5.07b Sign test: and Wilcoxon signed-rank

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.

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. [7]

Jo can use either of two different routes, A or B, for her journey to school. She believes that route A has shorter journey times. She measures how long her journey takes for 17 journeys by route A and 12 journeys by route B. She ranks the 29 journeys in increasing order of time taken, and she finds that the sum of the ranks of the journeys by route B is 219.

1. Test at the 10\% significance level whether route A has shorter journey times than route B. [8]
2. State an assumption about the 29 journeys which is necessary for the conclusion of the test to be valid. [1]

To qualify as a music examiner, a trainee must listen to a series of performances by 8 randomly chosen students. An experienced examiner and the trainee must both award scores for each of the 8 performances. In order for the trainee to qualify, there must not be a significant difference between the average scores given by the experienced examiner and the trainee.

1. Explain why the Wilcoxon signed rank test is appropriate. [2]

The scores awarded are shown below.

1. 1. Carry out an appropriate Wilcoxon signed rank test on this dataset, using a 5\% significance level.
   2. What conclusion should be reached about the suitability of the trainee to qualify? [9]

In country A, the median daily caffeine intake per student who drinks coffee is 120 mg. A university professor who oversees a foreign exchange programme believes that students visiting from country B drink more coffee and therefore have a greater daily caffeine intake from coffee. On a randomly chosen day, the caffeine intake, in mg, from coffee consumption by each of 15 randomly selected students from country B is given below. 136 \quad 149 \quad 202 \quad 0 \quad 110 \quad 0 \quad 100 \quad 180 0 \quad 187 \quad 0 \quad 0 \quad 138 \quad 197 \quad 115 The professor suspects that the students with zero caffeine intake do not drink coffee, and decides to ignore those students and instead focus on the coffee-drinking students.

1. Conduct an appropriate Wilcoxon test at a significance level as close to 5\% as possible. State your conclusion in context. [8]
2. State one limitation of this investigation. [1]

A medical student is investigating two different methods, A and B, of measuring a patient's blood pressure. He believes that Method B gives, on average, a higher reading than Method A so he defines the following hypotheses. \(H_0\): There is on average no difference in the readings obtained using Methods A and B; \(H_1\): The reading obtained using Method B is on average higher than the reading obtained using Method A. He selects 10 patients at random and he measures their blood pressures using both methods. He obtains the following results.

1. Carry out an appropriate Wilcoxon signed rank test on this data set, using a 5% significance level. [6]
2. State what conclusion the medical student should reach, justifying your answer. [2]

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]