Questions - OCR MEI S3 - A-Level Maths

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?

OCR MEI S3 2013 January Q3

3 In the manufacture of child car seats, a resin made up of three ingredients is used. The ingredients are two polymers and an impact modifier. The resin is prepared in batches. Each ingredient is supplied by a separate feeder and the amount supplied to each batch, in kg, is assumed to be Normally distributed with mean and standard deviation as shown in the table below. The three feeders are also assumed to operate independently of each other.

	Mean	Standard deviation
Polymer 1	2025	44.6
Polymer 2	1565	21.8
Impact modifier	1410	33.8

Find the probability that, in a randomly chosen batch of resin, there is no more than 2100 kg of polymer 1.
Find the probability that, in a randomly chosen batch of resin, the amount of polymer 1 exceeds the amount of polymer 2 by at least 400 kg .
Find the value of $b$ such that the total amount of the ingredients in a randomly chosen batch exceeds $b \mathrm {~kg} 95 \%$ of the time.
Polymer 1 costs $\pounds 1.20$ per kg, polymer 2 costs $\pounds 1.30$ per kg and the impact modifier costs $\pounds 0.80$ per kg. Find the mean and variance of the total cost of a batch of resin.
Each batch of resin is used to make a large number of car seats from which a random sample of 50 seats is selected in order that the tensile strength (in suitable units) of the resin can be measured. From one such sample, the $99 \%$ confidence interval for the true mean tensile strength of the resin in that batch was calculated as $( 123.72,127.38 )$. Find the mean and standard deviation of the sample.

OCR MEI S3 2013 January Q4

At a college, two examiners are responsible for marking, independently, the students' projects. Each examiner awards a mark out of 100 to each project. There is some concern that the examiners' marks do not agree, on average. Consequently a random sample of 12 projects is selected and the marks awarded to them are compared.
1. Describe how a random sample of projects should be chosen.
2. The marks given for the projects in the sample are as follows.
  Project 1 2 3 4 5 6 7 8 9 10 11 12
  Examiner A 58 37 72 78 67 77 62 41 80 60 65 70
  Examiner B 73 47 74 71 78 96 54 27 97 73 60 66
  Carry out a test at the $10 \%$ level of significance of the hypotheses $\mathrm { H } _ { 0 } : m = 0 , \mathrm { H } _ { 1 } : m \neq 0$, where $m$ is the population median difference.
A calculator has a built-in random number function which can be used to generate a list of random digits. If it functions correctly then each digit is equally likely to be generated. When it was used to generate 100 random digits, the frequencies of the digits were as follows.
Digit 0 1 2 3 4 5 6 7 8 9
Frequency 6 8 11 14 12 9 15 5 14 6
Use a goodness of fit test, with a significance level of $10 \%$, to investigate whether the random number function is generating digits with equal probability.

OCR MEI S3 2009 June Q1

1 Andy, a carpenter, constructs wooden shelf units for storing CDs. The wood used for the shelves has a thickness which is Normally distributed with mean 14 mm and standard deviation 0.55 mm . Andy works to a design which allows a gap of 145 mm between the shelves, but past experience has shown that the gap is Normally distributed with mean 144 mm and standard deviation 0.9 mm . Dimensions of shelves and gaps are assumed to be independent of each other.

Find the probability that a randomly chosen gap is less than 145 mm .
Find the probability that the combined height of a gap and a shelf is more than 160 mm . A complete unit has 7 shelves and 6 gaps.
Find the probability that the overall height of a unit lies between 960 mm and 965 mm . Hence find the probability that at least 3 out of 4 randomly chosen units are between 960 mm and 965 mm high.
I buy two randomly chosen CD units made by Andy. The probability that the difference in their heights is less than $h \mathrm {~mm}$ is 0.95 . Find $h$.

OCR MEI S3 2009 June Q2

2 Pat makes and sells fruit cakes at a local market. On her stall a sign states that the average weight of the cakes is 1 kg . A trading standards officer carries out a routine check of a random sample of 8 of Pat's cakes to ensure that they are not underweight, on average. The weights, in kg , that he records are as follows. $$\begin{array} { l l l l l l l l } 0.957 & 1.055 & 0.983 & 0.917 & 1.015 & 0.865 & 1.013 & 0.854 \end{array}$$

On behalf of the trading standards officer, carry out a suitable test at a $5 \%$ level of significance, stating your hypotheses clearly. Assume that the weights of Pat's fruit cakes are Normally distributed.
Find a 95\% confidence interval for the true mean weight of Pat's fruit cakes. Pat's husband, Tony, is the owner of a factory which makes and supplies fruit cakes to a large supermarket chain. A large random sample of $n$ of these cakes has mean weight $\bar { x } \mathrm {~kg}$ and variance $0.006 \mathrm {~kg} ^ { 2 }$.
Write down, in terms of $n$ and $\bar { x }$, a $95 \%$ confidence interval for the true mean weight of cakes produced in Tony's factory.
What is the size of the smallest sample that should be taken if the width of the confidence interval in part (iii) is to be 0.025 kg at most?

OCR MEI S3 2009 June Q3

3 A company which employs 600 staff wishes to improve its image by introducing new uniforms for the staff to wear. The human resources manager would like to obtain the views of the staff. She decides to do this by means of a systematic sample of $10 \%$ of the staff.

How should she go about obtaining such a sample, ensuring that all members of staff are equally likely to be selected? Explain whether this constitutes a simple random sample. At a later stage in the process, the choice of uniform has been reduced to two possibilities. Twelve members of staff are selected to take part in deciding which of the two uniforms to adopt. Each of the twelve assesses each uniform for comfort, appearance and practicality, giving it a total score out of 10. The scores are as follows.
Staff member 1 2 3 4 5 6 7 8 9 10 11 12
Uniform A 4.2 2.6 10.0 9.0 8.2 2.8 5.0 7.4 2.8 6.8 10.0 9.8
Uniform B 5.0 5.2 1.4 2.8 2.2 6.4 7.4 7.8 6.8 1.2 3.4 7.6
A Wilcoxon signed rank test is to be used to decide whether there is any evidence of a preference for one of the uniforms.
Explain why this test is appropriate in these circumstances and state the hypotheses that should be used.
Carry out the test at the $5 \%$ significance level.

OCR MEI S3 2009 June Q4

4 A random variable $X$ has probability density function $\mathrm { f } ( x ) = \frac { 2 x } { \lambda ^ { 2 } }$ for $0 < x < \lambda$, where $\lambda$ is a positive constant.

Show that, for any value of $\lambda , \mathrm { f } ( x )$ is a valid probability density function.
Find $\mu$, the mean value of $X$, in terms of $\lambda$ and show that $\mathrm { P } ( X < \mu )$ does not depend on $\lambda$.
Given that $\mathrm { E } \left( X ^ { 2 } \right) = \frac { \lambda ^ { 2 } } { 2 }$, find $\sigma ^ { 2 }$, the variance of $X$, in terms of $\lambda$. The random variable $X$ is used to model the depth of the space left by the filling machine at the top of a jar of jam. The model gives the following probabilities for $X$ (whatever the value of $\lambda$ ).
$0 < X \leqslant \mu - \sigma$ $\mu - \sigma < X \leqslant \mu$ $\mu < X \leqslant \mu + \sigma$ $\mu + \sigma < X < \lambda$
0.18573 0.25871 0.36983 0.18573
A sample of 50 random observations of $X$, classified in the same way, is summarised by the following frequencies.
4 11 20 15
Carry out a suitable test at the $5 \%$ level of significance to assess the goodness of fit of $X$ to these data. Explain briefly how your conclusion may be affected by the choice of significance level.

OCR MEI S3 2010 June Q1

The manager of a company that employs 250 travelling sales representatives wishes to carry out a detailed analysis of the expenses claimed by the representatives. He has an alphabetical (by surname) list of the representatives. He chooses a sample of representatives by selecting the 10th, 20th, 30th and so on. Name the type of sampling the manager is attempting to use. Describe a weakness in his method of using it, and explain how he might overcome this weakness. The representatives each use their own cars to drive to meetings with customers. The total distance, in miles, travelled by a representative in a month is Normally distributed with mean 2018 and standard deviation 96.
Find the probability that, in a randomly chosen month, a randomly chosen representative travels more than 2100 miles.
Find the probability that, in a randomly chosen 3-month period, a randomly chosen representative travels less than 6000 miles. What assumption is needed here? Give a reason why it may not be realistic.
Each month every representative submits a claim for travelling expenses plus commission. Travelling expenses are paid at the rate of 45 pence per mile. The commission is $10 \%$ of the value of sales in that month. The value, in $\pounds$, of the monthly sales has the distribution $\mathrm { N } \left( 21200,1100 ^ { 2 } \right)$. Find the probability that a randomly chosen claim lies between $\pounds 3000$ and $\pounds 3300$. William Sealy, a biochemistry student, is doing work experience at a brewery. One of his tasks is to monitor the specific gravity of the brewing mixture during the brewing process. For one particular recipe, an initial specific gravity of 1.040 is required. A random sample of 9 measurements of the specific gravity at the start of the process gave the following results. $$\begin{array} { l l l l l l l l l } 1.046 & 1.048 & 1.039 & 1.055 & 1.038 & 1.054 & 1.038 & 1.051 & 1.038 \end{array}$$
William has to test whether the specific gravity of the mixture meets the requirement. Why might a $t$ test be used for these data and what assumption must be made?
Carry out the test using a significance level of $10 \%$.
Find a 95\% confidence interval for the true mean specific gravity of the mixture and explain what is meant by a $95 \%$ confidence interval.

OCR MEI S3 2010 June Q3

In order to prevent and/or control the spread of infectious diseases, the Government has various vaccination programmes. One such programme requires people to receive a booster injection at the age of 18. It is felt that the proportion of people receiving this booster could be increased and a publicity campaign is undertaken for this purpose. In order to assess the effectiveness of this campaign, health authorities across the country are asked to report the percentage of 18-year-olds receiving the booster before and after the campaign. The results for a randomly chosen sample of 9 authorities are as follows.
Authority A B C D E F G H I
Before 76 98 88 81 86 84 83 93 80
After 82 97 93 77 83 95 91 95 89
This sample is to be tested to see whether the campaign appears to have been successful in raising the percentage receiving the booster.
1. Explain why the use of paired data is appropriate in this context.
2. Carry out an appropriate Wilcoxon signed rank test using these data, at the $5 \%$ significance level.
Benford's Law predicts the following probability distribution for the first significant digit in some large data sets.
Digit 1 2 3 4 5 6 7 8 9
Probability 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046
On one particular day, the first significant digits of the stock market prices of the shares of a random sample of 200 companies gave the following results.
Digit 1 2 3 4 5 6 7 8 9
Frequency 55 34 27 16 15 17 12 15 9
Test at the $10 \%$ level of significance whether Benford's Law provides a reasonable model in the context of share prices.

OCR MEI S3 2010 June Q4

4 A random variable $X$ has an exponential distribution with probability density function $\mathrm { f } ( x ) = \lambda \mathrm { e } ^ { - \lambda x }$ for $x \geqslant 0$, where $\lambda$ is a positive constant.

Verify that $\int _ { 0 } ^ { \infty } \mathrm { f } ( x ) \mathrm { d } x = 1$ and sketch $\mathrm { f } ( x )$.
In this part of the question you may use the following result. $$\int _ { 0 } ^ { \infty } x ^ { r } \mathrm { e } ^ { - \lambda x } \mathrm {~d} x = \frac { r ! } { \lambda ^ { r + 1 } } \quad \text { for } r = 0,1,2 , \ldots$$ Derive the mean and variance of $X$ in terms of $\lambda$. The random variable $X$ is used to model the lifetime, in years, of a particular type of domestic appliance. The manufacturer of the appliance states that, based on past experience, the mean lifetime is 6 years.
Let $\bar { X }$ denote the mean lifetime, in years, of a random sample of 50 appliances. Write down an approximate distribution for $\bar { X }$.
A random sample of 50 appliances is found to have a mean lifetime of 7.8 years. Does this cast any doubt on the model?

OCR MEI S3 2011 June Q1

1 Gerry runs 5000 -metre races for his local athletics club. His coach has been monitoring his practice times for several months and he believes that they can be modelled using a Normal distribution with mean 15.3 minutes. The coach suggests that Gerry should try running with a pacemaker in order to see if this can improve his times. Subsequently a random sample of 10 of Gerry's times with the pacemaker is collected to see if any reduction has been achieved. The sample of times (in minutes) is as follows. $$\begin{array} { l l l l l l l l l l } 14.86 & 15.00 & 15.62 & 14.44 & 15.27 & 15.64 & 14.58 & 14.30 & 15.08 & 15.08 \end{array}$$

Why might a $t$ test be used for these data?
Using a $5 \%$ significance level, carry out the test to see whether, on average, Gerry's times have been reduced.
What is meant by 'a $5 \%$ significance level'? What would be the consequence of decreasing the significance level?
Find a $95 \%$ confidence interval for the true mean of Gerry's times using a pacemaker.

OCR MEI S3 2011 June Q2

2 Scientists researching into the chemical composition of dust in space collect specimens using a specially designed spacecraft. The craft collects the particles of dust in trays that are made up of a large array of cells containing aerogel. The aerogel traps the particles that penetrate into the cells.

For a random sample of 100 cells, the number of particles of dust in each cell was counted, giving the following results.
Number of particles 0 1 2 3 4 5 6 7 8 9 $10 +$
Frequency 4 7 10 20 17 15 10 9 5 3 0
It is thought that the number of particles collected in each cell can be modelled using the distribution Poisson(4.2) since 4.2 is the sample mean for these data. Some of the calculations for a $\chi ^ { 2 }$ test are shown below. The cells for 8,9 and $10 +$ particles have been combined.
Number of particles
Observed frequency
Expected frequency
Contribution to $X ^ { 2 }$

5 6 7 $8 +$
15 10 9 8
16.33 11.44 6.86 6.39
0.1083 0.1813 0.6676 0.4056
Complete the calculations and carry out the test using a $10 \%$ significance level to see whether the number of particles per cell may be modelled in this way.
The diameters of the dust particles are believed to be distributed symmetrically about a median of 15 micrometres $( \mu \mathrm { m } )$. For a random sample of 20 particles, the sum of the signed ranks of the diameters of the particles smaller than $15 \mu \mathrm {~m} \left( W _ { - } \right)$is found to be 53 . Test at the $5 \%$ level of significance whether the median diameter appears to be more than $15 \mu \mathrm {~m}$.

OCR MEI S3 2011 June Q3

3 The time, in hours, until an electronic component fails is represented by the random variable $X$. In this question two models for $X$ are proposed.

In one model, $X$ has cumulative distribution function $$\mathrm { G } ( x ) = \begin{cases} 0 & x \leqslant 0
1 - \left( 1 + \frac { x } { 200 } \right) ^ { - 2 } & x > 0 \end{cases}$$ (A) Sketch $\mathrm { G } ( x )$.
(B) Find the interquartile range for this model. Hence show that a lifetime of more than 454 hours (to the nearest hour) would be classed as an outlier.
In the alternative model, $X$ has probability density function $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 200 } \mathrm { e } ^ { - \frac { 1 } { 200 } x } & x > 0
0 & \text { elsewhere. } \end{cases}$$ (A) For this model show that the cumulative distribution function of $X$ is $$\mathrm { F } ( x ) = \begin{cases} 0 & x \leqslant 0
1 - \mathrm { e } ^ { - \frac { 1 } { 200 } x } & x > 0 \end{cases}$$ (B) Show that $\mathrm { P } ( X > 50 ) = \mathrm { e } ^ { - 0.25 }$.
(C) It is observed that a particular component is still working after 400 hours. Find the conditional probability that it will still be working after a further 50 hours (i.e. after a total of 450 hours) given that it is still working after 400 hours.

OCR MEI S3 2011 June Q4

4 The weights of Avonley Blue cheeses made by a small producer are found to be Normally distributed with mean 10 kg and standard deviation 0.4 kg .

Find the probability that a randomly chosen cheese weighs less than 9.5 kg . One particular shop orders four Avonley Blue cheeses each week from the producer. From experience, the shopkeeper knows that the weekly demand from customers for Avonley Blue cheese is Normally distributed with mean 35 kg and standard deviation 3.5 kg . In the interests of food hygiene, no cheese is kept by the shopkeeper from one week to the next.
Find the probability that, in a randomly chosen week, demand from customers for Avonley Blue will exceed the supply. Following a campaign to promote Avonley Blue cheese, the shopkeeper finds that the weekly demand for it has increased by $30 \%$ (i.e. the mean and standard deviation are both increased by $30 \%$ ). Therefore the shopkeeper increases his weekly order by one cheese.
Find the probability that, in a randomly chosen week, demand will now exceed supply.
Following complaints, the cheese producer decides to check the mean weight of the Avonley Blue cheeses. For a random sample of 12 cheeses, she finds that the mean weight is 9.73 kg . Assuming that the population standard deviation of the weights is still 0.4 kg , find a $95 \%$ confidence interval for the true mean weight of the cheeses and comment on the result. Explain what is meant by a 95\% confidence interval. RECOGNISING ACHIEVEMENT

OCR MEI S3 2012 June Q1

1 Technologists at a company that manufactures paint are trying to develop a new type of gloss paint with a shorter drying time than the current product. In order to test whether the drying time has been reduced, the technologists paint a square metre of each of the new and old paints on each of 10 different surfaces. The lengths of time, in hours, that each square metre takes to dry are as follows.

Surface	A	B	C	D	E	F	G	H	I	J
Old paint	16.6	17.0	16.5	15.6	16.3	16.5	16.4	15.9	16.3	16.1
New paint	15.9	16.3	16.3	15.9	15.5	16.6	16.1	16.0	16.2	15.6

Explain why a paired sample is used in this context.
The mean reduction in drying time is to be investigated. Why might a $t$ test be appropriate in this context and what assumption needs to be made?
Using a significance level of $5 \%$, carry out a test to see if there appears to be any reduction in mean drying time.
Find a 95\% confidence interval for the true mean reduction in drying time.

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

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

Project	1	2	3	4	5	6	7	8	9	10	11	12
Examiner A	58	37	72	78	67	77	62	41	80	60	65	70
Examiner B	73	47	74	71	78	96	54	27	97	73	60	66

Staff member	1	2	3	4	5	6	7	8	9	10	11	12
Uniform A	4.2	2.6	10.0	9.0	8.2	2.8	5.0	7.4	2.8	6.8	10.0	9.8
Uniform B	5.0	5.2	1.4	2.8	2.2	6.4	7.4	7.8	6.8	1.2	3.4	7.6

\(0 < X \leqslant \mu - \sigma\)	\(\mu - \sigma < X \leqslant \mu\)	\(\mu < X \leqslant \mu + \sigma\)	\(\mu + \sigma < X < \lambda\)
0.18573	0.25871	0.36983	0.18573

Authority	A	B	C	D	E	F	G	H	I
Before	76	98	88	81	86	84	83	93	80
After	82	97	93	77	83	95	91	95	89

Digit	1	2	3	4	5	6	7	8	9
Probability	0.301	0.176	0.125	0.097	0.079	0.067	0.058	0.051	0.046

Number of particles	0	1	2	3	4	5	6	7	8	9	\(10 +\)
Frequency	4	7	10	20	17	15	10	9	5	3	0

5	6	7	\(8 +\)
15	10	9	8
16.33	11.44	6.86	6.39
0.1083	0.1813	0.6676	0.4056

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