5.07a Non-parametric tests: when to use

6 A biologist is studying the effect of nutrients on the heights to which plants grow. A random sample of 24 similar young plants is divided into two equal groups \(A\) and \(B\). The plants in group \(A\) are fed with nutrients and water and the plants in group \(B\) are given only water. After four weeks, the height, in cm, of each plant is measured and the results are as follows. The biologist decides to carry out a test at the \(5 \%\) significance level to test whether the nutrients have resulted in an increase in growth.

1. She carries out a Wilcoxon rank-sum test. Give a reason why this is an appropriate choice of test.
2. Carry out the Wilcoxon rank-sum test for these results.
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5 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.

1. 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.
2. Carry out a Wilcoxon matched-pairs signed-rank test at the \(10 \%\) significance level.

4 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 8 } \left( 1 + \frac { 1 } { x ^ { 2 } } \right) & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { E } ( \sqrt { X } )\).
   The random variable \(Y\) is given by \(Y = X ^ { 2 }\).
2. Find the probability density function of \(Y\).
3. Find the 40th percentile of \(Y\).

6 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 28 } \left( e ^ { \frac { 1 } { 2 } x } + 4 e ^ { - \frac { 1 } { 2 } x } \right) & 0 \leqslant x \leqslant 2 \ln 3 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the cumulative distribution function of \(X\).
   The random variable \(Y\) is defined by \(Y = e ^ { \frac { 1 } { 2 } ( X ) }\).
2. Find the probability density function of \(Y\).
3. Find the 30th percentile of \(Y\).
4. Find \(\mathrm { E } \left( Y ^ { 4 } \right)\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

1 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 6 } \left( x ^ { - \frac { 1 } { 3 } } - x ^ { - \frac { 2 } { 3 } } \right) & 1 \leqslant x \leqslant 27 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the cumulative distribution function of \(X\).
   The random variable \(Y\) is defined by \(Y = X ^ { \frac { 1 } { 3 } }\).
2. Find the probability density function of \(Y\).
3. Find the exact value of the median of \(Y\).

7 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { x } { 4 } \left( 4 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Find \(\operatorname { Var } ( \sqrt { X } )\).
   The continuous random variable \(Y\) is defined by \(Y = X ^ { 2 }\).
2. Find the probability density function of \(Y\).
3. Find the exact value of the median of \(Y\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

4 The levels of impurity in a particular alloy were measured using a random sample of 20 specimens. The results, in suitable units, were as follows.

1. Use the sign test, at the \(5 \%\) significance level, to decide if there is evidence that the population median level of impurity is greater than 2.70 .
2. State what other test might have been used, and give one advantage and one disadvantage this other test has over the sign test.

3 An engineering company buys a certain type of component from two suppliers, A and B. It is important that, on the whole, the strengths of these components are the same from both suppliers. The company can measure the strengths in its laboratory. Random samples of seven components from supplier A and five from supplier B give the following strengths, in a convenient unit. The underlying distributions of strengths are assumed to be Normal for both suppliers, with variances 2.45 for supplier A and 1.40 for supplier B.

1. Test at the \(5 \%\) level of significance whether it is reasonable to assume that the mean strengths from the two suppliers are equal.
2. Provide a two-sided 90\% confidence interval for the true mean difference.
3. Show that the test procedure used in part (i), with samples of sizes 7 and 5 and a \(5 \%\) significance level, leads to acceptance of the null hypothesis of equal means if \(- 1.556 < \bar { x } - \bar { y } < 1.556\), where \(\bar { x }\) and \(\bar { y }\) are the observed sample means from suppliers A and B . Hence find the probability of a Type II error for this test procedure if in fact the true mean strength from supplier A is 2.0 units more than that from supplier B.
4. A manager suggests that the Wilcoxon rank sum test should be used instead, comparing the median strengths for the samples of sizes 7 and 5 . Give one reason why this suggestion might be sensible and two why it might not.

3 At an agricultural research station, trials are being made of two fertilisers, A and B, to see whether they differ in their effects on the yield of a crop. Preliminary investigations have established that the underlying variances of the distributions of yields using the two fertilisers may be assumed equal. Scientific analysis of the fertilisers has suggested that fertiliser A may be inferior in that it leads, on the whole, to lower yield. A statistical analysis is being carried out to investigate this. The crop is grown in carefully controlled conditions in 14 experimental plots, 6 with fertiliser A and 8 with fertiliser B. The yields, in kg per plot, are as follows, arranged in ascending order for each fertiliser.

1. Carry out a Wilcoxon rank sum test at the \(5 \%\) significance level to examine appropriate hypotheses.
2. Carry out a \(t\) test at the \(5 \%\) significance level to examine appropriate hypotheses.
3. Goodness of fit tests based on more extensive data sets from other trials with these fertilisers have failed to reject hypotheses of underlying Normal distributions. Discuss the relative merits of the analyses in parts (i) and (ii).

3 At an agricultural research station, trials are being carried out to compare a standard variety of tomato with one that has been genetically modified (GM). The trials are concerned with the mean weight of the tomatoes and also with the aesthetic appearance of the tomatoes.

1. 1. Tomatoes of the standard and GM varieties are grown under similar conditions. The tomatoes are weighed and the data are summarised as follows.

      Variety	Sample size	Sum of weights \(( \mathrm { g } )\)	Sum of squares of weights \(\left( \mathrm { g } ^ { 2 } \right)\)
      Standard	30	3218.3	349257
      GM	26	2954.1	338691

      Carry out a test, using the Normal distribution, to investigate whether there is evidence, at the 5\% level of significance, that the two varieties of tomato differ in mean weight. State one assumption required for this test to be valid.
   2. The data in part (i) could have been used to carry out a test for the equality of means based on the \(t\) distribution. State two additional assumptions required for this test to be valid. Discuss briefly which test would be preferable in this case.
2. In order to judge whether, on the whole, GM tomatoes have a better aesthetic appearance than standard tomatoes, a trial is carried out as follows. 10 of each variety are chosen and consumer panel is asked to arrange the 20 tomatoes in order according to their appearance.
   1. State two important features of the way in which this trial should be designed. Comment briefly on how reliable the evidence from the trial is likely to be.
   2. The order in which the consumer panel arranges the tomatoes is as follows. The tomato with best appearance is listed first. \(G\) and \(S\) denote GM and standard tomatoes respectively. $$\begin{array} { c c c c c c c c c c c c c c c c c c c c } G & G & G & S & G & G & G & S & G & S & S & S & G & G & S & G & S & S & S & S \end{array}$$ Carry out an appropriate test at the \(1 \%\) level of significance.

3 A large department in a university wished to compare the standards of literacy and numeracy of its students. A random sample of 24 students was taken and sub-divided, randomly, into two groups of 12 . The students in one group took a literacy assessment (scores denoted by \(x\) ); the students in the other group took a numeracy assessment (scores denoted by \(y\) ). The two assessments were designed to give the same distributions of scores when taken by random samples from the general population. The scores obtained by the students on the two assessments are shown in the table. $$\sum x = 598 \quad \sum x ^ { 2 } = 31196 \quad \sum y = 707 \quad \sum y ^ { 2 } = 43543$$

1. Carry out an appropriate \(t\) test, at the \(5 \%\) level of significance, to compare the standards of literacy and numeracy.
2. State the distributional assumptions required for the \(t\) test to be valid. Name the test that you would use if the assumptions required for the \(t\) test are thought not to hold. State the hypotheses for this new test. Explain, in general terms, which of the two tests is more powerful, and why. A statistician at the university looked at the data and commented that a paired sample design would have been better.
3. Explain how a paired sample design would be applied in this context, and how the data would be analysed. Explain also why it would be better than the design used.

9 A rectangle of area \(A \mathrm {~m} ^ { 2 }\) has a perimeter of 20 m and each of the two shorter sides are of length \(X \mathrm {~m}\), where \(X\) is uniformly distributed between 0 and 2 .

1. Write down an expression for \(A\) in terms of \(X\), and hence show that \(A = 25 - ( X - 5 ) ^ { 2 }\).
2. Write down the probability density function of \(X\).
3. Show that the cumulative distribution function of \(A\) is $$\mathrm { F } ( a ) = \left\{ \begin{array} { l r } 0 & a < 0 , \\ \frac { 1 } { 2 } ( 5 - \sqrt { 25 - a } ) & 0 \leqslant a \leqslant 16 , \\ 1 & a > 16 . \end{array} \right.$$
4. Find the probability density function of \(A\). \section*{END OF QUESTION PAPER} \section*{OCR}

1 Ten archers shot at targets with two types of bow. Their scores out of 100 are shown in the table.

1. Use the sign test, at the \(5 \%\) level of significance, to test the hypothesis that bow type \(P\) is better than bow type \(Q\).
2. Why would a Wilcoxon signed rank test, if valid, be a better test than the sign test?

2 Low density lipoprotein (LDL) cholesterol is known as 'bad' cholesterol.
15 randomly chosen patients, each with an LDL level of 190 mg per decilitre of blood, were given one of two treatments, chosen at random. After twelve weeks their LDL levels, in mg per decilitre, were as follows. Use a Wilcoxon rank sum test, at the \(5 \%\) level of significance, to test whether the LDL levels of patients given treatment \(B\) are lower than the LDL levels of patients given treatment \(A\).

4 The heights of eleven randomly selected primary school children are measured. The results, in metres, are

1. Use a Wilcoxon rank-sum test, at the \(1 \%\) significance level, to test whether primary school girls are taller than primary school boys.
2. It is decided to repeat the test, using larger random samples. The heights of twenty girls and eighteen boys are measured. Find the greatest value of the test statistic \(W\) which will result in the conclusion that there is evidence, at the \(1 \%\) level of significance, that primary school girls are taller than primary school boys.

1

1. State briefly the circumstances under which a non-parametric test of significance should be used rather than a parametric test. The level of pollution in a river was measured at 12 randomly chosen locations. The results, in suitable units, are shown below, where higher values represent greater pollution.

   5.62

   5.73

   6.55

   6.81

   6.10

   5.75

   5.87

   6.47

   5.86

   6.26

   6.99

   5.91

2. Use a Wilcoxon signed-rank test to test whether the average pollution level in the river is more than 6.00. Use a \(5\%\) significance level.
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2 A company wishes to buy a new lathe for making chair legs. Two models of lathe, 'Allegro' and 'Vivace', were trialled. The company asked 12 randomly selected employees to make a particular type of chair leg on each machine. The times, in seconds, for each employee are shown in the table. The company wishes to test whether there is any difference in average times for the two machines.

1. State the circumstances under which a non-parametric test should be used.
2. Use two different non-parametric tests and show that they lead to different conclusions at the 5\% significance level.
3. State, with a reason, which conclusion is to be preferred.

5 In order to test whether the median salary of employees in a certain industry who had worked for three years was \(\pounds 19500\), the salaries \(x\), in thousands of pounds, of 50 randomly chosen employees were obtained.

1. The values \(| x - 19.5 |\) were calculated and ranked. No two values of \(x\) were identical and none was equal to 19.5 . The sum of the ranks corresponding to positive values of \(( x - 19.5 )\) was 867. Stating a required assumption, carry out a suitable test at the \(5 \%\) significance level.
2. If the assumption you stated in part (i) does not hold, what test could have been used?

10 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < 0 , \\ \frac { a } { 2 ^ { x } } & x \geqslant 0 , \end{cases}$$ where \(a\) is a positive constant. By expressing \(2 ^ { x }\) in the form \(\mathrm { e } ^ { k x }\), where \(k\) is a constant, show that \(X\) has a negative exponential distribution, and find the value of \(a\). State the value of \(\mathrm { E } ( X )\). The variable \(Y\) is related to \(X\) by \(Y = 2 ^ { X }\). Find the distribution function of \(Y\) and hence find its probability density function.

8 A student doing a school project wants to test a claim which she read in a newspaper that drinking a cup of tea will improve a person's arithmetic skills.
She chooses 13 students from her school and gets each of them to drink a cup of tea. She then gives each of them an arithmetic test. She knows that the average score for this test in students of the same age group as those she has chosen is 33.5.
The scores of the students she tests, arranged in ascending order, are as follows. \(\begin{array} { l l l l l l l l l l l l l } 26 & 28 & 29 & 30 & 31 & 32 & 34 & 42 & 49 & 54 & 55 & 56 & 61 \end{array}\) The student decides to use software to draw a Normal probability plot for these data, and to carry out a Normality test as shown in Fig. 8. \begin{figure}[h] \end{figure}

1. The student uses the output from the software to help in deciding on a suitable hypothesis test to use for investigating the claim about drinking tea.
   Explain what the student should conclude.
2. The student's teacher agrees with the student's choice of hypothesis test, but says that even this test may not be valid as there may be some unsatisfactory features in the student's project. Give three features that the teacher might identify as unsatisfactory.
3. Assuming that the student's procedures can be justified, carry out an appropriate test at the \(5 \%\) significance level to investigate the claim about drinking tea.

11 A particular dietary supplement, when taken for a period of 1 month, is claimed to increase lean body mass of adults by an average of 1 kg . A researcher believes that this claim overestimates the increase. She selects a random sample of 10 adults who then each take the supplement for a month. The increases in lean body masses in kg are as follows. $$\begin{array} { l l l l l l l l l l } - 0.84 & - 0.76 & - 0.16 & 0.43 & 1.31 & 1.32 & 1.47 & 1.64 & 1.93 & 2.14 \end{array}$$ A Normal probability plot and the \(p\)-value of the Kolmogorov-Smirnov test for these data are shown below. \includegraphics[max width=\textwidth, alt={}, center]{77eabbd6-a058-457f-9601-d66f3c2db005-09_575_1485_689_242}

1. The researcher decides to carry out a hypothesis test in order to investigate the claim. Comment on the type of hypothesis test that should be used. You should refer to

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.

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?

Nathan believes that shearers from Wales can shear more sheep, on average, in a given time than shearers from New Zealand. He takes a random sample of 8 shearers from Wales and 7 shearers from New Zealand. The numbers below indicate how many sheep were sheared in 45 minutes by the 15 shearers. Wales: \quad 60 \quad 53 \quad 42 \quad 38 \quad 37 \quad 36 \quad 31 \quad 28 New Zealand: \quad 39 \quad 35 \quad 27 \quad 26 \quad 17 \quad 16 \quad 15 Use a Mann-Whitney U test at the 1\% significance level to test whether Nathan is correct. You must state your hypotheses clearly and state the critical region. [7]

Alana is a PhD student researching language acquisition. She gives one group of randomly selected participants, Group A, 4 minutes to memorise 40 words that are similar in meaning. She gives a different, randomly selected group of participants, Group B, 4 minutes to memorise 40 words that are different in meaning. Alana believes that the students in Group B will do better than the students in Group A. The following results are the number of words recalled on testing the students from the two groups. Conduct a Mann-Whitney U test at a significance level as close as possible to 5\% to test Alana's belief. [6]