Questions S3 - A-Level Maths

	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.

OCR MEI S3 2012 June Q2

10 marks

1. Give two reasons why an investigator might need to take a sample in order to obtain information about a population.
2. State two requirements of a sample.
3. Discuss briefly the advantage of the sampling being random.
1. Under what circumstances might one use a Wilcoxon single sample test in order to test a hypothesis about the median of a population? What distributional assumption is needed for the test?
2. On a stretch of road leading out of the centre of a town, highways officials have been monitoring the speed of the traffic in case it has increased. Previously it was known that the median speed on this stretch was 28.7 miles per hour. For a random sample of 12 vehicles on the stretch, the following speeds were recorded. $$\begin{array} { l l l l l l l l l l l l } 32.0 & 29.1 & 26.1 & 35.2 & 34.4 & 28.6 & 32.3 & 28.5 & 27.0 & 33.3 & 28.2 & 31.9 \end{array}$$ Carry out a test, with a $5 \%$ significance level, to see whether the speed of the traffic on this stretch of road seems to have increased on the whole.
  [0pt] [10]

OCR MEI S3 2012 June Q3

3 The triathlon is a sports event in which competitors take part in three stages, swimming, cycling and running, one straight after the other. The winner is the competitor with the shortest overall time. In this question the times for the separate stages are assumed to be Normally distributed and independent of each other. For a particular triathlon event in which there was a very large number of competitors, the mean and standard deviation of the times, measured in minutes, for each stage were as follows.

For a randomly chosen competitor, find the probability that the swimming time is between 10 and 13 minutes.
For a randomly chosen competitor, find the probability that the running time exceeds the swimming time by more than 10 minutes.
For a randomly chosen competitor, find the probability that the swimming and running times combined exceed $\frac { 2 } { 3 }$ of the cycling time.
In a different triathlon event the total times, in minutes, for a random sample of 12 competitors were as follows. $$\begin{array} { l l l l l l l l l l l l } 103.59 & 99.04 & 85.03 & 81.34 & 106.79 & 89.14 & 98.55 & 98.22 & 108.87 & 116.29 & 102.51 & 92.44 \end{array}$$ Find a 95\% confidence interval for the mean time of all competitors in this event.
Discuss briefly whether the assumptions of Normality and independence for the stages of triathlon events are reasonable.

OCR MEI S3 2012 June Q4

4 The numbers of call-outs per day received by a fire station for a random sample of 255 weekdays were recorded as follows.

Number of call-outs	0	1	2	3	4	5 or more
Frequency (days)	145	79	22	6	3	0

The mean number of call-outs per day for these data is 0.6 . A Poisson model, using this sample mean of 0.6 , is fitted to the data, and gives the following expected frequencies (correct to 3 decimal places).

Number of call-outs	0	1	2	3	4	5 or more
Expected frequency	139.947	83.968	25.190	5.038	0.756	0.101

Using a $5 \%$ significance level, carry out a test to examine the goodness of fit of the model to the data. The time $T$, measured in days, that elapses between successive call-outs can be modelled using the exponential distribution for which $\mathrm { f } ( t )$, the probability density function, is $$\mathrm { f } ( t ) = \begin{cases} 0 & t < 0 ,
\lambda \mathrm { e } ^ { - \lambda t } & t \geqslant 0 , \end{cases}$$ where $\lambda$ is a positive constant.
For the distribution above, it can be shown that $\mathrm { E } ( T ) = \frac { 1 } { \lambda }$. Given that the mean time between successive call-outs is $\frac { 5 } { 3 }$ days, write down the value of $\lambda$.
Find $\mathrm { F } ( t )$, the cumulative distribution function.
Find the probability that the time between successive call-outs is more than 1 day.
Find the median time that elapses between successive call-outs.

OCR MEI S3 2013 June Q1

1 In the past, the times for workers in a factory to complete a particular task had a known median of 7.4 minutes. Following a review, managers at the factory wish to know if the median time to complete the task has been reduced.

A random sample of 12 times, in minutes, gives the following results. $$\begin{array} { l l l l l l l l l l l l } 6.90 & 7.23 & 6.54 & 7.62 & 7.04 & 7.33 & 6.74 & 6.45 & 7.81 & 7.71 & 7.50 & 6.32 \end{array}$$ Carry out an appropriate test using a $5 \%$ level of significance.
Some time later, a much larger random sample of times gives the following results. $$n = 80 \quad \sum x = 555.20 \quad \sum x ^ { 2 } = 3863.9031$$ Find a $95 \%$ confidence interval for the true mean time for the task. Justify your choice of which distribution to use.
Describe briefly one advantage and one disadvantage of having a $99 \%$ confidence interval instead of a $95 \%$ confidence interval.

OCR MEI S3 2013 June Q2

2 A company supplying cattle feed to dairy farmers claims that its new brand of feed will increase average milk yields by 10 litres per cow per week. A farmer thinks the increase will be less than this and decides to carry out a statistical investigation using a paired $t$ test. A random sample of 10 dairy cows are given the new feed and then their milk yields are compared with their yields when on the old feed. The yields, in litres per week, for the 10 cows are as follows.

Cow	A	B	C	D	E	F	G	H	I	J
Old feed	144	130	132	146	137	140	140	149	138	133
New feed	148	139	138	159	138	148	146	156	147	145

Why is it sensible to use a paired test?
State the condition necessary for a paired $t$ test.
Assuming the condition stated in part (ii) is met, carry out the test, using a significance level of $5 \%$, to see whether it appears that the company's claim is justified.
Find a 95\% confidence interval for the mean increase in the milk yield using the new feed.

OCR MEI S3 2013 June Q3

3 The random variable $X$ has the following probability density function, $\mathrm { f } ( x )$. $$f ( x ) = \begin{cases} k x ( x - 5 ) ^ { 2 } & 0 \leqslant x < 5
0 & \text { elsewhere } \end{cases}$$

Sketch $\mathrm { f } ( x )$.
Find, in terms of $k$, the cumulative distribution function, $\mathrm { F } ( x )$.
Hence show that $k = \frac { 12 } { 625 }$. The random variable $X$ is proposed as a model for the amount of time, in minutes, lost due to stoppages during a football match. The times lost in a random sample of 60 matches are summarised in the table. The table also shows some of the corresponding expected frequencies given by the model.
Time (minutes) $0 \leqslant x < 1$ $1 \leqslant x < 2$ $2 \leqslant x < 3$ $3 \leqslant x < 4$ $4 \leqslant x < 5$
Observed frequency 5 15 23 11 6
Expected frequency 17.76 9.12 1.632
Find the remaining expected frequencies.
Carry out a goodness of fit test, using a significance level of $2.5 \%$, to see if the model might be suitable in this context.

OCR MEI S3 2013 June Q4

4 A company that makes meat pies includes a "small" size in its product range. These pies consist of a pastry case and meat filling, the weights of which are independent of each other. The weight of the pastry case, $C$, is Normally distributed with mean 96 g and variance $21 \mathrm {~g} ^ { 2 }$. The weight of the meat filling, $M$, is Normally distributed with mean 57 g and variance $14 \mathrm {~g} ^ { 2 }$.

Find the probability that, in a randomly chosen pie, the weight of the pastry case is between 90 and 100 g .
The wrappers on the pies state that the weight is 145 g . Find the proportion of pies that are underweight.
The pies are sold in packs of 4 . Find the value of $w$ such that, in $95 \%$ of packs, the total weight of the 4 pies in a randomly chosen pack exceeds $w \mathrm {~g}$.
It is required that the weight of the meat filling in a pie should be at least $35 \%$ of the total weight. Show that this means that $0.65 M - 0.35 C \geqslant 0$. Hence find the probability that, in a randomly chosen pie, this requirement is met.

OCR MEI S3 2014 June Q1

Let $X$ be a random variable with variance $\sigma ^ { 2 }$. The independent random variables $X _ { 1 }$ and $X _ { 2 }$ are both distributed as $X$. Write down the variances of $X _ { 1 } + X _ { 2 }$ and $2 X$; explain why they are different. A large company has produced an aptitude test which consists of three parts. The parts are called mathematical ability, spatial awareness and communication. The scores obtained by candidates in the three parts are continuous random variables $X , Y$ and $W$ which have been found to have independent Normal distributions with means and standard deviations as shown in the table.
Mean Standard deviation
Mathematical ability, $X$ 30.1 5.1
Spatial awareness, $Y$ 25.4 4.2
Communication, $W$ 28.2 3.9
Find the probability that a randomly selected candidate obtains a score of less than 22 in the mathematical ability part of the test.
Find the probability that a randomly selected candidate obtains a total score of at least 100 in the whole test.
For a particular role in the company, the score $2 X + Y$ is calculated. Find the score that is exceeded by only $2 \%$ of candidates.
For a different role, a candidate must achieve a score in communication which is at least $60 \%$ of the score obtained in mathematical ability. What proportion of candidates do not achieve this?

OCR MEI S3 2014 June Q2

Explain what is meant by a simple random sample. A manufacturer produces tins of paint which nominally contain 1 litre. The quantity of paint delivered by the machine that fills the tins can be assumed to be a Normally distributed random variable. The machine is designed to deliver an average of 1.05 litres to each tin. However, over time paint builds up in the delivery nozzle of the machine, reducing the quantity of paint delivered. Random samples of 10 tins are taken regularly from the production process. If a significance test, carried out at the $5 \%$ level, suggests that the average quantity of paint delivered is less than 1.02 litres, the machine is cleaned.
By carrying out an appropriate test, determine whether or not the sample below leads to the machine being cleaned. $$\begin{array} { l l l l l l l l l l } 0.994 & 1.010 & 1.021 & 1.015 & 1.016 & 1.022 & 1.009 & 1.007 & 1.011 & 1.026 \end{array}$$ Each time the machine has been cleaned, a random sample of 10 tins is taken to determine whether or not the average quantity of paint delivered has returned to 1.05 litres.
On one occasion after the machine has been cleaned, the quality control manager thinks that the distribution of the quantity of paint is symmetrical but not necessarily Normal. The sample on this occasion is as follows.
1.055 1.064 1.063 1.043 1.062 1.070 1.059 1.044 1.054
1.053
By carrying out an appropriate test at the $5 \%$ level of significance, determine whether or not this sample supports the conclusion that the average quantity of paint delivered is 1.05 litres.

OCR MEI S3 2014 June Q3

A personal trainer believes that drinking a glass of beetroot juice an hour before exercising enables endurance tests to be completed more quickly. To test his belief he takes a random sample of 12 of his trainees and, on two occasions, asks them to carry out 100 repetitions of a particular exercise as quickly as possible. Each trainee drinks a glass of water on one occasion and a glass of beetroot juice on the other occasion. The times in seconds taken by the trainees are given in the table.
Trainee Water Beetroot juice
A 75.1 72.9
B 86.2 79.9
C 77.3 71.6
D 89.1 90.2
E 67.9 68.2
F 101.5 95.2
G 82.5 76.5
H 83.3 80.2
I 102.5 99.1
J 91.3 82.2
K 92.5 90.1
L 77.2 77.9
The trainer wishes to test his belief using a paired $t$ test at the $1 \%$ level of significance. Assuming any necessary assumptions are valid, carry out a test of the hypotheses $\mathrm { H } _ { 0 } : \mu _ { D } = 0 , \mathrm { H } _ { 1 } : \mu _ { D } < 0$, where $\mu _ { D }$ is the population mean difference in times (time with beetroot juice minus time with water).
An ornithologist believes that the number of birds landing on the bird feeding station in her garden in a given interval of time during the morning should follow a Poisson distribution. In order to test her belief, she makes the following observations in 60 randomly chosen minutes one morning.
Number of birds 0 1 2 3 4 5 6 $\geqslant 7$
Frequency 2 5 10 17 14 7 4 1
Given that the data in the table have a mean value of 3.3, use a goodness of fit test, with a significance level of $5 \%$, to investigate whether the ornithologist is justified in her belief.

OCR MEI S3 2014 June Q4

4 The probability density function of a random variable $X$ is given by $$\mathrm { f } ( x ) = \begin{cases} k x & 0 \leqslant x \leqslant a
k ( 2 a - x ) & a < x \leqslant 2 a
0 & \text { otherwise } \end{cases}$$ where $a$ and $k$ are positive constants.

Sketch $\mathrm { f } ( x )$. Hence explain why $\mathrm { E } ( X ) = a$.
Show that $k = \frac { 1 } { a ^ { 2 } }$.
Find $\operatorname { Var } ( X )$ in terms of $a$. In order to estimate the value of $a$, a random sample of size 50 is taken from the distribution. It is found that the sample mean and standard deviation are $\bar { x } = 1.92$ and $s = 0.8352$.
Construct a symmetrical $95 \%$ confidence interval for $a$. Give one reason why the answer is only approximate.
A non-statistician states that the probability that $a$ lies in the interval found in part (iv) is 0.95 . Comment on this statement. \section*{END OF QUESTION PAPER} \section*{OCR $^ { \text {® } }$}

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

Time (minutes)	\(0 \leqslant x < 1\)	\(1 \leqslant x < 2\)	\(2 \leqslant x < 3\)	\(3 \leqslant x < 4\)	\(4 \leqslant x < 5\)
Observed frequency	5	15	23	11	6
Expected frequency			17.76	9.12	1.632

	Mean	Standard deviation
Mathematical ability, \(X\)	30.1	5.1
Spatial awareness, \(Y\)	25.4	4.2
Communication, \(W\)	28.2	3.9

Trainee	Water	Beetroot juice
A	75.1	72.9
B	86.2	79.9
C	77.3	71.6
D	89.1	90.2
E	67.9	68.2
F	101.5	95.2
G	82.5	76.5
H	83.3	80.2
I	102.5	99.1
J	91.3	82.2
K	92.5	90.1
L	77.2	77.9

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

1.055	1.064	1.063	1.043	1.062	1.070	1.059	1.044	1.054
1.053

Number of birds	0	1	2	3	4	5	6	\(\geqslant 7\)
Frequency	2	5	10	17	14	7	4	1

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