed rank test at a significance level as close to \(1 \%\) as possible and give your conclusion in context.