Questions S2 - A-Level Maths

Find the probability that a randomly selected bottle contains at least 750 ml .
A case of wine consists of 6 bottles. Find the probability that all 6 bottles in a case contain at least 750 ml .
Find the probability that, in a random sample of 25 cases, there are at least 2 cases in which all 6 bottles contain at least 750 ml . It is decided to increase the proportion of bottles which contain at least 750 ml to $98 \%$.
This can be done by changing the value of $\mu$, but retaining the original value of $\sigma$. Find the required value of $\mu$.
An alternative is to change the value of $\sigma$, but retain the original value of $\mu$. Find the required value of $\sigma$.
Comment briefly on which method might be easier to implement and which might be preferable to the vineyard owners.

OCR MEI S2 2012 June Q4

17 marks Standard +0.3

Mary is opening a cake shop. As part of her market research, she carries out a survey into which type of cake people like best. She offers people 4 types of cake to taste: chocolate, carrot, lemon and ginger. She selects a random sample of 150 people and she classifies the people as children and adults. The results are as follows.

\multirow{2}{*}{}		Classification of person		\multirow{2}{*}{Row totals}
		Child	Adult
\multirow{4}{*}{Type of cake}	Chocolate	34	23	57
	Carrot	16	18	34
	Lemon	4	18	22
	Ginger	13	24	37
Column totals		67	83	150

The contributions to the test statistic for the usual $\chi ^ { 2 }$ test are shown in the table below.

Classification of person

\cline { 3 - 4 } \multicolumn{2}{|c|}{}

The sum of these contributions, correct to 2 decimal places, is 12.86 .

Calculate the expected frequency for children preferring chocolate cake. Verify the corresponding contribution, 2.8646, to the test statistic.
Carry out the test at the $1 \%$ level of significance.

Mary buys flour in bags which are labelled as containing 5 kg . She suspects that the average contents of these bags may be less than 5 kg . In order to test this, she selects a random sample of 8 bags and weighs their contents. Assuming that weights are Normally distributed with standard deviation 0.0072 kg , carry out a test at the $5 \%$ level, given that the weights of the 8 bags in kg are as follows.
4.992
4.981
4.982
4.996
4.991
5.006
5.009
5.003
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OCR MEI S2 2013 June Q1

18 marks Standard +0.3

1 Salbutamol is a drug used to improve lung function. In a medical trial, a random sample of 60 people with impaired lung function was selected. The forced expiratory volume in one second (FEV1) was measured for each person, both before being given salbutamol and again after a two-week course of the drug. The variables $x$ and $y$, measured in suitable units, represent FEV1 before and after the two-week course respectively. The data are illustrated in the scatter diagram below, together with the summary statistics for these data.
\includegraphics[max width=\textwidth, alt={}, center]{f3690bc0-3392-4f29-86f7-797d33fab4f1-2_682_1024_502_516} Summary statistics: $$n = 60 , \quad \sum x = 43.62 , \quad \sum y = 55.15 , \quad \sum x ^ { 2 } = 32.68 , \quad \sum y ^ { 2 } = 51.44 , \quad \sum x y = 40.66$$

Calculate the sample product moment correlation coefficient.
Carry out a hypothesis test at the $5 \%$ significance level to investigate whether there is positive correlation between FEV1 before and after the course.
State the distributional assumption which is necessary for this test to be valid. State, with a reason, whether the assumption appears to be valid.
Explain the meaning of the term 'significance level'.
Calculate the values of the summary statistics if the data point $x = 0.55 , y = 1.00$ had been incorrectly recorded as $x = 1.00 , y = 0.55$.

OCR MEI S2 2013 June Q2

18 marks Standard +0.3

2 Suppose that 3\% of the population of a large city have red hair.

A random sample of 10 people from the city is selected. Find the probability that there is at least one person with red hair in this sample. A random sample of 60 people from the city is selected. The random variable $X$ represents the number of people in this sample who have red hair.
Explain why the distribution of $X$ may be approximated by a Poisson distribution. Write down the mean of this Poisson distribution.
Hence find
(A) $\mathrm { P } ( X = 2 )$,
(B) $\mathrm { P } ( X > 2 )$.
Discuss whether or not it would be appropriate to model $X$ using a Normal approximating distribution. A random sample of 5000 people from the city is selected.
State the exact distribution of the number of people with red hair in the sample.
Use a suitable Normal approximating distribution to find the probability that there are at least 160 people with red hair in the sample.

OCR MEI S2 2013 June Q3

18 marks Standard +0.3

3 The scores, $X$, in Paper 1 of an English examination have an underlying Normal distribution with mean 76 and standard deviation 12. The scores are reported as integer marks. So, for example, a score for which $75.5 \leqslant X < 76.5$ is reported as 76 marks.

Find the probability that a candidate's reported mark is 76 .
Find the probability that a candidate's reported mark is at least 80 .
Three candidates are chosen at random. Find the probability that exactly one of these three candidates' reported marks is at least 80 . The proportion of candidates who receive an A* grade (the highest grade) must not exceed $10 \%$ but should be as close as possible to $10 \%$.
Find the lowest reported mark that should be awarded an A* grade. The scores in Paper 2 of the examination have an underlying Normal distribution with mean $\mu$ and standard deviation 12.
Given that $20 \%$ of candidates receive a reported mark of 50 or less, find the value of $\mu$.

OCR MEI S2 2013 June Q4

18 marks Standard +0.3

4 An art gallery is holding an exhibition. A random sample of 150 visitors to the exhibition is selected. The visitors are asked which of four artists they prefer. Their preferences, classified according to whether the visitor is female or male, are given in the table.

		Artist preferred
\cline { 3 - 6 } \multicolumn{2}{\|c\|}{}	Monet	Renoir	Degas	Cézanne
\multirow{2}{*}{Sex}	Male	8	25	18	19
\cline { 2 - 6 }	Female	18	35	10	17

Carry out a test at the $10 \%$ significance level to examine whether there is any association between artist preferred and sex of visitor. Your working should include a table showing the contributions of each cell to the test statistic.
For each artist, comment briefly on how the preferences of each sex compare with what would be expected if there were no association.

OCR MEI S2 2014 June Q1

18 marks Standard +0.3

1 A medical student is investigating the claim that young adults with high diastolic blood pressure tend to have high systolic blood pressure. The student measures the diastolic and systolic blood pressures of a random sample of ten young adults. The data are shown in the table and illustrated in the scatter diagram.

Diastolic blood pressure	60	61	62	63	73	76	84	87	90	95
Systolic blood pressure	98	121	118	114	108	112	132	130	134	139

\includegraphics[max width=\textwidth, alt={}, center]{17e474c4-f5be-4ca1-b7c3-e444b46c3bec-2_865_809_593_628}

Calculate the value of Spearman's rank correlation coefficient for these data.
Carry out a hypothesis test at the $5 \%$ significance level to examine whether there is positive association between diastolic blood pressure and systolic blood pressure in the population of young adults.
Explain why, in the light of the scatter diagram, it might not be valid to carry out a test based on the product moment correlation coefficient. The product moment correlation coefficient between the diastolic and systolic blood pressures of a random sample of 10 athletes is 0.707 .
Carry out a hypothesis test at the $1 \%$ significance level to investigate whether there appears to be positive correlation between these two variables in the population of athletes. You may assume that in this case such a test is valid.

OCR MEI S2 2014 June Q2

17 marks Standard +0.3

2 Manufacturing defects occur in a particular type of aluminium sheeting randomly, independently and at a constant average rate of 1.7 defects per square metre.

Explain the meaning of the term 'independently' and name the distribution that models this situation.
Find the probability that there are exactly 2 defects in a sheet of area 1 square metre.
Find the probability that there are exactly 12 defects in a sheet of area 7 square metres. In another type of aluminium sheet, defects occur randomly, independently and at a constant average rate of 0.8 defects per square metre.
A large box is made from 2 square metres of the first type of sheet and 2 square metres of the second type of sheet, chosen independently. Show that the probability that there are at least 8 defects altogether in the box is 0.1334 . A random sample of 100 of these boxes is selected.
State the exact distribution of the number of boxes which have at least 8 defects.
Use a suitable approximating distribution to find the probability that there are at least 20 boxes in the sample which have at least 8 defects.

OCR MEI S2 2014 June Q3

19 marks Standard +0.3

3 The wing lengths of native English male blackbirds, measured in mm , are Normally distributed with mean 130.5 and variance 11.84.

Find the probability that a randomly selected native English male blackbird has a wing length greater than 135 mm .
Given that $1 \%$ of native English male blackbirds have wing length more than $k \mathrm {~mm}$, find the value of $k$.
Find the probability that a randomly selected native English male blackbird has a wing length which is 131 mm correct to the nearest millimetre. It is suspected that Scandinavian male blackbirds have, on average, longer wings than native English male blackbirds. A random sample of 20 Scandinavian male blackbirds has mean wing length 132.4 mm . You may assume that wing lengths in this population are Normally distributed with variance $11.84 \mathrm {~mm} ^ { 2 }$.
Carry out an appropriate hypothesis test, at the $5 \%$ significance level.
Discuss briefly one advantage and one disadvantage of using a $10 \%$ significance level rather than a $5 \%$ significance level in hypothesis testing in general.

OCR MEI S2 2014 June Q4

18 marks Standard +0.3

4 A researcher at a large company thinks that there may be some relationship between the numbers of working days lost due to illness per year and the ages of the workers in the company. The researcher selects a random sample of 190 workers. The ages of the workers and numbers of days lost for a period of 1 year are summarised below.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	Working days lost
\cline { 3 - 5 } \multicolumn{2}{c\|}{}	0 to 4	5 to 9	10 or more
\multirow{3}{*}{Age}	Under 35	31	27	4
\cline { 2 - 5 }	35 to 50	28	32	8
\cline { 2 - 5 }	Over 50	16	28	16

Carry out a test at the $1 \%$ significance level to investigate whether the researcher's belief appears to be true. Your working should include a table showing the contributions of each cell to the test statistic.
For the 'Over 50' age group, comment briefly on how the working days lost compare with what would be expected if there were no association.
A student decides to reclassify the 'working days lost' into two groups, ' 0 to 4 ' and ' 5 or more', but leave the age groups as before. The test statistic with this classification is 7.08 . Carry out the test at the $1 \%$ level with this new classification, using the same hypotheses as for the original test.
Comment on the results of the two tests. \section*{END OF QUESTION PAPER}

OCR MEI S2 2015 June Q1

17 marks Moderate -0.5

1 A random sample of wheat seedlings is planted and their growth is measured. The table shows their average growth, $y \mathrm {~mm}$, at half-day intervals.

Time $t$ days	0	0.5	1	1.5	2	2.5	3
Average growth $y \mathrm {~mm}$	0	7	21	33	45	56	62

Draw a scatter diagram to illustrate these data.
Calculate the equation of the regression line of $y$ on $t$.
Calculate the value of the residual for the data point at which $t = 2$.
Use the equation of the regression line to calculate an estimate of the average growth after 5 days for wheat seedlings. Comment on the reliability of this estimate. It is suggested that it would be better to replace the regression line by a line which passes through the origin. You are given that the equation of such a line is $y = a t$, where $a = \frac { \sum y t } { \sum t ^ { 2 } }$.
Find the equation of this line and plot the line on your scatter diagram.

OCR MEI S2 2015 June Q2

19 marks Moderate -0.3

2 It was stated in 2012 that $3 \%$ of $\pounds 1$ coins were fakes. Throughout this question, you should assume that this is still the case.

Find the probability that, in a random selection of $25 \pounds 1$ coins, there is exactly one fake coin. A random sample of $250 \pounds 1$ coins is selected.
Explain why a Poisson distribution is an appropriate approximating distribution for the number of fake coins in the sample.
Use a Poisson distribution to find the probability that, in this sample, there are
(A) exactly 10 fake coins,
(B) at least 10 fake coins.
Use a suitable approximating distribution to find the probability that there are at least 50 fake coins in a sample of 2000 coins. It is known that $0.2 \%$ of another type of coin are fakes.
A random sample of size $n$ of these coins is taken. Using a Poisson approximating distribution, show that the probability of at most one fake coin in the sample is equal to $\mathrm { e } ^ { - \lambda } + \lambda \mathrm { e } ^ { - \lambda }$, where $\lambda = 0.002 n$.
Use the approximation $\mathrm { e } ^ { - \lambda } + \lambda \mathrm { e } ^ { - \lambda } \approx 1 - \frac { \lambda ^ { 2 } } { 2 }$ for small values of $\lambda$ to estimate the value of $n$ for which the probability in part ( $\mathbf { v }$ ) is equal to 0.995 .

OCR MEI S2 2015 June Q3

16 marks Moderate -0.3

3 The random variable $X$ represents the weight in kg of a randomly selected male dog of a particular breed. $X$ is Normally distributed with mean 30.7 and standard deviation 3.5.

Find
(A) $\mathrm { P } ( X < 30 )$,
(B) $P ( 25 < X < 35 )$.
Five of these dogs are chosen at random. Find the probability that each of them weighs at least 30 kg .
The weights of females of the same breed of dog are Normally distributed with mean 26.8 kg . Given that $5 \%$ of female dogs of this breed weigh more than 30 kg , find the standard deviation of their weights.
Sketch the distributions of the weights of male and female dogs of this breed on a single diagram.

OCR MEI S2 2015 June Q4

20 marks Standard +0.3

As part of an investigation into smoking, a random sample of 120 students was selected. The students were asked whether they were smokers, and also whether either of their parents were smokers. The results are summarised in the table below. Test, at the $5 \%$ significance level, whether there is any association between the smoking habits of the students and their parents.
At least one
parent smokes
Neither parent
smokes
Student smokes 21 27
Student does not smoke 17 55
The manufacturer of a particular brand of cigarette claims that the nicotine content of these cigarettes is Normally distributed with mean 0.87 mg . A researcher suspects that the mean nicotine content of this brand is higher than the value claimed by the manufacturer. The nicotine content, $x \mathrm { mg }$, is measured for a random sample of 100 cigarettes. The data are summarised as follows. $$\sum x = 88.20 \quad \sum x ^ { 2 } = 78.68$$ Carry out a test at the $1 \%$ significance level to investigate the researcher's belief. \section*{END OF QUESTION PAPER}

OCR MEI S2 2016 June Q1

18 marks Standard +0.3

1 A researcher believes that there may be negative association between the quantity of fertiliser used and the percentage of the population who live in rural areas in different countries. The data below show the percentage of the population who live in rural areas and the fertiliser use measured in kg per hectare, for a random sample of 11 countries.

Percentage of population	33	6	58	35	81	69	61	7	74	71	17
Fertiliser use	76	44	6	68	3	10	7	176	5	137	157

Draw a scatter diagram to illustrate the data.
Explain why it might not be valid to carry out a test based on the product moment correlation coefficient in this case.
Calculate the value of Spearman's rank correlation coefficient.
Carry out a hypothesis test at the $1 \%$ significance level to investigate the researcher's belief.
Explain the meaning of ' $1 \%$ significance level'.
In order to carry out a test based on Spearman's rank correlation coefficient, what modelling assumptions, if any, are required about the underlying distribution?

OCR MEI S2 2016 June Q2

16 marks Standard +0.3

2 When a genetic sequence of plant DNA is given a dose of radiation, some of the genes may mutate. The probability that a gene mutates is 0.012 . Mutations occur randomly and independently.

Explain the meanings of the terms 'randomly' and 'independently' in this context. A short stretch of DNA containing 20 genes is given a dose of radiation.
Find the probability that exactly 1 out of the 20 genes mutates. A longer stretch of DNA containing 500 genes is given a dose of radiation.
Explain why a Poisson distribution is an appropriate approximating distribution for the number of genes that mutate.
Use this Poisson distribution to find the probability that there are
(A) exactly two genes that mutate,
(B) at least two genes that mutate. A third stretch of DNA containing 50000 genes is given a dose of radiation.
Use a suitable approximating distribution to find the probability that there are at least 650 genes that mutate.

OCR MEI S2 2016 June Q3

18 marks Moderate -0.3

3 Many types of computer have cooling fans. The random variable $X$ represents the lifetime in hours of a particular model of cooling fan. $X$ is Normally distributed with mean 50600 and standard deviation 3400.

Find $\mathrm { P } ( 50000 < X < 55000 )$.
The manufacturers claim that at least $95 \%$ of these fans last longer than 45000 hours. Is this claim valid?
Find the value of $h$ for which $99.9 \%$ of these fans last $h$ hours or more.
The random variable $Y$ represents the lifetime in hours of a different model of cooling fan. $Y$ is Normally distributed with mean $\mu$ and standard deviation $\sigma$. It is known that $\mathrm { P } ( Y < 60000 ) = 0.6$ and $\mathrm { P } ( Y > 50000 ) = 0.9$. Find the values of $\mu$ and $\sigma$.
Sketch the distributions of lifetimes for both types of cooling fan on a single diagram.

OCR MEI S2 2016 June Q4

20 marks Moderate -0.3

A random sample of 80 GCSE students was selected to take part in an investigation into whether attitudes to mathematics differ between girls and boys. The students were asked if they agreed with the statement 'Mathematics is one of my favourite subjects'. They were given three options 'Agree', 'Disagree', 'Neither agree nor disagree'. The results, classified according to sex, are summarised in the table below.
Agree Disagree Neither
Male 17 13 8
Female 12 11 19
The contributions to the test statistic for the usual $\chi ^ { 2 }$ test are shown in the table below.
Agree Disagree Neither
Male 0.7550 0.2246 1.8153
Female 0.6831 0.2032 1.6424
1. Calculate the expected frequency for females who agree. Verify the corresponding contribution, 0.6831 , to the test statistic.
2. Carry out the test at the $5 \%$ level of significance.
The level of radioactivity in limpets (a type of shellfish) in the sea near to a nuclear power station is regularly monitored. Over a period of years it has been found that the level (measured in suitable units) is Normally distributed with mean 5.64. Following an incident at the power station, a researcher suspects that the mean level of radioactivity in limpets may have increased. The researcher selects a random sample of 60 limpets. Their levels of radioactivity, $x$ (measured in the same units), are summarised as follows. $$\sum x = 373 \quad \sum x ^ { 2 } = 2498$$ Carry out a test at the $5 \%$ significance level to investigate the researcher's belief.

OCR MEI S2 Q3

18 marks Standard +0.3

3 In a triathlon, competitors have to swim 600 metres, cycle 40 kilometres and run 10 kilometres. To improve her strength, a triathlete undertakes a training programme in which she carries weights in a rucksack whilst running. She runs a specific course and notes the total time taken for each run. Her coach is investigating the relationship between time taken and weight carried. The times taken with eight different weights are illustrated on the scatter diagram below, together with the summary statistics for these data. The variables $x$ and $y$ represent weight carried in kilograms and time taken in minutes respectively.
\includegraphics[max width=\textwidth, alt={}, center]{d138173d-c70c-46db-b9b9-d5f19334c5f1-04_627_1536_630_281} Summary statistics: $n = 8 , \Sigma x = 36 , \Sigma y = 214.8 , \Sigma x ^ { 2 } = 204 , \Sigma y ^ { 2 } = 5775.28 , \Sigma x y = 983.6$.

Calculate the equation of the regression line of $y$ on $x$. On one of the eight runs, the triathlete was carrying 4 kilograms and took 27.5 minutes. On this run she was delayed when she tripped and fell over.
Calculate the value of the residual for this weight.
The coach decides to recalculate the equation of the regression line without the data for this run. Would it be preferable to use this recalculated equation or the equation found in part (i) to estimate the delay when the triathlete tripped and fell over? Explain your answer. The triathlete's coach claims that there is positive correlation between cycling and swimming times in triathlons. The product moment correlation coefficient of the times of twenty randomly selected competitors in these two sections is 0.209 .
Carry out a hypothesis test at the $5 \%$ level to examine the coach's claim, explaining your conclusions clearly.
What distributional assumption is necessary for this test to be valid? How can you use a scatter diagram to decide whether this assumption is likely to be true?

Edexcel S2 2014 January Q1

8 marks Moderate -0.3

The probability of a leaf cutting successfully taking root is 0.05

Find the probability that, in a batch of 10 randomly selected leaf cuttings, the number taking root will be

1. exactly 1
2. more than 2 A second random sample of 160 leaf cuttings is selected.
Using a suitable approximation, estimate the probability of at least 10 leaf cuttings taking root.

Edexcel S2 2014 January Q2

10 marks Moderate -0.3

2. Bill owns a restaurant. Over the next four weeks Bill decides to carry out a sample survey to obtain the customers’ opinions.

Suggest a suitable sampling frame for the sample survey.
Identify the sampling units.
Give one advantage and one disadvantage of taking a census rather than a sample survey. Bill believes that only $30 \%$ of customers would like a greater choice on the menu. He takes a random sample of 50 customers and finds that 20 of them would like a greater choice on the menu.
Test, at the $5 \%$ significance level, whether or not the percentage of customers who would like a greater choice on the menu is more than Bill believes. State your hypotheses clearly.

Edexcel S2 2014 January Q3

11 marks Standard +0.3

The continuous random variable $X$ has cumulative distribution function given by

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0 \\ \frac { 1 } { 6 } x ( x + 1 ) & 0 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{array} \right.$$

Find the value of $a$ such that $\mathrm { P } ( X > a ) = 0.4$ Give your answer to 3 significant figures.
Use calculus to find (i) $\mathrm { E } ( X )$
(ii) $\operatorname { Var } ( X )$.

Edexcel S2 2014 January Q4

7 marks Standard +0.3

The number of telephone calls per hour received by a business is a random variable with distribution $\operatorname { Po } ( \lambda )$.

Charlotte records the number of calls, $C$, received in 4 hours. A test of the null hypothesis $\mathrm { H } _ { 0 } : \lambda = 1.5$ is carried out.
$\mathrm { H } _ { 0 }$ is rejected if $C > 10$

Write down the alternative hypothesis.
Find the significance level of the test. Given that $\mathrm { P } ( C > 10 ) < 0.1$
find the largest possible value of $\lambda$ that can be found by using the tables.

Edexcel S2 2014 January Q5

12 marks Standard +0.8

5. A school photocopier breaks down randomly at a rate of 15 times per year.

Find the probability that there will be exactly 3 breakdowns in the next month.
Show that the probability that there will be at least 2 breakdowns in the next month is 0.355 to 3 decimal places.
Find the probability of at least 2 breakdowns in each of the next 4 months. The teachers would like a new photocopier. The head teacher agrees to monitor the situation for the next 12 months. The head teacher decides he will buy a new photocopier if there is more than 1 month when the photocopier has at least 2 breakdowns.
Find the probability that the head teacher will buy a new photocopier.

Edexcel S2 2014 January Q6

15 marks Standard +0.3

The continuous random variable $X$ has probability density function given by

$$f ( x ) = \left\{ \begin{array} { c c } k ( x + 1 ) ^ { 2 } & - 1 \leqslant x \leqslant 1 \\ k ( 6 - 2 x ) & 1 < x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$ where $k$ is a positive constant.

Sketch the graph of $\mathrm { f } ( x )$.
Show that the value of $k$ is $\frac { 3 } { 20 }$
Define fully the cumulative distribution function $\mathrm { F } ( x )$.
Find the median of $X$, giving your answer to 3 significant figures.

Time \(t\) days	0	0.5	1	1.5	2	2.5	3
Average growth \(y \mathrm {~mm}\)	0	7	21	33	45	56	62

	Agree	Disagree	Neither
Male	0.7550	0.2246	1.8153
Female	0.6831	0.2032	1.6424

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