Questions S4 - A-Level Maths

Show by calculation that the test can never result in rejection of the null hypothesis when $n = 4$. The coach of a college swimming team expects Elena, the best 50 m freestyle swimmer, to have a median time less than 30 seconds. Elena found from records of her previous 72 swims that 44 were less than 30 seconds and 28 were greater than 30 seconds.
Stating a necessary assumption, test at the $5 \%$ significance level whether Elena's median time for the 50 m freestyle is less than 30 seconds.

OCR S4 2012 June Q6

6 The random variables $S$ and $T$ are independent and have joint probability distribution given in the table.

	$S$
\cline { 2 - 5 }		0	1	2
\cline { 2 - 5 }	1	$a$	0.18	$b$
2	0.08	0.12	0.20
\cline { 2 - 5 }
\cline { 2 - 5 }

Show that $a = 0.12$ and find the value of $b$.
Find $\mathrm { P } ( T - S = 1 )$.
Find $\operatorname { Var } ( T - S )$.

OCR S4 2012 June Q7

7 The continuous random variable $X$ has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 4 } ( 1 + a x ) & - 2 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where $a$ is a constant.

Show that $| a | \leqslant \frac { 1 } { 2 }$.
Find $\mathrm { E } ( X )$ in terms of $a$.
Construct an unbiased estimator $T _ { 1 }$ of $a$ based on one observation $X _ { 1 }$ of $X$.
A second observation $X _ { 2 }$ is taken. Show that $T _ { 2 }$, where $T _ { 2 } = \frac { 3 } { 8 } \left( X _ { 1 } + X _ { 2 } \right)$, is also an unbiased estimator of a.
Given that $\operatorname { Var } ( X ) = \sigma ^ { 2 }$, determine which of $T _ { 1 }$ and $T _ { 2 }$ is the better estimator.

OCR S4 2012 June Q8

8 Events $A$ and $B$ are such that $\mathrm { P } ( A ) = 0.3$ and $\mathrm { P } ( A \mid B ) = 0.6$.

Show that $\mathrm { P } ( B ) \leqslant 0.5$.
Given also that $\mathrm { P } ( A \cup B ) = x$, find $\mathrm { P } ( B )$ in terms of $x$.

OCR S4 2013 June Q1

	$S$
	0	1	2
0	$\frac { 1 } { 8 }$	$\frac { 1 } { 8 }$	0
1	$\frac { 1 } { 8 }$	$\frac { 1 } { 4 }$	$\frac { 1 } { 8 }$
2	0	$\frac { 1 } { 8 }$	$\frac { 1 } { 8 }$

An unbiased coin is tossed three times. The random variables $F$ and $S$ denote the total number of heads that occur in the first two tosses and the total number of heads that occur in the last two tosses respectively. The table above shows the joint probability distribution of $F$ and $S$.

Show how the entry $\frac { 1 } { 4 }$ in the table is obtained.
Find $\operatorname { Cov } ( F , S )$.

OCR S4 2013 June Q2

2 Two drugs, I and II, for alleviating hay fever are trialled in a hospital on each of 12 volunteer patients. Each received drug I on one day and drug II on a different day. After receiving a drug, the number of times each patient sneezed over a period of one hour was noted. The results are given in the table.

Patient	1	2	3	4	5	6	7	8	9	10	11	12
Drug I	11	34	19	16	10	29	6	17	20	13	4	25
Drug II	12	20	10	18	3	21	9	13	10	19	9	12

The patients may be considered to be a random sample of all hay fever sufferers.
A researcher believes that patients taking drug II sneeze less than patients taking drug I.
Test this belief using the Wilcoxon signed rank test at the $5 \%$ significance level.

OCR S4 2013 June Q3

3 The continuous random variable $X$ has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 4 } x \mathrm { e } ^ { - \frac { 1 } { 2 } x } & x \geqslant 0
0 & \text { otherwise } . \end{cases}$$

Show that the moment generating function of $X$ is $( 1 - 2 t ) ^ { - 2 }$ for $t < \frac { 1 } { 2 }$, and state why the condition $t < \frac { 1 } { 2 }$ is necessary.
Use the moment generating function to find $\operatorname { Var } ( X )$.

OCR S4 2013 June Q4

4 The effect of water salinity on the growth of a type of grass was studied by a biologist. A random sample of 22 seedlings was divided into two groups $A$ and $B$, each of size 11 .
Group $A$ was treated with water of $0 \%$ salinity and group $B$ was treated with water of $0.5 \%$ salinity. After three weeks the height (in cm) of each seedling was measured with the following results, which are ordered for convenience.

Group $A$	8.6	9.4	9.7	9.8	10.1	10.5	11.0	11.2	11.8	12.7
Group $B$	7.4	8.4	8.5	8.8	9.2	9.3	9.5	9.9	10.0	11.1

Jeffery was asked to test whether the two treatments resulted, on average, in a difference in growth. He chose the Wilcoxon rank sum test.

Justify Jeffery's choice of test.
Carry out the test at the $5 \%$ significance level.

OCR S4 2013 June Q5

5 The discrete random variable $U$ has probability distribution given by $$\mathrm { P } ( U = r ) = \begin{cases} \frac { 1 } { 16 } \binom { 4 } { r } & r = 0,1,2,3,4
0 & \text { otherwise } \end{cases}$$

Find and simplify the probability generating function (pgf) of $U$.
Use the pgf to find $\mathrm { E } ( U )$ and $\operatorname { Var } ( U )$.
Identify the distribution of $U$, giving the values of any parameters.
Obtain the pgf of $Y$, where $Y = U ^ { 2 }$.
State, giving a reason, whether you can obtain the pgf of $U + Y$ by multiplying the pgf of $U$ by the pgf of $Y$.

OCR S4 2013 June Q6

6 The continuous random variable $X$ has mean $\mu$ and variance $\sigma ^ { 2 }$, and the independent continuous random variable $Y$ has mean $2 \mu$ and variance $3 \sigma ^ { 2 }$. Two observations of $X$ and three observations of $Y$ are taken and are denoted by $X _ { 1 } , X _ { 2 } , Y _ { 1 } , Y _ { 2 }$ and $Y _ { 3 }$ respectively.

Find the expectation of the sum of these 5 observations and hence construct an unbiased estimator, $T _ { 1 }$, of $\mu$.
The estimator $T _ { 2 }$, where $T _ { 2 } = X _ { 1 } + X _ { 2 } + c \left( Y _ { 1 } + Y _ { 2 } + Y _ { 3 } \right)$, is an unbiased estimator of $\mu$. Find the value of the constant $c$.
Determine which of $T _ { 1 }$ and $T _ { 2 }$ is more efficient.
Find the values of the constants $a$ and $b$ for which $$a \left( X _ { 1 } ^ { 2 } + X _ { 2 } ^ { 2 } \right) + b \left( Y _ { 1 } ^ { 2 } + Y _ { 2 } ^ { 2 } + Y _ { 3 } ^ { 2 } \right)$$ is an unbiased estimator of $\sigma ^ { 2 }$.

OCR S4 2013 June Q7

7 Each question on a multiple-choice examination paper has $n$ possible responses, only one of which is correct. Joni takes the paper and has probability $p$, where $0 < p < 1$, of knowing the correct response to any question, independently of any other. If she knows the correct response she will choose it, otherwise she will choose randomly from the $n$ possibilities. The events $K$ and $A$ are 'Joni knows the correct response' and 'Joni answers correctly' respectively.

Show that $\mathrm { P } ( A ) = \frac { q + n p } { n }$, where $q = 1 - p$.
Find $P ( K \mid A )$. A paper with 100 questions has $n = 4$ and $p = 0.5$. Each correct response scores 1 and each incorrect response scores - 1 .
(a) Joni answers all the questions on the paper and scores 40 . How many questions did she answer correctly?
(b) By finding the distribution of the number of correct answers, or otherwise, find the probability that Joni scores at least 40 on the paper using her strategy.

OCR S4 2014 June Q1

1 A teacher believes that the calculator paper in a GCSE Mathematics examination was easier than the non-calculator paper. The marks of a random sample of ten students are shown in the table.

Student	A	B	C	D	E	F	G	H	I	J
Mark on paper 1 (non-calculator)	66	79	58	87	67	55	75	62	50	84
Mark on paper 2 (calculator)	57	84	70	90	75	42	82	72	65	82

Use a Wilcoxon signed-rank test, at the $5 \%$ significance level, to test the teacher's belief.
State the assumption necessary for this test to be applied.

OCR S4 2014 June Q2

2 During an outbreak of a disease, it is known that $68 \%$ of people do not have the disease. Of people with the disease, $96 \%$ react positively to a test for diagnosing it, as do $m \%$ of people who do not have the disease.

In the case $m = 8$, find the probability that a randomly chosen person has the disease, given that the person reacts positively to the test.
What value of $m$ would be required for the answer to part (i) to be 0.95 ?

OCR S4 2014 June Q3

3 The discrete random variable $X$ has probability generating function $\frac { t } { a - b t }$, where $a$ and $b$ are constants.

Find a relationship between $a$ and $b$.
Use the probability generating function to find $\mathrm { E } ( X )$ in terms of $a$, giving your answer as simply as possible.
Expand the probability generating function as a power series, as far as the term in $t ^ { 3 }$, giving the coefficients in terms of $a$ and $b$.
Name the distribution for which $\frac { t } { a - b t }$ is the probability generating function, and state its parameter(s) in terms of $a$.

OCR S4 2014 June Q4

4 The continuous random variable $X$ has probability density function $$f ( x ) = \left\{ \begin{array} { c c } x & 0 \leqslant x \leqslant 1
2 - x & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{array} \right.$$

Show that the moment generating function of $X$ is $\frac { \left( \mathrm { e } ^ { t } - 1 \right) ^ { 2 } } { t ^ { 2 } }$.
$Y _ { 1 }$ and $Y _ { 2 }$ are independent observations of a random variable $Y$. The moment generating function of $Y _ { 1 } + Y _ { 2 }$ is $\frac { \left( \mathrm { e } ^ { t } - 1 \right) ^ { 2 } } { t ^ { 2 } }$.
Write down the moment generating function of $Y$.
Use the expansion of $\mathrm { e } ^ { t }$ to find $\operatorname { Var } ( Y )$.
Deduce the value of $\operatorname { Var } ( X )$.

OCR S4 2014 June Q5

5 Two discrete random variables $X$ and $Y$ have a joint probability distribution defined by $$\mathrm { P } ( X = x , Y = y ) = a ( x + y + 1 ) \quad \text { for } x = 0,1,2 \text { and } y = 0,1,2 ,$$ where $a$ is a constant.

Show that $a = \frac { 1 } { 27 }$.
Find $\mathrm { E } ( X )$.
Find $\operatorname { Cov } ( X , Y )$.
Are $X$ and $Y$ independent? Give a reason for your answer.
Find $\mathrm { P } ( X = 1 \mid Y = 2 )$.

OCR S4 2014 June Q6

6 A Wilcoxon rank-sum test with samples of sizes 11 and 12 is carried out.

What is the least possible value of the test statistic $W$ ?
The null hypothesis is that the two samples came from identical populations. Given that the null hypothesis was rejected at the $1 \%$ level using a 2 -tail test, find the set of possible values of $W$.

OCR S4 2014 June Q7

7 The continuous random variable $X$ has probability density function $$f ( x ) = \left\{ \begin{array} { c l } \frac { k } { ( x + \theta ) ^ { 5 } } & \text { for } x \geqslant 0
0 & \text { otherwise } \end{array} \right.$$ where $k$ is a positive constant and $\theta$ is a parameter taking positive values.

Find an expression for $k$ in terms of $\theta$.
Show that $\mathrm { E } ( X ) = \frac { 1 } { 3 } \theta$. You are given that $\operatorname { Var } ( X ) = \frac { 2 } { 9 } \theta ^ { 2 }$. A random sample $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ of $n$ observations of $X$ is obtained. The estimator $T _ { 1 }$ is defined as $T _ { 1 } = \frac { 3 } { n } \sum _ { i = 1 } ^ { n } X _ { i }$.
Show that $T _ { 1 }$ is an unbiased estimator of $\theta$, and find the variance of $T _ { 1 }$.
A second unbiased estimator $T _ { 2 }$ is defined by $T _ { 2 } = \frac { 1 } { 3 } \left( X _ { 1 } + 3 X _ { 2 } + 5 X _ { 3 } \right)$. For the case $n = 3$, which of $T _ { 1 }$ and $T _ { 2 }$ is more efficient? \section*{OCR}

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.

	\(S\)
	0	1	2
0	\(\frac { 1 } { 8 }\)	\(\frac { 1 } { 8 }\)	0
1	\(\frac { 1 } { 8 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 8 }\)
2	0	\(\frac { 1 } { 8 }\)	\(\frac { 1 } { 8 }\)

Group \(A\)	8.6	9.4	9.7	9.8	10.1	10.5	11.0	11.2	11.8	12.7
Group \(B\)	7.4	8.4	8.5	8.8	9.2	9.3	9.5	9.9	10.0	11.1

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

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