Two jigsaw puzzles have the same number of pieces with identical shapes but have different pictures printed on them. One puzzle has a seaside picture and the other has a cartoon picture. A researcher believes that children will complete the cartoon puzzle more quickly. To test this belief, 10 children are randomly selected. The time taken in seconds for each child to complete each puzzle is recorded below.

(b) Show that using a paired-sample sign test at the \(5 \%\) significance level would result in the opposite conclusion to that found in part (a).

It was later discovered that the experiment had been conducted such that each child completed the seaside puzzle first followed by the cartoon puzzle.
(c) Comment on the validity of using this experiment to test the researcher's belief.

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.
(a) 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.
(b) Explain, with numerical justification, whether or not the conclusion of the test in part (a) remains the same.

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.

(b) State an assumption that you have made in carrying out the test in part (a).

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.

(a) 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.

(b) 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.

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.

(a) 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.

(b) Carry out a Wilcoxon matched-pairs signed-rank test at the \(10\%\) significance level.

A company is developing a new flavour of chocolate by varying the quantities of the ingredients. A random selection of 9 flavours of chocolate are judged by two tasters who each give marks out of 100 to each flavour of chocolate.

Carry out a Wilcoxon matched-pairs signed-rank test at the \(10 \%\) significance level to investigate whether, on average, there is a difference between marks awarded by the two tasters.

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.

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.
(b) By considering how this change affects the test statistic, explain why the conclusion of the test carried out in part (a) remains the same.

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.
(a) Carry out a Wilcoxon matched-pairs signed-rank test, at the \(5 \%\) significance level, to test whether the data supports the claim.

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.
(b) Determine whether this correction affects the conclusion of the test carried out in part (a).