Questions S3 - A-Level Maths

\cline { 2 - 5 }		In favour	Neutral	Against
\cline { 2 - 5 }	Democrat	58	16	16
\cline { 2 - 5 } Party	Independent	25	4	11
\cline { 2 - 5 }	Republican	17	20	33
\cline { 2 - 5 }
\cline { 2 - 5 }

Carry out a $\chi ^ { 2 }$ test, at the $1 \%$ level of significance, to investigate whether there is an association between party supported and attitude to the proposed form of taxation.

OCR S3 2016 June Q3

A company packages butter. Of 50 randomly selected packs, 8 were found to have damaged wrappers. Find an approximate $95 \%$ confidence interval for the proportion of packs with damaged wrappers.
The mass of a pack has a normal distribution with standard deviation 8.5 g . In a random sample of 10 packs the masses, in g , are as follows. $$\begin{array} { l l l l l l l l l l } 220 & 225 & 218 & 223 & 224 & 220 & 229 & 228 & 226 & 228 \end{array}$$ Find a 99\% confidence interval for the mean mass of a pack.

OCR S3 2016 June Q4

4 A group of students were tested in geography before and after a fieldwork course. The marks of 10 randomly selected students are shown in the table.

Student	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$	$J$
Mark before fieldwork	19	84	84	99	59	19	29	49	54	69
Mark after fieldwork	23	98	83	88	68	33	28	53	58	88

Use a suitable $t$-test, at the $5 \%$ level of significance, to test whether the students' performance has improved.
State the necessary assumption in applying the test.

OCR S3 2016 June Q5

5 The independent random variables $X$ and $Y$ have distributions $\mathrm { N } \left( 30 , \sigma ^ { 2 } \right)$ and $\mathrm { N } \left( 20 , \sigma ^ { 2 } \right)$ respectively. The random variable $a X + b Y$, where $a$ and $b$ are constants, has the distribution $\mathrm { N } \left( 410,130 \sigma ^ { 2 } \right)$.

Given that $a$ and $b$ are integers, find the value of $a$ and the value of $b$.
Given that $\mathrm { P } ( X > Y ) = 0.966$, find $\sigma ^ { 2 }$.

OCR S3 2016 June Q6

6 The masses at birth, in kg, of random samples of babies were recorded for each of the years 1970 and 2010. The table shows the sample mean and an unbiased estimate of the population variance for each year.

A researcher tests the null hypothesis that babies born in 2010 are 0.04 kg heavier, on average, than babies born in 1970, against the alternative hypothesis that they are more than 0.04 kg heavier on average. Show that, at the $5 \%$ level of significance, the null hypothesis is not rejected.
Another researcher chooses samples of equal size, $n$, for the two years. Using the same hypothesis as in part (i), she finds that the null hypothesis is rejected at the $5 \%$ level of significance. Assuming that the sample means and unbiased estimates of population variance for the two years are as given in the table, find the smallest possible value of $n$.

OCR S3 2016 June Q7

7 A continuous random variable $X$ has probability density function $$f ( x ) = \begin{cases} a x ^ { 3 } & 0 \leqslant x \leqslant 1
a x ^ { 2 } & 1 < x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where $a$ is a constant.

Show that $a = \frac { 12 } { 31 }$.
Find $\mathrm { E } ( X )$. It is thought that the time taken by a student to complete a task can be well modelled by $X$. The times taken by 992 randomly chosen students are summarised in the table, together with some of the expected frequencies.
Time $0 \leqslant x < 0.5$ $0.5 \leqslant x < 1$ $1 \leqslant x < 1.5$ $1.5 \leqslant x \leqslant 2$
Observed frequency 8 92 279 613
Expected frequency 6 90
Find the other expected frequencies and test, at the $5 \%$ level of significance, whether the data can be well modelled by $X$.

OCR S3 2016 June Q8

8 The radius, $R$, of a sphere is a random variable with a continuous uniform distribution between 0 and 10 .

Find the cumulative distribution function and probability density function of $A$, the surface area of the sphere.
Find $\mathrm { P } ( \mathrm { A } \leqslant 200 \pi )$. \section*{END OF QUESTION PAPER}

OCR MEI S3 2009 January Q1

A continuous random variable $X$ has probability density function $$\mathrm { f } ( x ) = \lambda x ^ { c } , \quad 0 \leqslant x \leqslant 1 ,$$ where $c$ is a constant and the parameter $\lambda$ is greater than 1 .
1. Find $c$ in terms of $\lambda$.
2. Find $\mathrm { E } ( X )$ in terms of $\lambda$.
3. Show that $\operatorname { Var } ( X ) = \frac { \lambda } { ( \lambda + 2 ) ( \lambda + 1 ) ^ { 2 } }$.
Every day, Godfrey does a puzzle from the newspaper and records the time taken in minutes. Last year, his median time was 32 minutes. His times for a random sample of 12 puzzles this year are as follows. $$\begin{array} { l l l l l l l l l l l l } 40 & 20 & 18 & 11 & 47 & 36 & 38 & 35 & 22 & 14 & 12 & 21 \end{array}$$ Use an appropriate test, with a 5\% significance level, to examine whether Godfrey's times this year have decreased on the whole.

OCR MEI S3 2009 January Q2

2 A factory manufactures paperweights consisting of glass mounted on a wooden base. The volume of glass, in $\mathrm { cm } ^ { 3 }$, in a paperweight has a Normal distribution with mean 56.5 and standard deviation 2.9. The volume of wood, in $\mathrm { cm } ^ { 3 }$, also has a Normal distribution with mean 38.4 and standard deviation 1.1. These volumes are independent of each other. For the purpose of quality control, paperweights for testing are chosen at random from the factory's output.

Find the probability that the volume of glass in a randomly chosen paperweight is less than $60 \mathrm {~cm} ^ { 3 }$.
Find the probability that the total volume of a randomly chosen paperweight is more than $100 \mathrm {~cm} ^ { 3 }$. The glass has a mass of 3.1 grams per $\mathrm { cm } ^ { 3 }$ and the wood has a mass of 0.8 grams per $\mathrm { cm } ^ { 3 }$.
Find the probability that the total mass of a randomly chosen paperweight is between 200 and 220 grams.
The factory manager introduces some modifications intended to reduce the mean mass of the paperweights to 200 grams or less. The variance is also affected but not the Normality. Subsequently, for a random sample of 10 paperweights, the sample mean mass is 205.6 grams and the sample standard deviation is 8.51 grams. Is there evidence, at the $5 \%$ level of significance, that the intended reduction of the mean mass has not been achieved?

OCR MEI S3 2009 January Q3

3 Pathology departments in hospitals routinely analyse blood specimens. Ideally the analysis should be done while the specimens are fresh to avoid any deterioration, but this is not always possible. A researcher decides to study the effect of freezing specimens for later analysis by measuring the concentrations of a particular hormone before and after freezing. He collects and divides a sample of 15 specimens. One half of each specimen is analysed immediately, the other half is frozen and analysed a month later. The concentrations of the particular hormone (in suitable units) are as follows.

Immediately	15.21	13.36	15.97	21.07	12.82	10.80	11.50	12.05
After freezing	15.96	10.65	13.38	15.00	12.11	12.65	12.48	8.49

Immediately	10.90	18.48	13.43	13.16	16.62	14.91	17.08
After freezing	9.13	15.53	11.84	8.99	16.24	14.03	16.13

A $t$ test is to be used in order to see if, on average, there is a reduction in hormone concentration as a result of being frozen.

Explain why a paired test is appropriate in this situation.
State the hypotheses that should be used, together with any necessary assumptions.
Carry out the test using a $1 \%$ significance level.
A $p \%$ confidence interval for the true mean reduction in hormone concentration is found to be ( $0.4869,2.8131$ ). Determine the value of $p$.

OCR MEI S3 2009 January Q4

Explain the meaning of 'opportunity sampling'. Give one reason why it might be used and state one disadvantage of using it. A market researcher is conducting an 'on-street' survey in a busy city centre, for which he needs to stop and interview 100 people. For each interview the researcher counts the number of people he has to ask until one agrees to be interviewed. The data collected are as follows.
No. of people asked 1 2 3 4 5 6 7 or more
Frequency 26 19 17 13 11 8 6
A model for these data is proposed as follows, where $p$ (assumed constant throughout) is the probability that a person asked agrees to be interviewed, and $q = 1 - p$.
No. of people asked 1 2 3 4 5 6 7 or more
Probability $p$ $p q$ $p q ^ { 2 }$ $p q ^ { 3 }$ $p q ^ { 4 }$ $p q ^ { 5 }$ $q ^ { 6 }$
Verify that these probabilities add to 1 whatever the value of $p$.
Initially it is thought that on average 1 in 4 people asked agree to be interviewed. Test at the $10 \%$ level of significance whether it is reasonable to suppose that the model applies with $p = 0.25$.
Later an estimate of $p$ obtained from the data is used in the analysis. The value of the test statistic (with no combining of cells) is found to be 9.124 . What is the outcome of this new test? Comment on your answer in relation to the outcome of the test in part (iii).

OCR MEI S3 2010 January Q1

1 Coastal wildlife wardens are monitoring populations of herring gulls. Herring gulls usually lay 3 eggs per nest and the wardens wish to model the number of eggs per nest that hatch. They assume that the situation can be modelled by the binomial distribution $\mathrm { B } ( 3 , p )$ where $p$ is the probability that an egg hatches. A random sample of 80 nests each containing 3 eggs has been observed with the following results.

Number of eggs hatched	0	1	2	3
Number of nests	7	23	29	21

Initially it is assumed that the value of $p$ is $\frac { 1 } { 2 }$. Test at the $5 \%$ level of significance whether it is reasonable to suppose that the model applies with $p = \frac { 1 } { 2 }$.
The model is refined by estimating $p$ from the data. Find the mean of the observed data and hence an estimate of $p$.
Using the estimated value of $p$, the value of the test statistic $X ^ { 2 }$ turns out to be 2.3857 . Is it reasonable to suppose, at the $5 \%$ level of significance, that this refined model applies?
Discuss the reasons for the different outcomes of the tests in parts (i) and (iii).

OCR MEI S3 2010 January Q2

A continuous random variable, $X$, has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 72 } \left( 8 x - x ^ { 2 } \right) & 2 \leqslant x \leqslant 8
0 & \text { otherwise } \end{cases}$$
1. Find $\mathrm { F } ( x )$, the cumulative distribution function of $X$.
2. Sketch $\mathrm { F } ( x )$.
3. The median of $X$ is $m$. Show that $m$ satisfies the equation $m ^ { 3 } - 12 m ^ { 2 } + 148 = 0$. Verify that $m \approx 4.42$.
The random variable in part (a) is thought to model the weights, in kilograms, of lambs at birth. The birth weights, in kilograms, of a random sample of 12 lambs, given in ascending order, are as follows. $$\begin{array} { l l l l l l l l l l l l } 3.16 & 3.62 & 3.80 & 3.90 & 4.02 & 4.72 & 5.14 & 6.36 & 6.50 & 6.58 & 6.68 & 6.78 \end{array}$$ Test at the 5\% level of significance whether a median of 4.42 is consistent with these data.

OCR MEI S3 2010 January Q3

3 Cholesterol is a lipid (fat) which is manufactured by the liver from the fatty foods that we eat. It plays a vital part in allowing the body to function normally. However, when high levels of cholesterol are present in the blood there is a risk of arterial disease. Among the factors believed to assist with achieving and maintaining low cholesterol levels are weight loss and exercise. A doctor wishes to test the effectiveness of exercise in lowering cholesterol levels. For a random sample of 12 of her patients, she measures their cholesterol levels before and after they have followed a programme of exercise. The measurements obtained are as follows.

Patient	A	B	C	D	E	F	G	H	I	J	K	L
Before	5.7	5.7	4.0	6.8	7.4	5.5	6.7	6.4	7.2	7.2	7.1	4.4
After	5.8	4.0	5.2	5.7	6.0	5.0	5.8	4.2	7.3	5.2	6.4	4.1

A $t$ test is to be used in order to see if, on average, the exercise programme seems to be effective in lowering cholesterol levels. State the distributional assumption necessary for the test, and carry out the test using a $1 \%$ significance level.
A second random sample of 12 patients gives a $95 \%$ confidence interval of $( - 0.5380,1.4046 )$ for the true mean reduction (before - after) in cholesterol level. Find the mean and standard deviation for this sample. How might the doctor interpret this interval in relation to the exercise programme?

OCR MEI S3 2010 January Q4

4 The weights of a particular variety (A) of tomato are known to be Normally distributed with mean 80 grams and standard deviation 11 grams.

Find the probability that a randomly chosen tomato of variety A weighs less than 90 grams. The weights of another variety (B) of tomato are known to be Normally distributed with mean 70 grams. These tomatoes are packed in sixes using packaging that weighs 15 grams.
The probability that a randomly chosen pack of 6 tomatoes of variety B , including packaging, weighs less than 450 grams is 0.8463 . Show that the standard deviation of the weight of single tomatoes of variety B is 6 grams, to the nearest gram.
Tomatoes of variety A are packed in fives using packaging that weighs 25 grams. Find the probability that the total weight of a randomly chosen pack of variety A is greater than the total weight of a randomly chosen pack of variety B .
A new variety (C) of tomato is introduced. The weights, $c$ grams, of a random sample of 60 of these tomatoes are measured giving the following results. $$\Sigma c = 3126.0 \quad \Sigma c ^ { 2 } = 164223.96$$ Find a $95 \%$ confidence interval for the true mean weight of these tomatoes.

OCR MEI S3 2011 January Q1

1 Each month the amount of electricity, measured in kilowatt-hours ( kWh ), used by a particular household is Normally distributed with mean 406 and standard deviation 12.

Find the probability that, in a randomly chosen month, less than 420 kWh is used. The charge for electricity used is 14.6 pence per kWh .
Write down the distribution of the total charge for the amount of electricity used in any one month. Hence find the probability that, in a randomly chosen month, the total charge is more than $\pounds 60$.
The household receives a bill every three months. Assume that successive months may be regarded as independent of each other. Find the value of $b$ such that the probability that a randomly chosen bill is less than $\pounds b$ is 0.99 . In a different household, the amount of electricity used per month was Normally distributed with mean 432 kWh . This household buys a new washing machine that is claimed to be cheaper to run than the old one. Over the next six months the amounts of electricity used, in kWh , are as follows. $$\begin{array} { l l l l l l } 404 & 433 & 420 & 423 & 413 & 440 \end{array}$$
Treating this as a random sample, carry out an appropriate test, with a $5 \%$ significance level, to see if there is any evidence to suggest that the amount of electricity used per month by this household has decreased on average.

OCR MEI S3 2011 January Q2

1. What is stratified sampling? Why would it be used?
2. A local authority official wishes to conduct a survey of households in the borough. He decides to select a stratified sample of 2000 households using Council Tax property bands as the strata. At the time of the survey there are 79368 households in the borough. The table shows the numbers of households in the different tax bands.
  Tax band A - B C - D E - F G - H
  Number of households 32298 33211 9739 4120
  Calculate the number of households that the official should choose from each stratum in order to obtain his sample of 2000 households so that each stratum is represented proportionally.
1. What assumption needs to be made when using a Wilcoxon single sample test?
2. As part of an investigation into trends in local authority spending, one of the categories of expenditure considered was 'Highways and the Environment'. For a random sample of 10 local authorities, the percentages of their total expenditure spent on Highways and the Environment in 1999 and then in 2009 are shown in the table.
  Local authority A B C D E F G H I J
  1999 9.60 8.40 8.67 9.32 9.89 9.35 7.91 8.08 9.61 8.55
  2009 8.94 8.42 7.87 8.41 10.17 10.11 8.31 9.76 9.54 9.67
  Use a Wilcoxon test, with a significance level of $10 \%$, to determine whether there appears to be any change to the average percentage of total expenditure spent on Highways and the Environment between 1999 and 2009.

OCR MEI S3 2011 January Q3

3 The masses, in kilograms, of a random sample of 100 chickens on sale in a large supermarket were recorded as follows.

Mass $( m \mathrm {~kg} )$	$m < 1.6$	$1.6 \leqslant m < 1.8$	$1.8 \leqslant m < 2.0$	$2.0 \leqslant m < 2.2$	$2.2 \leqslant m < 2.4$	$2.4 \leqslant m < 2.6$	$2.6 \leqslant m$
Frequency	2	8	30	42	11	5	2

Assuming that the first and last classes are the same width as the other classes, calculate an estimate of the sample mean and show that the corresponding estimate of the sample standard deviation is 0.2227 kg . A Normal distribution using the mean and standard deviation found in part (i) is to be fitted to these data. The expected frequencies for the classes are as follows.

Mass $( m \mathrm {~kg} )$

$m < 1.6$

$1.6 \leqslant m < 1.8$

$1.8 \leqslant m < 2.0$

$2.0 \leqslant m < 2.2$

$2.2 \leqslant m < 2.4$

$2.4 \leqslant m < 2.6$

Use the Normal distribution to find $f$.
Carry out a goodness of fit test of this Normal model using a significance level of 5\%.
Discuss the outcome of the test with reference to the contributions to the test statistic and to the possibility of other significance levels.

OCR MEI S3 2011 January Q4

4 A timber supplier cuts wooden fence posts from felled trees. The posts are of length $( k + X ) \mathrm { cm }$ where $k$ is a constant and $X$ is a random variable which has probability density function $$f ( x ) = \begin{cases} 1 + x & - 1 \leqslant x < 0
1 - x & 0 \leqslant x \leqslant 1
0 & \text { elsewhere } \end{cases}$$

Sketch $\mathrm { f } ( x )$.
Write down the value of $\mathrm { E } ( X )$ and find $\operatorname { Var } ( X )$.
Write down, in terms of $k$, the approximate distribution of $\bar { L }$, the mean length of a random sample of 50 fence posts. Justify your choice of distribution.
In a particular sample of 50 posts, the mean length is 90.06 cm . Find a $95 \%$ confidence interval for the true mean length of the fence posts.
Explain whether it is reasonable to suppose that $k = 90$.

OCR MEI S3 2012 January Q1

Define simple random sampling. Describe briefly one difficulty associated with simple random sampling.
Freeze-drying is an economically important process used in the production of coffee. It improves the retention of the volatile aroma compounds. In order to maintain the quality of the coffee, technologists need to monitor the drying rate, measured in suitable units, at regular intervals. It is known that, for best results, the mean drying rate should be 70.3 units and anything substantially less than this would be detrimental to the coffee. Recently, a random sample of 12 observations of the drying rate was as follows. $$\begin{array} { l l l l l l l l l l l l } 66.0 & 66.1 & 59.8 & 64.0 & 70.9 & 71.4 & 66.9 & 76.2 & 65.2 & 67.9 & 69.2 & 68.5 \end{array}$$
1. Carry out a test to investigate at the $5 \%$ level of significance whether the mean drying rate appears to be less than 70.3. State the distributional assumption that is required for this test.
2. Find a 95\% confidence interval for the true mean drying rate.

OCR MEI S3 2012 January Q2

2 In a particular chain of supermarkets, one brand of pasta shapes is sold in small packets and large packets. Small packets have a mean weight of 505 g and a standard deviation of 11 g . Large packets have a mean weight of 1005 g and a standard deviation of 17 g . It is assumed that the weights of packets are Normally distributed and are independent of each other.

Find the probability that a randomly chosen large packet weighs between 995 g and 1020 g .
Find the probability that the weights of two randomly chosen small packets differ by less than 25 g .
Find the probability that the total weight of two randomly chosen small packets exceeds the weight of a randomly chosen large packet.
Find the probability that the weight of one randomly chosen small packet exceeds half the weight of a randomly chosen large packet by at least 5 g .
A different brand of pasta shapes is sold in packets of which the weights are assumed to be Normally distributed with standard deviation 14 g . A random sample of 20 packets of this pasta is found to have a mean weight of 246 g . Find a $95 \%$ confidence interval for the population mean weight of these packets.

OCR MEI S3 2012 January Q3

A medical researcher is looking into the delay, in years, between first and second myocardial infarctions (heart attacks). The following table shows the results for a random sample of 225 patients.
Delay (years) $0 -$ $1 -$ $2 -$ $3 -$ $4 - 10$
Number of patients 160 40 13 9 3
The mean of this sample is used to construct a model which gives the following expected frequencies.
Delay (years) $0 -$ $1 -$ $2 -$ $3 -$ $4 - 10$
Number of patients 142.23 52.32 19.25 7.08 4.12
Carry out a test, using a $2.5 \%$ level of significance, of the goodness of fit of the model to the data.
A further piece of research compares the incidence of myocardial infarction in men aged 55 to 70 with that in women aged 55 to 70 . Incidence is measured by the number of infarctions per 10000 of the population. For a random sample of 8 health authorities across the UK, the following results for the year 2010 were obtained.
Health authority A B C D E F G H
Incidence in men 47 56 15 51 45 54 50 32
Incidence in women 36 30 30 47 54 55 27 27
A Wilcoxon paired sample test, using the hypotheses $\mathrm { H } _ { 0 } : m = 0$ and $\mathrm { H } _ { 1 } : m \neq 0$ where $m$ is the population median difference, is to be carried out to investigate whether there is any difference between men and women on the whole.
1. Explain why a paired test is being used in this context.
2. Carry out the test using a $10 \%$ level of significance.

OCR MEI S3 2012 January Q4

4 At the school summer fair, one of the games involves throwing darts at a circular dartboard of radius $a$ lying on the ground some distance away. Only darts that land on the board are counted. The distance from the centre of the board to the point where a dart lands is modelled by the random variable $R$. It is assumed that the probability that a dart lands inside a circle of radius $r$ is proportional to the area of the circle.

By considering $\mathrm { P } ( R < r )$ show that $\mathrm { F } ( r )$, the cumulative distribution function of $R$, is given by $$\mathrm { F } ( r ) = \begin{cases} 0 & r < 0 ,
\frac { r ^ { 2 } } { a ^ { 2 } } & 0 \leqslant r \leqslant a ,
1 & r > a . \end{cases}$$
Find $\mathrm { f } ( r )$, the probability density function of $R$.
Find $\mathrm { E } ( R )$ and show that $\operatorname { Var } ( R ) = \frac { a ^ { 2 } } { 18 }$. The radius $a$ of the dartboard is 22.5 cm .
Let $\bar { R }$ denote the mean distance from the centre of the board of a random sample of 100 darts. Write down an approximation to the distribution of $\bar { R }$.
A random sample of 100 darts is found to give a mean distance of 13.87 cm . Does this cast any doubt on the modelling?

OCR MEI S3 2013 January Q1

1 A certain industrial process requires a supply of water. It has been found that, for best results, the mean water pressure in suitable units should be 7.8. The water pressure is monitored by taking measurements at regular intervals. On a particular day, a random sample of the measurements is as follows. $$\begin{array} { l l l l l l l l l } 7.50 & 7.64 & 7.68 & 7.51 & 7.70 & 7.85 & 7.34 & 7.72 & 7.74 \end{array}$$ These data are to be used to carry out a hypothesis test concerning the mean water pressure.

Why is a test based on the Normal distribution not appropriate in this case?
What distributional assumption is needed for a test based on the $t$ distribution?
Carry out a $t$ test, with a $2 \%$ level of significance, to see whether it is reasonable to assume that the mean pressure is 7.8 .
Explain what is meant by a $95 \%$ confidence interval.
Find a $95 \%$ confidence interval for the actual mean water pressure.

OCR MEI S3 2013 January Q2

2 A particular species of reed that grows up to 2 metres in length is used for thatching. The lengths in metres of the reeds when harvested are modelled by the random variable $X$ which has the following probability density function, $\mathrm { f } ( x )$. $$f ( x ) = \begin{cases} \frac { 3 } { 16 } \left( 4 x - x ^ { 2 } \right) & \text { for } 0 \leqslant x \leqslant 2
0 & \text { elsewhere } \end{cases}$$

Sketch $\mathrm { f } ( x )$.
Show that $\mathrm { E } ( X ) = \frac { 5 } { 4 }$ and find the standard deviation of the lengths of the harvested reeds.
Find the standard error of the mean length for a random sample of 100 reeds. Once the harvested reeds have been collected, any that are shorter than 1 metre are discarded.
Find the proportion of reeds that should be discarded according to the model.
Reeds are harvested from a large area which is divided into several reed beds. A sample of the harvested reeds is required for quality control. How might the method of cluster sampling be used to obtain it?

No. of people asked	1	2	3	4	5	6	7 or more
Probability	\(p\)	\(p q\)	\(p q ^ { 2 }\)	\(p q ^ { 3 }\)	\(p q ^ { 4 }\)	\(p q ^ { 5 }\)	\(q ^ { 6 }\)

Mass \(( m \mathrm {~kg} )\)	\(m < 1.6\)	\(1.6 \leqslant m < 1.8\)	\(1.8 \leqslant m < 2.0\)	\(2.0 \leqslant m < 2.2\)	\(2.2 \leqslant m < 2.4\)	\(2.4 \leqslant m < 2.6\)	\(2.6 \leqslant m\)
Frequency	2	8	30	42	11	5	2

Student	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
Mark before fieldwork	19	84	84	99	59	19	29	49	54	69
Mark after fieldwork	23	98	83	88	68	33	28	53	58	88

Time	\(0 \leqslant x < 0.5\)	\(0.5 \leqslant x < 1\)	\(1 \leqslant x < 1.5\)	\(1.5 \leqslant x \leqslant 2\)
Observed frequency	8	92	279	613
Expected frequency	6	90

Tax band	A - B	C - D	E - F	G - H
Number of households	32298	33211	9739	4120

Local authority	A	B	C	D	E	F	G	H	I	J
1999	9.60	8.40	8.67	9.32	9.89	9.35	7.91	8.08	9.61	8.55
2009	8.94	8.42	7.87	8.41	10.17	10.11	8.31	9.76	9.54	9.67

Delay (years)	\(0 -\)	\(1 -\)	\(2 -\)	\(3 -\)	\(4 - 10\)
Number of patients	160	40	13	9	3

Health authority	A	B	C	D	E	F	G	H
Incidence in men	47	56	15	51	45	54	50	32
Incidence in women	36	30	30	47	54	55	27	27

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