5.07d Paired vs two-sample: selection

1. Students studying for their Mathematics GCSE are assessed by two examination papers. A teacher believes that on average the score on paper I is more than 1 mark higher than the score on paper II. To test this belief the scores of 8 randomly selected students are recorded. The results are given in the table below.

Assuming that the scores are normally distributed and stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence to support the teacher's belief.

1. In a trial for a new cough medicine, a random sample of 8 healthy patients were given steadily increasing doses of a pepper extract until they started coughing. The level of pepper that triggered the coughing was recorded. Each patient completed the trial after taking a standard cough medicine and, at a later time, after taking the new medicine. The results are given in the table below.

1. Using a suitable test, at the \(5 \%\) level of significance, state whether or not, on the basis of this trial, you would recommend using the new medicine. State your hypotheses clearly.
2. State an assumption needed to carry out this test.
   

1. A random sample of 8 people were given a new drug designed to help people sleep.

In a two-week period the drug was given for one week and a placebo (a tablet that contained no drug) was given for one week. In the first week 4 people, selected at random, were given the drug and the other 4 people were given the placebo. Those who were given the drug in the first week were given the placebo in the second week. Those who were given the placebo in the first week were given the drug in the second week. The mean numbers of hours of sleep per night for each of the people are shown in the table.

1. State one assumption that needs to be made in order to carry out a paired \(t\)-test.
2. Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not the drug increases the mean number of hours of sleep per night by more than 10 minutes. State the critical value for this test.
   

1. The Sales Manager of a large chain of convenience stores is studying the sale of lottery tickets in her stores. She randomly selects 8 of her stores. From these stores she collects data for the total sales of lottery tickets in the previous January and July. The data are shown below

1. Use a paired \(t\)-test to determine whether or not there is evidence, at the \(5 \%\) level of significance, that the mean sales of lottery tickets in this chain's stores are higher in July than in January. You should state your hypotheses and show your working clearly.
2. State what assumption the Sales Manager needs to make about the sales of lottery tickets in her stores for the test in part (a) to be valid.

1. A new diet has been designed. Its designers claim that following the diet for a month will result in a mean weight loss of more than 2 kg . In a trial, a random sample of 10 people followed the new diet for a month. Their weights, in kg, before starting the diet and their weights after following the diet for a month were recorded. The results are given in the table below.

1. Using a suitable \(t\)-test, at the \(5 \%\) level of significance, state whether or not the trial supports the designers' claim. State your hypotheses and show your working clearly.
2. State an assumption necessary for the test in part (a).
   

6. A triathlon race organiser wishes to know whether competitors who are members of a triathlon club race more frequently than competitors who are not members of a triathlon club. Six competitors from a triathlon club and six competitors who are not members of a triathlon club are selected at random. The table below shows the number of triathlon races they each entered last year.

1. Use a Mann-Whitney U test at a significance level as close to \(5 \%\) as possible to carry out the race organiser's investigation.
2. Briefly explain why a Wilcoxon signed-rank test is not appropriate in this case.

5 Alexa believes that students are equally likely to achieve the same percentage score on each of two tests, paper I and paper II. She randomly selects 8 students and gives them each paper I and paper II. The percentage scores for each paper are recorded. The following paired data are collected. Test, at the \(1 \%\) significance level, whether or not there is evidence to support Alexa's belief. State your hypotheses clearly and show your working.

8 In an experiment to investigate the effect of background music in carrying out work, ten students were each given a task. Five of the students did the task in silence and the other five did the task with background music. The scores on the tasks were as follows.

1. Use a Wilcoxon rank-sum test to test at the 10\% level whether the presence of background music affects scores.
2. A statistician suggests that the experiment is redesigned so that each student takes one task in silence and another task with background music. The differences in the test scores would then be analysed using a paired-sample method. State an advantage in redesigning the experiment in this way.

3. A statistics teacher wants to investigate whether students from the north of a county and students from the south of the same county feel similarly stressed about examinations. The teacher carries out a psychometric test on 10 randomly selected students to give a score between 0 (low stress) and 100 (high stress) to measure their stress levels before a set of examinations. The results are shown in the table below.

1. State one reason why a Mann-Whitney test is appropriate.
2. Conduct a Mann-Whitney test at a significance level as close to \(5 \%\) as possible. State your conclusion clearly.
3. How could this investigation be improved?

A motoring organisation wishes to determine whether or not the petrol consumption of two different car models A and B are the same. A trial is therefore carried out in which 6 cars of each model are given 10 litres of petrol and driven at a predetermined speed around a track until the petrol is used up. The distances travelled, in miles, are shown below Model A: \(86.3 \quad 84.2 \quad 85.8 \quad 83.1 \quad 84.7 \quad 85.3\) Model B: \(84.9 \quad 85.9 \quad 84.8 \quad 86.5 \quad 85.2 \quad 85.5\) It is proposed to use a test with significance level 5% based on the Mann-Whitney statistic \(U\).

1. State suitable hypotheses. [2]
2. Find the critical region for the test. [3]
3. Determine the value of \(U\) for the above data and state your conclusion in context. You must justify your answer. [4]

A university course was taught by two different professors. Students could choose whether to attend the lectures given by Professor \(Q\) or the lectures given by Professor \(R\). At the end of the course all the students took the same examination. The examination marks of a random sample of 30 students taught by Professor \(Q\) and a random sample of 24 students taught by Professor \(R\) were ranked. The sum of the ranks of the students taught by Professor \(Q\) was 726. Test at the 5% significance level whether there is a difference in the ranks of the students taught by the two professors. [10]

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]