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.
| 22.4 | 20.9 | 22.8 | 21.5 | 23.2 | 22.9 | 23.9 | 21.7 | 19.8 | 23.6 | 22.6 | 23.0 |
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.
| 21.2 | 23.5 | 22.9 | 18.6 | 19.4 |
| 22.1 | 26.5 | 20.2 | 25.7 | 20.6 |
| 22.3 | 17.4 | 22.2 | 27.0 | 23.9 |
| 28.2 | 22.6 | 27.2 | 23.0 | 23.7 |
| 19.8 | 22.7 | 23.3 | 21.5 | 24.3 |
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.
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.
| 18.2 | 17.5 | 16.4 | 15.1 | 20.5 | 26.5 | 19.2 | 23.2 |
| 17.9 | 18.8 | 25.8 | 19.9 | 17.7 | 16.2 | 17.3 | 16.6 |
| 17.1 | 20.1 | 20.3 | 12.6 | 16.0 | 21.4 | 22.7 | 18.4 |
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.
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.
| 5.56 | 5.45 | 5.47 | 5.58 | 5.54 | 5.52 | 5.60 | 5.35 | 5.59 | 5.51 | 5.62 |
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.
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.
| 6.44 | 6.16 | 5.62 | 5.82 | 6.51 | 6.62 | 6.19 | 6.42 | 6.34 | 6.28 |
It is claimed that the median time to complete the task is 6.4 milliseconds.
(a) Carry out a Wilcoxon signed-rank test at the \(5 \%\) significance level to test this claim.
(b) State an underlying assumption that is made when using a Wilcoxon signed-rank test.
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.
| 210 | 186 | 188 | 208 | 184 | 191 | 215 | 198 | 196 |
A scientist believes that the median breaking strength of metal rods produced by this factory is less than 200 tonnes.
(a) Use a Wilcoxon signed-rank test, at the \(5 \%\) significance level, to test whether there is evidence to support the scientist's belief.
(b) Give a reason why a Wilcoxon signed-rank test is preferable to a sign test, when both are valid.
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.
51.3 | 52.0 | 53.4 | 49.2 | 49.3 | 51.1 | 52.2 | 47.2 |
53.0 | 48.5 | 49.4 | 50.3 | 50.8 | 51.6 | 49.1 | 52.3 |
51.8 | 52.4 | 47.9 | 48.9 | 50.6 | 51.9 |
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.
The level of sound produced by a particular type of machine was measured for a random sample of 11 such machines. The results, in suitable units, are shown below.
| Machine | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | \(J\) | \(K\) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Sound level | 7.66 | 8.48 | 8.21 | 7.98 | 8.01 | 7.77 | 8.25 | 8.11 | 8.03 | 8.16 | 7.92 |
(a) Use a Wilcoxon signed-rank test to test whether the average sound level produced by this type of machine is more than 8.00. Use a \(5 \%\) significance level.
(b) Give a reason why a Wilcoxon signed-rank test may be more appropriate than a \(t\)-test in this case.
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.
Child | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | \(J\) |
|---|---|---|---|---|---|---|---|---|---|---|
Seaside | 182 | 130 | 193 | 181 | 192 | 204 | 184 | 192 | 180 | 189 |
Cartoon | 161 | 111 | 195 | 159 | 202 | 200 | 168 | 165 | 145 | 160 |
(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.
Company | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | \(J\) | \(K\) | \(L\) | \(M\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Number in 2018 | 104 | 19 | 126 | 234 | 970 | 514 | 35 | 149 | 429 | 12 | 86 | 304 | 1104 |
Number in 2022 | 106 | 24 | 127 | 228 | 1012 | 525 | 32 | 156 | 449 | 24 | 78 | 294 | 1154 |
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.
Student | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) |
|---|---|---|---|---|---|---|---|---|---|
Paper 1 | 46 | 73 | 55 | 64 | 86 | 42 | 66 | 68 | 60 |
Paper 2 | 41 | 66 | 61 | 63 | 90 | 40 | 58 | 42 | 70 |
(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.
| Student | A | B | C | D | E | F | G | H | I | J | K | L | M | N |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Judge 1 | 79 | 54 | 63 | 74 | 69 | 52 | 50 | 57 | 55 | 42 | 63 | 55 | 56 | 48 |
| Judge 2 | 75 | 62 | 60 | 72 | 76 | 41 | 31 | 51 | 45 | 55 | 49 | 50 | 65 | 36 |
(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.
| Employee | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | \(J\) | \(K\) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Uniform \(X\) | 82 | 74 | 42 | 59 | 60 | 73 | 94 | 98 | 62 | 36 | 50 |
| Uniform \(Y\) | 78 | 75 | 63 | 56 | 67 | 82 | 99 | 90 | 72 | 48 | 61 |
(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.
Chocolate | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) |
|---|---|---|---|---|---|---|---|---|---|
Taster 1 | 72 | 86 | 75 | 92 | 98 | 79 | 87 | 60 | 62 |
Taster 2 | 84 | 72 | 74 | 95 | 85 | 87 | 82 | 75 | 68 |
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.
Student | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | \(J\) | \(K\) | \(L\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
Judge 1 | 62 | 74 | 52 | 48 | 68 | 55 | 56 | 64 | 37 | 70 | 81 | 59 |
Judge 2 | 65 | 70 | 47 | 49 | 76 | 74 | 67 | 54 | 50 | 77 | 72 | 75 |
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.
Student | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | \(J\) | \(K\) | \(L\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
Teacher 1 | 36 | 38 | 40 | 36 | 22 | 34 | 45 | 44 | 48 | 35 | 28 | 30 |
Teacher 2 | 38 | 42 | 32 | 41 | 32 | 41 | 42 | 50 | 36 | 44 | 42 | 41 |
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).
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.
Over 50 | 198 | 212 | 217 | 229 | 235 | 242 | ||
|---|---|---|---|---|---|---|---|---|
Under 25 | 178 | 181 | 183 | 192 | 203 | 209 | 223 | 231 |
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.
| 21.4 | 24.6 | 25.3 | 22.7 | 20.8 | 21.5 | 22.9 | 21.3 | 22.3 |
The scores for a random sample of 10 applicants who took assessment \(Y\) are as follows.
| 20.9 | 23.5 | 24.8 | 21.9 | 23.4 | 24.0 | 23.8 | 24.1 | 25.1 | 25.8 |
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.
Machine \(X\) | 4.0 | 4.6 | 4.7 | 4.8 | 5.0 | 5.2 | 5.6 | 5.8 | |
|---|---|---|---|---|---|---|---|---|---|
Machine \(Y\) | 4.5 | 4.9 | 5.1 | 5.3 | 5.4 | 5.7 | 5.9 | 6.3 | 6.4 |
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.