Questions - OCR MEI S3 - A-Level Maths

OCR MEI S3 2012 June Q2

10 marks

2

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.
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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.

Mean

Standard

deviation

Swimming

11.07

2.36

Cycling

57.33

8.76

Running

24.23

3.75

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

1

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

2

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

3

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 $^ { \text {® } }$}

OCR MEI S3 2016 June Q1

1 A game consists of 20 rounds. Each round is denoted as either a starter, middle or final round. The times taken for each round are independently and Normally distributed with the following parameters (given in seconds).

Type of round	Mean	Standard deviation
Starter	200	15
Middle	220	25
Final	250	20

The game consists of 4 starter, 12 middle and 4 final rounds. Find the probability that

the mean time per round for the 4 final rounds will exceed 260 seconds,
all 20 rounds will be completed in a total time of 75 minutes or less,
the 12 middle rounds will take at least 3.5 times as long in total as the 4 starter rounds,
the mean time per round for the 12 middle rounds will be at least 25 seconds less than the mean time per round for the 4 final rounds.

OCR MEI S3 2016 June Q2

2

A genetic model involving body colour and eye colour of fruit flies predicts that offspring will consist of four phenotypes in the ratio $9 : 3 : 3 : 1$. A random sample of 200 such offspring is taken. Their phenotypes are found to be as follows.
Phenotype Brown body Red eye Brown body Brown eye Black body Red eye Black body Brown eye
Frequency 125 37 32 6
Relative proportion from model 9 3 3 1
Carry out a test, using a $2.5 \%$ level of significance, of the goodness of fit of the genetic model to these data.
The median length of European fruit flies is 2.5 mm . South American fruit flies are believed to be larger than European fruit flies. A random sample of 12 South American fruit flies is taken. The flies are found to have the following lengths (in mm).
$1.7 \quad 1.4$
$3.1 \quad 3.5$
3.8
4.2
2.2
2.9
4.4
2.6
$3.9 \quad 3.2$ Carry out a Wilcoxon signed rank test, using a $5 \%$ level of significance, to test this belief.

OCR MEI S3 2016 June Q3

3 The random variable $X$ has the following probability density function: $$\mathrm { f } ( x ) = \begin{cases} k \left( 1 - x ^ { 2 } \right) & - 1 \leqslant x \leqslant 1
0 & \text { elsewhere } \end{cases}$$ where $k$ is a positive constant.

Calculate the value of $k$.
Sketch the probability density function.
Calculate $\operatorname { Var } ( X )$.
Find a cubic equation satisfied by the upper quartile $q$, and hence verify that $q = 0.35$ to 2 decimal places.
A random sample of 40 values of $X$ is taken. Using a suitable approximating distribution, calculate the probability that the mean of these values is greater than 0.125 . Justify your choice of distribution.

OCR MEI S3 2016 June Q4

4 An insurance company is investigating a new system designed to reduce the average time taken to process claim forms. The company has decided to use 10 experienced employees to process claims using the old system and the new system. Two procedures for comparing the systems are proposed.
Procedure $A$ There are two sets of claim forms, set 1 and set 2. Each contains the same number of forms. Each employee processes set 1 on the old system and set 2 on the new system. The times taken are compared. Procedure $B$ There is just one set of claim forms which each employee processes firstly on the old system and then on the new system. The times taken are compared.

State one weakness of each of these procedures. In fact a third procedure which avoids these two weaknesses is adopted. In this procedure each employee is given a randomly selected set of claim forms. Each set contains the same number of forms. The employees each process their set of claim forms on both systems. The times taken, in minutes, are shown in the table.
Employee 1 2 3 4 5 6 7 8 9 10
Old system 40.5 42.9 52.8 51.7 77.2 66.7 65.2 49.2 55.6 58.3
New system 39.2 40.7 50.6 50.7 71.4 70.5 71.1 47.7 52.1 55.5
Carry out a paired $t$ test at the $5 \%$ level of significance to investigate whether the mean length of time taken to process a set of forms has reduced using the new system.
State fully the usual conditions for a paired $t$ test.
Construct a $99 \%$ confidence interval for the mean reduction in time taken to process a set of forms using the new system.

OCR MEI S3 2008 January Q1

1

The time (in milliseconds) taken by my computer to perform a particular task is modelled by the random variable $T$. The probability that it takes more than $t$ milliseconds to perform this task is given by the expression $\mathrm { P } ( T > t ) = \frac { k } { t ^ { 2 } }$ for $t \geqslant 1$, where $k$ is a constant.
1. Write down the cumulative distribution function of $T$ and hence show that $k = 1$.
2. Find the probability density function of $T$.
3. Find the mean time for the task.
For a different task, the times (in milliseconds) taken by my computer on 10 randomly chosen occasions were as follows. $$\begin{array} { c c c c c c c c c c } 6.4 & 5.9 & 5.0 & 6.2 & 6.8 & 6.0 & 5.2 & 6.5 & 5.7 & 5.3 \end{array}$$ From past experience it is thought that the median time for this task is 5.4 milliseconds. Carry out a test at the $5 \%$ level of significance to investigate this, stating your hypotheses carefully.

OCR MEI S3 2008 January Q2

2 In the vegetable section of a local supermarket, leeks are on sale either loose (and unprepared) or prepared in packs of 4 . The weights of unprepared leeks are modelled by the random variable $X$ which has the Normal distribution with mean 260 grams and standard deviation 24 grams. The prepared leeks have had $40 \%$ of their weight removed, so that their weights, $Y$, are modelled by $Y = 0.6 X$.

Find the probability that a randomly chosen unprepared leek weighs less than 300 grams.
Find the probability that a randomly chosen prepared leek weighs more than 175 grams.
Find the probability that the total weight of 4 randomly chosen prepared leeks in a pack is less than 600 grams.
What total weight of prepared leeks in a randomly chosen pack of 4 is exceeded with probability 0.975 ?
Sandie is making soup. She uses 3 unprepared leeks and 2 onions. The weights of onions are modelled by the Normal distribution with mean 150 grams and standard deviation 18 grams. Find the probability that the total weight of her ingredients is more than 1000 grams.
A large consignment of unprepared leeks is delivered to the supermarket. A random sample of 100 of them is taken. Their weights have sample mean 252.4 grams and sample standard deviation 24.6 grams. Find a $99 \%$ confidence interval for the true mean weight of the leeks in this consignment.

OCR MEI S3 2008 January Q3

3 Engineers in charge of a chemical plant need to monitor the temperature inside a reaction chamber. Past experience has shown that when functioning correctly the temperature inside the chamber can be modelled by a Normal distribution with mean $380 ^ { \circ } \mathrm { C }$. The engineers are concerned that the mean operating temperature may have fallen. They decide to test the mean using the following random sample of 12 recent temperature readings.

374.0	378.1	363.0	357.0	377.9	388.4
379.6	372.4	362.4	377.3	385.2	370.6

Give three reasons why a $t$ test would be appropriate.
Carry out the test using a $5 \%$ significance level. State your hypotheses and conclusion carefully.
Find a 95\% confidence interval for the true mean temperature in the reaction chamber.
Describe briefly one advantage and one disadvantage of having a 99\% confidence interval instead of a 95\% confidence interval.

OCR MEI S3 2008 January Q4

4

In Germany, towards the end of the nineteenth century, a study was undertaken into the distribution of the sexes in families of various sizes. The table shows some data about the numbers of girls in 500 families, each with 5 children. It is thought that the binomial distribution $\mathrm { B } ( 5 , p )$ should model these data.
Number of girls Number of families
0 32
1 110
2 154
3 125
4 63
5 16
1. Use this information to calculate an estimate for the mean number of girls per family of 5 children. Hence show that 0.45 can be taken as an estimate of $p$.
2. Investigate at a $5 \%$ significance level whether the binomial model with $p$ estimated as 0.45 fits the data. Comment on your findings and also on the extent to which the conditions for a binomial model are likely to be met.
A researcher wishes to select 50 families from the 500 in part (a) for further study. Suggest what sort of sample she might choose and describe how she should go about choosing it.

OCR MEI S3 Q2

2 Geoffrey is a university lecturer. He has to prepare five questions for an examination. He knows by experience that it takes about 3 hours to prepare a question, and he models the time (in minutes) taken to prepare one by the Normally distributed random variable $X$ with mean 180 and standard deviation 12, independently for all questions.

One morning, Geoffrey has a gap of 2 hours 50 minutes ( 170 minutes) between other activities. Find the probability that he can prepare a question in this time.
One weekend, Geoffrey can devote 14 hours to preparing the complete examination paper. Find the probability that he can prepare all five questions in this time. A colleague, Helen, has to check the questions.
She models the time (in minutes) to check a question by the Normally distributed random variable $Y$ with mean 50 and standard deviation 6, independently for all questions and independently of $X$. Find the probability that the total time for Geoffrey to prepare a question and Helen to check it exceeds 4 hours.
When working under pressure of deadlines, Helen models the time to check a question in a different way. She uses the Normally distributed random variable $\frac { 1 } { 4 } X$, where $X$ is as above. Find the length of time, as given by this model, which Helen needs to ensure that, with probability 0.9 , she has time to check a question. Ian, an educational researcher, suggests that a better model for the time taken to prepare a question would be a constant $k$ representing "thinking time" plus a random variable $T$ representing the time required to write the question itself, independently for all questions.
Taking $k$ as 45 and $T$ as Normally distributed with mean 120 and standard deviation 10 (all units are minutes), find the probability according to Ian's model that a question can be prepared in less than 2 hours 30 minutes. Juliet, an administrator, proposes that the examination should be reduced in time and shorter questions should be used.
Juliet suggests that Ian's model should be used for the time taken to prepare such shorter questions but with $k = 30$ and $T$ replaced by $\frac { 3 } { 5 } T$. Find the probability as given by this model that a question can be prepared in less than $1 \frac { 3 } { 4 }$ hours.

OCR MEI S3 Q4