Non-parametric tests

1. A company has 800 employees.

The manager of the company is going to take a sample of 80 employees.

1. Explain how this sample can be taken using systematic sampling. The company has offices in London, Edinburgh and Cardiff. The table shows the number of employees in each city.

   City	London	Edinburgh	Cardiff
   Number of employees	430	250	120

   The president of the company is going to take a sample of 100 employees to determine the average time employees spend in front of a computer each week.
2. Explain how this sample can be taken using stratified sampling.
3. Explain an advantage of using stratified sampling rather than simple random sampling.

1. The names of the 400 employees of a company are listed alphabetically in a book.

The chairperson of the company wishes to select a sample of 8 employees.
The chairperson numbers the employees from 001 to 400

1. Describe how the list of numbers can be used to select a systematic sample of 8 employees.
2. State one disadvantage of systematic sampling in this case.
3. Write down the probability that the sample includes both the first name (employee 001) and the last name (employee 400) in the list.

4

1. Describe, with the aid of a specific example, an experimental situation for which a Latin square design is appropriate, indicating carefully the features which show that a completely randomised or randomised blocks design would be inappropriate.
2. The model for the one-way analysis of variance may be written, in a customary notation, as $$x _ { i j } = \mu + \alpha _ { i } + e _ { i j }$$ State the distributional assumptions underlying \(e _ { i j }\) in this model. What is the interpretation of the term \(\alpha _ { i }\) ?
3. An experiment for comparing 5 treatments is carried out, with a total of 20 observations. A partial one-way analysis of variance table for the analysis of the results is as follows.

   Source of variation	Sums of squares	Degrees of freedom	Mean squares	Mean square ratio
   Between treatments
   Residual	68.76
   Total	161.06

   Copy and complete the table, and carry out the appropriate test using a \(1 \%\) significance level.

1. Explain how to obtain a sample from a population using
   1. stratified sampling,
   2. quota sampling.
   Give one advantage and one disadvantage of each sampling method.

4 A wood contains a large number of mature beech trees. The diameters in centimetres of a random sample of 10 of these trees are as follows. \(\begin{array} { l l l l l l l l l l } 82.6 & 79.2 & 77.8 & 38.4 & 88.1 & 32.2 & 26.5 & 23.4 & 94.3 & 104.2 \end{array}\) A tree surgeon wants to know if the average diameter of mature beech trees in this wood is 50 cm . The tree surgeon produces a Normal probability plot for these data. \includegraphics[max width=\textwidth, alt={}, center]{4caa7409-cb32-41da-ad64-012a45753296-4_796_1230_589_230}

1. Explain why the tree surgeon should not carry out a test based on the \(t\) distribution.
2. Carry out a suitable test at the \(5 \%\) significance level to investigate whether the average diameter of mature beech trees in this wood is 50 cm .

4

1. Explain the advantages of randomisation and replication in a statistically designed experiment.
2. The usual statistical model underlying the one-way analysis of variance is given, in the usual notation, by $$x _ { i j } = \mu + \alpha _ { i } + e _ { i j }$$ where \(x _ { i j }\) denotes the \(j\) th observation on the \(i\) th treatment. Define carefully all the terms in this model and state the properties of the term that represents experimental error.
3. A trial of five fertilisers is carried out at an agricultural research station according to a completely randomised design in which each fertiliser is applied to four experimental plots of a crop (so that there are 20 experimental units altogether). The sums of squares in a one-way analysis of variance of the resulting data on yields of the crop are as follows.

   Source of variation	Sum of squares
   Between fertilisers	219.2
   Residual	304.5
   Total	523.7

   State the customary null and alternative hypotheses that are tested. Provide the degrees of freedom for each sum of squares. Hence copy and complete the analysis of variance table and carry out the test at the 5\% level.

1. (a) Explain what you understand by a census.

Each cooker produced at GT Engineering is stamped with a unique serial number. GT Engineering produces cookers in batches of 2000. Before selling them, they test a random sample of 5 to see what electric current overload they will take before breaking down.
(b) Give one reason, other than to save time and cost, why a sample is taken rather than a census.
(c) Suggest a suitable sampling frame from which to obtain this sample.
(d) Identify the sampling units.

2. (a) Explain what is meant by a simple random sample.
(b) Explain briefly how you could use a table of random numbers to select a simple random sample of size 12 from a list of the 70 junior members of a tennis club.
(c) Give an example of a situation in which you might choose to take a stratified sample and explain why.

6 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.

1. Carry out a Wilcoxon rank-sum test at the \(5 \%\) significance level to test whether the manager's claim is supported by the data.
2. Explain whether a paired sample \(t\)-test would be appropriate to test the manager's claim if earnings are normally distributed.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

5 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. 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.

2 A large number of students are taking a Physics course. They are assessed by a practical examination and a written examination. The marks out of 100 obtained by a random sample of 15 students in each of the examinations are as follows. Use a sign test, at the \(10 \%\) significance level, to test whether, on average, the practical examination marks are higher than the written examination marks.

1 A meteorologist claims that the median daily rainfall in London is 2.2 mm . A single sample sign test is to be used to test the claim, using the following hypotheses: \(\mathrm { H } _ { 0 }\) : a sample comes from a population with median 2.2, \(\mathrm { H } _ { 1 }\) : the sample does not come from a population with median 2.2.
30 randomly selected observations of daily rainfall in London are compared with 2.2, and given a '+' sign if greater than 2.2 and a '-' sign if less than 2.2. (You may assume that no data values are exactly equal to 2.2.) The test is to be carried out at the \(5 \%\) level of significance. Let the number of ' + ' signs be \(k\). Find, in terms of \(k\), the critical region for the test showing the values of any relevant probabilities.