5.07c Single-sample tests

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.

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

3 At a factory, two production lines are in use for making steel rods. A critical dimension is the diameter of a rod. For the first production line, it is assumed from experience that the diameters are Normally distributed with standard deviation 1.2 mm . For the second production line, it is assumed from experience that the diameters are Normally distributed with standard deviation 1.4 mm . It is desired to test whether the mean diameters for the two production lines, \(\mu _ { 1 }\) and \(\mu _ { 2 }\), are equal. A random sample of 8 rods is taken from the first production line and, independently, a random sample of 10 rods is taken from the second production line.

1. Find the acceptance region for the customary test based on the Normal distribution for the null hypothesis \(\mu _ { 1 } = \mu _ { 2 }\), against the alternative hypothesis \(\mu _ { 1 } \neq \mu _ { 2 }\), at the \(5 \%\) level of significance.
2. The sample means are found to be 25.8 mm and 24.4 mm respectively. What is the result of the test? Provide a two-sided \(99 \%\) confidence interval for \(\mu _ { 1 } - \mu _ { 2 }\). The production lines are modified so that the diameters may be assumed to be of equal (but unknown) variance. However, they may no longer be Normally distributed. A two-sided test of the equality of the population medians is required, at the \(5 \%\) significance level.
3. The diameters in independent random samples of sizes 6 and 8 are as follows, in mm .

   First production line	25.9	25.8	25.3	24.7	24.4	25.4
   Second production line	23.8	25.6	24.0	23.5	24.1	24.5	24.3	25.1

   Use an appropriate procedure to carry out the test.

1 Ten archers shot at targets with two types of bow. Their scores out of 100 are shown in the table.

1. Use the sign test, at the \(5 \%\) level of significance, to test the hypothesis that bow type \(P\) is better than bow type \(Q\).
2. Why would a Wilcoxon signed rank test, if valid, be a better test than the sign test?

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.

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.

11 A particular dietary supplement, when taken for a period of 1 month, is claimed to increase lean body mass of adults by an average of 1 kg . A researcher believes that this claim overestimates the increase. She selects a random sample of 10 adults who then each take the supplement for a month. The increases in lean body masses in kg are as follows. $$\begin{array} { l l l l l l l l l l } - 0.84 & - 0.76 & - 0.16 & 0.43 & 1.31 & 1.32 & 1.47 & 1.64 & 1.93 & 2.14 \end{array}$$ A Normal probability plot and the \(p\)-value of the Kolmogorov-Smirnov test for these data are shown below. \includegraphics[max width=\textwidth, alt={}, center]{77eabbd6-a058-457f-9601-d66f3c2db005-09_575_1485_689_242}

1. The researcher decides to carry out a hypothesis test in order to investigate the claim. Comment on the type of hypothesis test that should be used. You should refer to

4 Sheena travels to school by bus. She records the number of minutes, \(T\), that her bus is late on each of 32 days. She believes that on average \(T\) is greater than 5, and she carries out a significance test at the \(5 \%\) level.

1. State a condition needed for a Wilcoxon test to be valid in this case. Assume now that this condition is satisfied.
2. State an advantage of using a Wilcoxon test rather than a sign test.
3. Calculate the critical region for the test, in terms of a variable which should be defined.

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.

Elder twin	65	37	60	79	39	40	88
Younger twin	58	39	61	62	50	26	84

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. [6]
2. Explain the advantage in this case of a Wilcoxon test over a sign test. [1]