5.07e Test medians

6 The manager of a technology company \(A\) claims that his employees earn more per year than the employees at technology company \(B\). The amounts earned per year, in hundreds of dollars, by a random sample of 12 employees from company \(A\) and an independent random sample of 12 employees from company \(B\) are shown below.

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. Explain whether a paired sample \(t\)-test would be appropriate to test the manager's claim if earnings are normally distributed.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

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

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.

8 A student doing a school project wants to test a claim which she read in a newspaper that drinking a cup of tea will improve a person's arithmetic skills.
She chooses 13 students from her school and gets each of them to drink a cup of tea. She then gives each of them an arithmetic test. She knows that the average score for this test in students of the same age group as those she has chosen is 33.5.
The scores of the students she tests, arranged in ascending order, are as follows. \(\begin{array} { l l l l l l l l l l l l l } 26 & 28 & 29 & 30 & 31 & 32 & 34 & 42 & 49 & 54 & 55 & 56 & 61 \end{array}\) The student decides to use software to draw a Normal probability plot for these data, and to carry out a Normality test as shown in Fig. 8. \begin{figure}[h] \end{figure}

1. The student uses the output from the software to help in deciding on a suitable hypothesis test to use for investigating the claim about drinking tea.
   Explain what the student should conclude.
2. The student's teacher agrees with the student's choice of hypothesis test, but says that even this test may not be valid as there may be some unsatisfactory features in the student's project. Give three features that the teacher might identify as unsatisfactory.
3. Assuming that the student's procedures can be justified, carry out an appropriate test at the \(5 \%\) significance level to investigate the claim about drinking tea.

7 An environmental investigator wants to check whether the level of selenium in carrots in fields near a mine is different from the usual level in the country, which is \(9.4 \mathrm { ng } / \mathrm { g }\) (nanograms per gram). She takes a random sample of 10 carrots from fields near the mine and measures the selenium level of each of them in \(\mathrm { ng } / \mathrm { g }\), with results as follows. \(\begin{array} { l l l l l l l l l l } 6.20 & 10.72 & 11.42 & 16.32 & 15.33 & 10.56 & 8.83 & 9.21 & 7.78 & 14.32 \end{array}\)

1. Find estimates of each of the following.
   The investigator produces a Normal probability plot and carries out a Kolmogorov-Smirnov test for these data as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{bab116b3-6e5f-44db-ac86-670e4040d649-06_583_1499_959_242}
2. Comment on what the Normal probability plot and the \(p\)-value of the test suggest about the data.
3. State the null hypothesis for the Kolmogorov-Smirnov test for Normality.
4. In this question you must show detailed reasoning. Carry out a test at the \(5 \%\) significance level to investigate whether the mean selenium level in carrots from fields near the mine is different from \(9.4 \mathrm { ng } / \mathrm { g }\).
5. If the \(p\)-value of the Kolmogorov-Smirnov test for Normality had been 0.007, explain what procedure you could have used to investigate the selenium level in carrots from fields near the mine.

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.