Students of the same age from two schools, school \(A\) and school \(B\), take a large number of quizzes throughout the year and are each awarded a mark out of 1000 . The marks of 123 students in school \(A\) and 147 students in school \(B\) are ranked from lowest (rank 1) to highest (rank 270). The sum of the ranks of the students from school \(A\) is 15355 . (a) Carry out a Wilcoxon rank-sum test at the \(5 \%\) significance level to investigate whether there is a difference in average marks between the students in school \(A\) and school \(B\). (b) State an assumption that is required for the Wilcoxon rank-sum test to be valid.

A researcher claims that older people take longer to react to a sudden loud noise than younger people. To investigate this, the researcher randomly selects 6 people over 50 years old and 8 people under 25 years old and records their reaction times, in milliseconds, to a sudden loud noise. The reaction times are as follows.

Carry out a Wilcoxon rank-sum test at the \(5 \%\) significance level to test the researcher's claim.

A college uses two assessments, \(X\) and \(Y\), when interviewing applicants for research posts at the college. These assessments have been used for a large number of applicants this year.

The scores for a random sample of 9 applicants who took assessment \(X\) are as follows.

The scores for a random sample of 10 applicants who took assessment \(Y\) are as follows.

The interviewer believes that the population median score from assessment \(X\) is lower than the population median score from assessment \(Y\).

Carry out a Wilcoxon rank-sum test, at the \(1 \%\) significance level, to test whether the interviewer's belief is supported by the data.

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 \(Y\).
(a) Carry out a Wilcoxon rank-sum test at the \(5 \%\) significance level to test whether the manager's claim is supported by the data.
(b) 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.

(c) 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.

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.

(a) Carry out a Wilcoxon rank-sum test at the 5% significance level to test whether the manager's claim is supported by the data.

(b) Explain whether a paired sample \(t\)-test would be appropriate to test the manager's claim if earnings are normally distributed.

The blood cholesterol levels, measured in suitable units, of a random sample of 11 women and a random sample of 12 men are shown below.

Carry out a Wilcoxon rank-sum test, at the \(5 \%\) significance level, to test whether, on average, there is a difference in cholesterol levels between women and men.

Applicants for a particular college take a written test when they attend for interview. There are two different written tests, \(A\) and \(B\), and each applicant takes one or the other. The interviewer wants to determine whether the medians of the distribution of marks obtained in the two tests are equal. The marks obtained by a random sample of 8 applicants who took test \(A\) and a random sample of 8 applicants who took test \(B\) are as follows.

The interviewer considers using the given information to carry out a paired sample \(t\)-test to determine whether there is a difference in the population means for the two tests.
(b) Give two reasons why it is not appropriate to use this test.

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.

(a) She carries out a Wilcoxon rank-sum test. Give a reason why this is an appropriate choice of test.

(b) Carry out the Wilcoxon rank-sum test for these results.