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OCR MEI S3 2008 June Q3

A tea grower is testing two types of plant for the weight of tea they produce. A trial is set up in which each type of plant is grown at each of 8 sites. The total weight, in grams, of tea leaves harvested from each plant is measured and shown below.
Site A B C D E F G H
Type I 225.2 268.9 303.6 244.1 230.6 202.7 242.1 247.5
Type II 215.2 242.1 260.9 241.7 245.5 204.7 225.8 236.0
1. The grower intends to perform a $t$ test to examine whether there is any difference in the mean yield of the two types of plant. State the hypotheses he should use and also any necessary assumption.
2. Carry out the test using a $5 \%$ significance level.
The tea grower deals with many types of tea and employs tasters to rate them. The tasters do this by giving each tea a score out of 100. The tea grower wishes to compare the scores given by two of the tasters. Their scores for a random selection of 10 teas are as follows.
Tea Q R S T U V W X Y Z
Taster 1 69 79 85 63 81 65 85 86 89 77
Taster 2 74 75 99 66 75 64 96 94 96 86
Use a Wilcoxon test to examine, at the $5 \%$ level of significance, whether it appears that, on the whole, the scores given to teas by these two tasters differ.

OCR MEI S3 2008 June Q4

A researcher is investigating the feeding habits of bees. She sets up a feeding station some distance from a beehive and, over a long period of time, records the numbers of bees arriving each minute. For a random sample of 100 one-minute intervals she obtains the following results.
Number of bees 0 1 2 3 4 5 6 7 $\geqslant 8$
Number of intervals 6 16 19 18 17 14 6 4 0
1. Show that the sample mean is 3.1 and find the sample variance. Do these values support the possibility of a Poisson model for the number of bees arriving each minute? Explain your answer.
2. Use the mean in part (i) to carry out a test of the goodness of fit of a Poisson model to the data.
The researcher notes the length of time, in minutes, that each bee spends at the feeding station. The times spent are assumed to be Normally distributed. For a random sample of 10 bees, the mean is found to be 1.465 minutes and the standard deviation is 0.3288 minutes. Find a $95 \%$ confidence interval for the overall mean time.

Find the $x$-coordinate of P .
Show that Q lies on the line $y = x$.
Differentiate $x \sin 3 x$. Hence prove that the line $y = x$ touches the curve at Q .
Show that the area of the region bounded by the curve and the line $y = x$ is $\frac { 1 } { 72 } \left( \pi ^ { 2 } - 8 \right)$.

OCR MEI C3 2005 June Q9

9 The function $\mathrm { f } ( x ) = \ln \left( 1 + x ^ { 2 } \right)$ has domain $- 3 \leqslant x \leqslant 3$.
Fig. 9 shows the graph of $y = \mathrm { f } ( x )$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3efea8db-9fa1-47a8-89b8-e4888f87a313-4_540_943_477_550} \captionsetup{labelformat=empty} \caption{Fig. 9}

\end{figure}

Show algebraically that the function is even. State how this property relates to the shape of the curve.
Find the gradient of the curve at the point $\mathrm { P } ( 2 , \ln 5 )$.
Explain why the function does not have an inverse for the domain $- 3 \leqslant x \leqslant 3$. The domain of $\mathrm { f } ( x )$ is now restricted to $0 \leqslant x \leqslant 3$. The inverse of $\mathrm { f } ( x )$ is the function $\mathrm { g } ( x )$.
Sketch the curves $y = \mathrm { f } ( x )$ and $y = \mathrm { g } ( x )$ on the same axes. State the domain of the function $\mathrm { g } ( x )$. Show that $\mathrm { g } ( x ) = \sqrt { \mathrm { e } ^ { x } - 1 }$.
Differentiate $\mathrm { g } ( x )$. Hence verify that $\mathrm { g } ^ { \prime } ( \ln 5 ) = 1 \frac { 1 } { 4 }$. Explain the connection between this result and your answer to part (ii).

OCR MEI S4 2006 June Q1

1 A parcel is weighed, independently, on two scales. The weights are given by the random variables $W _ { 1 }$ and $W _ { 2 }$ which have underlying Normal distributions as follows. $$W _ { 1 } \sim \mathrm {~N} \left( \mu , \sigma _ { 1 } ^ { 2 } \right) , \quad W _ { 2 } \sim \mathrm {~N} \left( \mu , \sigma _ { 2 } ^ { 2 } \right) ,$$ where $\mu$ is an unknown parameter and $\sigma _ { 1 } ^ { 2 }$ and $\sigma _ { 2 } ^ { 2 }$ are taken as known.

Show that the maximum likelihood estimator of $\mu$ is $$\hat { \mu } = \frac { \sigma _ { 2 } ^ { 2 } } { \sigma _ { 1 } ^ { 2 } + \sigma _ { 2 } ^ { 2 } } W _ { 1 } + \frac { \sigma _ { 1 } ^ { 2 } } { \sigma _ { 1 } ^ { 2 } + \sigma _ { 2 } ^ { 2 } } W _ { 2 } .$$ [You may quote the probability density function of the general Normal distribution from page 9 in the MEI Examination Formulae and Tables Booklet (MF2).]
Show that $\hat { \mu }$ is an unbiased estimator of $\mu$.
Obtain the variance of $\hat { \mu }$.
A simpler estimator $T = \frac { 1 } { 2 } \left( W _ { 1 } + W _ { 2 } \right)$ is proposed. Write down the variance of $T$ and hence show that the relative efficiency of $T$ with respect to $\hat { \mu }$ is $$y = \left( \frac { 2 \sigma _ { 1 } \sigma _ { 2 } } { \sigma _ { 1 } ^ { 2 } + \sigma _ { 2 } ^ { 2 } } \right) ^ { 2 }$$
Show that $y \leqslant 1$ for all values of $\sigma _ { 1 } ^ { 2 }$ and $\sigma _ { 2 } ^ { 2 }$. Explain why this means that $\hat { \mu }$ is preferable to $T$ as an estimator of $\mu$.

OCR MEI S4 2006 June Q2

8 marks

2 [In this question, you may use the result $\int _ { 0 } ^ { \infty } u ^ { m } \mathrm { e } ^ { - u } \mathrm {~d} u = m$ ! for any non-negative integer $m$.]
The random variable $X$ has probability density function $$\mathrm { f } ( x ) = \begin{cases} \frac { \lambda ^ { k + 1 } x ^ { k } \mathrm { e } ^ { - \lambda x } } { k ! } , & x > 0
0 , & \text { elsewhere } \end{cases}$$ where $\lambda > 0$ and $k$ is a non-negative integer.

Show that the moment generating function of $X$ is $\left( \frac { \lambda } { \lambda - \theta } \right) ^ { k + 1 }$.
The random variable $Y$ is the sum of $n$ independent random variables each distributed as $X$. Find the moment generating function of $Y$ and hence obtain the mean and variance of $Y$. [8]
State the probability density function of $Y$.
For the case $\lambda = 1 , k = 2$ and $n = 5$, it may be shown that the definite integral of the probability density function of $Y$ between limits 10 and $\infty$ is 0.9165 . Calculate the corresponding probability that would be given by a Normal approximation and comment briefly.

OCR MEI S4 2006 June Q3

3 The human resources department of a large company is investigating two methods, A and B, for training employees to carry out a certain complicated and intricate task.

Two separate random samples of employees who have not previously performed the task are taken. The first sample is of size 10 ; each of the employees in it is trained by method A. The second sample is of size 12; each of the employees in it is trained by method B. After completing the training, the time for each employee to carry out the task is measured, in controlled conditions. The times are as follows, in minutes.
Employees trained by method A: 35.2 47.8 25.8 38.0 53.6 31.0 33.9
35.4 21.6 42.5
Employees trained by method B: 43.0 57.5 68.6 20.9 31.4 44.9 62.8
27.6 41.8 46.1 39.8 61.6
Stating appropriate assumptions concerning the underlying populations, use a $t$ test at the $5 \%$ significance level to examine whether either training method is better in respect of leading, on the whole, to a lower time to carry out the task.
A further trial of method B is carried out to see if the performance of experienced and skilled workers can be improved by re-training them. A random sample of 8 such workers is taken. The times in minutes, under controlled conditions, for each worker to carry out the task before and after re-training are as follows.
Worker $W _ { 1 }$ $W _ { 2 }$ $W _ { 3 }$ $W _ { 4 }$ $W _ { 5 }$ $W _ { 6 }$ $W _ { 7 }$ $W _ { 8 }$
Time before 32.6 28.5 22.9 27.6 34.9 28.8 34.2 31.3
Time after 26.2 24.1 19.0 28.6 29.3 20.0 36.0 19.2
Stating an appropriate assumption, use a $t$ test at the $5 \%$ significance level to examine whether the re-training appears, on the whole, to lead to a lower time to carry out the task.
Explain how the test procedure in part (ii) is enhanced by designing it as a paired comparison.

OCR MEI S4 2006 June Q4

12 marks

4 An experiment is carried out to compare five industrial paints, A, B, C, D, E, that are intended to be used to protect exterior surfaces in polluted urban environments. Five different types of surface (I, II, III, IV, V) are to be used in the experiment, and five specimens of each type of surface are available. Five different external locations ( $1,2,3,4,5$ ) are used in the experiment. The paints are applied to the specimens of the surfaces which are then left in the locations for a period of six months. At the end of this period, a "score" is given to indicate how effective the paint has been in protecting the surface.

Name a suitable experimental design for this trial and give an example of an experimental layout. Initial analysis of the data indicates that any differences between the types of surface are negligible, as also are any differences between the locations. It is therefore decided to analyse the data by one-way analysis of variance.
State the usual model, including the accompanying distributional assumptions, for the one-way analysis of variance. Interpret the terms in the model.
The data for analysis are as follows. Higher scores indicate better performance.
Paint A Paint B Paint C Paint D Paint E
64 66 59 65 64
58 68 56 78 52
73 76 69 69 56
60 70 60 72 61
67 71 63 71 58
[The sum of these data items is 1626 and the sum of their squares is 106838 .]
Construct the usual one-way analysis of variance table. Carry out the appropriate test, using a 5\% significance level. Report briefly on your conclusions.
[0pt] [12]

OCR MEI S4 2007 June Q1

1 The random variable $X$ has the continuous uniform distribution with probability density function $$\mathrm { f } ( x ) = \frac { 1 } { \theta } , \quad 0 \leqslant x \leqslant \theta$$ where $\theta ( \theta > 0 )$ is an unknown parameter.
A random sample of $n$ observations from $X$ is denoted by $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$, with sample mean $\bar { X } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } X _ { i }$.

Show that $2 \bar { X }$ is an unbiased estimator of $\theta$.
Evaluate $2 \bar { X }$ for a case where, with $n = 5$, the observed values of the random sample are $0.4,0.2$, 1.0, 0.1, 0.6. Hence comment on a disadvantage of $2 \bar { X }$ as an estimator of $\theta$. For a general random sample of size $n$, let $Y$ represent the sample maximum, $Y = \max \left( X _ { 1 } , X _ { 2 } , \ldots , X _ { n } \right)$. You are given that the probability density function of $Y$ is $$g ( y ) = \frac { n y ^ { n - 1 } } { \theta ^ { n } } , \quad 0 \leqslant y \leqslant \theta$$
An estimator $k Y$ is to be used to estimate $\theta$, where $k$ is a constant to be chosen. Show that the mean square error of $k Y$ is $$k ^ { 2 } \mathrm { E } \left( Y ^ { 2 } \right) - 2 k \theta \mathrm { E } ( Y ) + \theta ^ { 2 }$$ and hence find the value of $k$ for which the mean square error is minimised.
Comment on whether $k Y$ with the value of $k$ found in part (iii) suffers from the disadvantage identified in part (ii).

OCR MEI S4 2007 June Q2

2 The random variable $X$ has the binomial distribution with parameters $n$ and $p$, i.e. $X \sim \mathrm {~B} ( n , p )$.

Show that the probability generating function of $X$ is $\mathrm { G } ( t ) = ( q + p t ) ^ { n }$, where $q = 1 - p$.
Hence obtain the mean $\mu$ and variance $\sigma ^ { 2 }$ of $X$.
Write down the mean and variance of the random variable $Z = \frac { X - \mu } { \sigma }$.
Write down the moment generating function of $X$ and use the linear transformation result to show that the moment generating function of $Z$ is $$\mathrm { M } _ { Z } ( \theta ) = \left( q \mathrm { e } ^ { - \frac { p \theta } { \sqrt { n p q } } } + p \mathrm { e } ^ { \frac { q \theta } { \sqrt { n p q } } } \right) ^ { n } .$$
By expanding the exponential terms in $\mathrm { M } _ { Z } ( \theta )$, show that the limit of $\mathrm { M } _ { Z } ( \theta )$ as $n \rightarrow \infty$ is $\mathrm { e } ^ { \theta ^ { 2 } / 2 }$. You may use the result $\lim _ { n \rightarrow \infty } \left( 1 + \frac { y + \mathrm { f } ( n ) } { n } \right) ^ { n } = \mathrm { e } ^ { y }$ provided $\mathrm { f } ( n ) \rightarrow 0$ as $n \rightarrow \infty$.
What does the result in part (v) imply about the distribution of $Z$ as $n \rightarrow \infty$ ? Explain your reasoning briefly.
What does the result in part (vi) imply about the distribution of $X$ as $n \rightarrow \infty$ ?

OCR MEI S4 2007 June Q3

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.

Supplier A	25.8	27.4	26.2	23.5	28.3	26.4	27.2
Supplier B	25.6	24.9	23.7	25.8	26.9

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.

Test at the $5 \%$ level of significance whether it is reasonable to assume that the mean strengths from the two suppliers are equal.
Provide a two-sided 90\% confidence interval for the true mean difference.
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.
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.

OCR MEI S4 2007 June Q4

4 An agricultural company conducts a trial of five fertilisers (A, B, C, D, E) in an experimental field at its research station. The fertilisers are applied to plots of the field according to a completely randomised design. The yields of the crop from the plots, measured in a standard unit, are analysed by the one-way analysis of variance, from which it appears that there are no real differences among the effects of the fertilisers. A statistician notes that the residual mean square in the analysis of variance is considerably larger than had been anticipated from knowledge of the general behaviour of the crop, and therefore suspects that there is some inadequacy in the design of the trial.

Explain briefly why the statistician should be suspicious of the design.
Explain briefly why an inflated residual leads to difficulty in interpreting the results of the analysis of variance, in particular that the null hypothesis is more likely to be accepted erroneously. Further investigation indicates that the soil at the west side of the experimental field is naturally more fertile than that at the east side, with a consistent 'fertility gradient' from west to east.
What experimental design can accommodate this feature? Provide a simple diagram of the experimental field indicating a suitable layout. The company decides to conduct a new trial in its glasshouse, where experimental conditions can be controlled so that a completely randomised design is appropriate. The yields are as follows.
Fertiliser A Fertiliser B Fertiliser C Fertiliser D Fertiliser E
23.6 26.0 18.8 29.0 17.7
18.2 35.3 16.7 37.2 16.5
32.4 30.5 23.0 32.6 12.8
20.8 31.4 28.3 31.4 20.4
[The sum of these data items is 502.6 and the sum of their squares is 13610.22 .]
Construct the usual one-way analysis of variance table. Carry out the appropriate test, using a $5 \%$ significance level. Report briefly on your conclusions.
State the assumptions about the distribution of the experimental error that underlie your analysis in part (iv).

OCR MEI S4 2008 June Q1

1 The random variable $X$ has the Poisson distribution with parameter $\theta$ so that its probability function is $$\mathrm { P } ( X = x ) = \frac { \mathrm { e } ^ { - \theta } \theta ^ { x } } { x ! } , \quad x = 0,1,2 , \ldots$$ where $\theta ( \theta > 0 )$ is unknown. A random sample of $n$ observations from $X$ is denoted by $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$.

Find $\hat { \theta }$, the maximum likelihood estimator of $\theta$. The value of $\mathrm { P } ( X = 0 )$ is denoted by $\lambda$.
Write down an expression for $\lambda$ in terms of $\theta$.
Let $R$ denote the number of observations in the sample with value zero. By considering the binomial distribution with parameters $n$ and $\mathrm { e } ^ { - \theta }$, write down $\mathrm { E } ( R )$ and $\operatorname { Var } ( R )$. Deduce that the observed proportion of observations in the sample with value zero, denoted by $\tilde { \lambda }$, is an unbiased estimator of $\lambda$ with variance $\frac { \mathrm { e } ^ { - \theta } \left( 1 - \mathrm { e } ^ { - \theta } \right) } { n }$.
In large samples, the variance of the maximum likelihood estimator of $\lambda$ may be taken as $\frac { \theta \mathrm { e } ^ { - 2 \theta } } { n }$. Use this and the appropriate result from part (iii) to show that the relative efficiency of $\tilde { \lambda }$ with respect to the maximum likelihood estimator is $\frac { \theta } { \mathrm { e } ^ { \theta } - 1 }$. Show that this expression is always less than 1 . Show also that it is near 1 if $\theta$ is small and near 0 if $\theta$ is large.

OCR MEI S4 2008 June Q2

2 Independent trials, on each of which the probability of a 'success' is $p ( 0 < p < 1 )$, are being carried out. The random variable $X$ counts the number of trials up to and including that on which the first success is obtained. The random variable $Y$ counts the number of trials up to and including that on which the $n$th success is obtained.

Write down an expression for $\mathrm { P } ( X = x )$ for $x = 1,2 , \ldots$. Show that the probability generating function of $X$ is $$\mathrm { G } ( t ) = p t ( 1 - q t ) ^ { - 1 }$$ where $q = 1 - p$, and hence that the mean and variance of $X$ are $$\mu = \frac { 1 } { p } \quad \text { and } \quad \sigma ^ { 2 } = \frac { q } { p ^ { 2 } }$$ respectively.
Explain why the random variable $Y$ can be written as $$Y = X _ { 1 } + X _ { 2 } + \ldots + X _ { n }$$ where the $X _ { i }$ are independent random variables each distributed as $X$. Hence write down the probability generating function, the mean and the variance of $Y$.
State an approximation to the distribution of $Y$ for large $n$.
The aeroplane used on a certain flight seats 140 passengers. The airline seeks to fill the plane, but its experience is that not all the passengers who buy tickets will turn up for the flight. It uses the random variable $Y$ to model the situation, with $p = 0.8$ as the probability that a passenger turns up. Find the probability that it needs to sell at least 160 tickets to get 140 passengers who turn up. Suggest a reason why the model might not be appropriate.

OCR MEI S4 2008 June Q3

Explain the meaning of the following terms in the context of hypothesis testing: Type I error, Type II error, operating characteristic. A machine fills salt containers that will be sold in shops. The containers are supposed to contain 750 g of salt. The machine operates in such a way that the amount of salt delivered to each container is a Normally distributed random variable with standard deviation 20 g . The machine should be calibrated in such a way that the mean amount delivered, $\mu$, is 750 g . Each hour, a random sample of 9 containers is taken from the previous hour's output and the sample mean amount of salt is determined. If this is between 735 g and 765 g , the previous hour's output is accepted. If not, the previous hour's output is rejected and the machine is recalibrated.
Find the probability of rejecting the previous hour's output if the machine is properly calibrated. Comment on your result.
Find the probability of accepting the previous hour's output if $\mu = 725 \mathrm {~g}$. Comment on your result.
Obtain an expression for the operating characteristic of this testing procedure in terms of the cumulative distribution function $\Phi ( z )$ of the standard Normal distribution. Evaluate the operating characteristic for the following values (in g) of $\mu$ : 720, 730, 740, 750, 760, 770, 780.

OCR MEI S4 2008 June Q4

State the usual model, including the accompanying distributional assumptions, for the one-way analysis of variance. Interpret the terms in the model.
An examinations authority is considering using an external contractor for the typesetting and printing of its examination papers. Four contractors are being investigated. A random sample of 20 examination papers over the entire range covered by the authority is selected and 5 are allocated at random to each contractor for preparation. The authority carefully checks the printed papers for errors and assigns a score to each to indicate the overall quality (higher scores represent better quality). The scores are as follows.
Contractor A Contractor B Contractor C Contractor D
41 54 56 41
49 45 45 36
50 50 54 46
44 50 50 38
56 47 49 35
[The sum of these data items is 936 and the sum of their squares is 44544 .]
Construct the usual one-way analysis of variance table. Carry out the appropriate test, using a $5 \%$ significance level. Report briefly on your conclusions.
The authority thinks that there might be differences in the ways the contractors cope with the preparation of examination papers in different subject areas. For this purpose, the subject areas are broadly divided into mathematics, sciences, languages, humanities, and others. The authority wishes to design a further investigation, ensuring that each of these subject areas is covered by each contractor. Name the experimental design that should be used and describe briefly the layout of the investigation.

OCR MEI S4 2010 June Q1

1 The random variable $X$ has probability density function $$\mathrm { f } ( x ) = \frac { x \mathrm { e } ^ { - x / \lambda } } { \lambda ^ { 2 } } \quad ( x > 0 )$$ where $\lambda$ is a parameter $( \lambda > 0 ) . X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ are $n$ independent observations on $X$, and $\bar { X } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } X _ { i }$ is their mean.

Obtain $\mathrm { E } ( X )$ and deduce that $\hat { \lambda } = \frac { 1 } { 2 } \bar { X }$ is an unbiased estimator of $\lambda$.
$\operatorname { Obtain } \operatorname { Var } ( \hat { \lambda } )$.
Explain why the results in parts (i) and (ii) indicate that $\hat { \lambda }$ is a good estimator of $\lambda$ in large samples.
Suppose that $n = 3$ and consider the alternative estimator $$\tilde { \lambda } = \frac { 1 } { 8 } X _ { 1 } + \frac { 1 } { 4 } X _ { 2 } + \frac { 1 } { 8 } X _ { 3 } .$$ Show that $\tilde { \lambda }$ is an unbiased estimator of $\lambda$. Find the relative efficiency of $\tilde { \lambda }$ compared with $\hat { \lambda }$. Which estimator do you prefer in this case?

OCR MEI S4 2010 June Q2

2 The random variable $X$ has the Poisson distribution with parameter $\lambda$.

Show that the probability generating function of $X$ is $\mathrm { G } ( t ) = \mathrm { e } ^ { \lambda ( t - 1 ) }$.
Hence obtain the mean $\mu$ and variance $\sigma ^ { 2 }$ of $X$.
Write down the mean and variance of the random variable $Z = \frac { X - \mu } { \sigma }$.
Write down the moment generating function of $X$. State the linear transformation result for moment generating functions and use it to show that the moment generating function of $Z$ is $$\mathrm { M } _ { Z } ( \theta ) = \mathrm { e } ^ { \mathrm { f } ( \theta ) } \quad \text { where } \mathrm { f } ( \theta ) = \lambda \left( \mathrm { e } ^ { \theta / \sqrt { \lambda } } - \frac { \theta } { \sqrt { \lambda } } - 1 \right)$$
Show that the limit of $\mathrm { M } _ { Z } ( \theta )$ as $\lambda \rightarrow \infty$ is $\mathrm { e } ^ { \theta ^ { 2 } / 2 }$.
Explain briefly why this implies that the distribution of $Z$ tends to $\mathrm { N } ( 0,1 )$ as $\lambda \rightarrow \infty$. What does this imply about the distribution of $X$ as $\lambda \rightarrow \infty$ ?

Site	A	B	C	D	E	F	G	H
Type I	225.2	268.9	303.6	244.1	230.6	202.7	242.1	247.5
Type II	215.2	242.1	260.9	241.7	245.5	204.7	225.8	236.0

Tea	Q	R	S	T	U	V	W	X	Y	Z
Taster 1	69	79	85	63	81	65	85	86	89	77
Taster 2	74	75	99	66	75	64	96	94	96	86

Number of bees	0	1	2	3	4	5	6	7	\(\geqslant 8\)
Number of intervals	6	16	19	18	17	14	6	4	0

Employees trained by method A:	35.2	47.8	25.8	38.0	53.6	31.0	33.9
	35.4	21.6	42.5
Employees trained by method B:	43.0	57.5	68.6	20.9	31.4	44.9	62.8
	27.6	41.8	46.1	39.8	61.6

Worker	\(W _ { 1 }\)	\(W _ { 2 }\)	\(W _ { 3 }\)	\(W _ { 4 }\)	\(W _ { 5 }\)	\(W _ { 6 }\)	\(W _ { 7 }\)	\(W _ { 8 }\)
Time before	32.6	28.5	22.9	27.6	34.9	28.8	34.2	31.3
Time after	26.2	24.1	19.0	28.6	29.3	20.0	36.0	19.2

Paint A	Paint B	Paint C	Paint D	Paint E
64	66	59	65	64
58	68	56	78	52
73	76	69	69	56
60	70	60	72	61
67	71	63	71	58

Fertiliser A	Fertiliser B	Fertiliser C	Fertiliser D	Fertiliser E
23.6	26.0	18.8	29.0	17.7
18.2	35.3	16.7	37.2	16.5
32.4	30.5	23.0	32.6	12.8
20.8	31.4	28.3	31.4	20.4

Contractor A	Contractor B	Contractor C	Contractor D
41	54	56	41
49	45	45	36
50	50	54	46
44	50	50	38
56	47	49	35

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