Questions S2 - A-Level Maths

For the continuous random variable $V$, it is known that $\mathrm { E } ( V ) = 72.0$. The mean of a random sample of 40 observations of $V$ is denoted by $\bar { V }$. Given that $\mathrm { P } ( \bar { V } < 71.2 ) = 0.35$, estimate the value of $\operatorname { Var } ( V )$.
Explain why you need to use the Central Limit Theorem in part (i), and why its use is justified.

OCR S2 2012 June Q3

7 marks Moderate -0.3

3 It is known that on average one person in three prefers the colour of a certain object to be blue. In a psychological test, 12 randomly chosen people were seated in a room with blue walls, and asked to state independently which colour they preferred for the object. Seven of the 12 people said that they preferred blue. Carry out a significance test, at the $5 \%$ level, of whether the statement "on average one person in three prefers the colour of the object to be blue" is true for people who are seated in a room with blue walls.

OCR S2 2012 June Q4

11 marks Moderate -0.3

4 In a rock, small crystal formations occur at a constant average rate of 3.2 per cubic metre.

State a further assumption needed to model the number of crystal formations in a fixed volume of rock by a Poisson distribution. In the remainder of the question, you should assume that a Poisson model is appropriate.
Calculate the probability that in one cubic metre of rock there are exactly 5 crystal formations.
Calculate the probability that in 0.74 cubic metres of rock there are at least 3 crystal formations.
Use a suitable approximation to calculate the probability that in 10 cubic metres of rock there are at least 36 crystal formations.

OCR S2 2012 June Q5

11 marks Moderate -0.3

5 The acidity $A$ (measured in pH ) of soil of a particular type has a normal distribution. The pH values of a random sample of 80 soil samples from a certain region can be summarised as $$\Sigma a = 496 , \quad \Sigma a ^ { 2 } = 3126 .$$ Test, at the $10 \%$ significance level, whether in this region the mean pH of soil is 6.1 .

OCR S2 2012 June Q6

11 marks Moderate -0.3

6 At a tourist car park, a survey is made of the regions from which cars come.

It is given that $40 \%$ of cars come from the London region. Use a suitable approximation to find the probability that, in a random sample of 32 cars, more than 17 come from the London region. Justify your approximation.
It is given that $1 \%$ of cars come from France. Use a suitable approximation to find the probability that, in a random sample of 90 cars, exactly 3 come from France.

OCR S2 2012 June Q7

12 marks Standard +0.3

7 The continuous random variable $X$ has probability density function $$f ( x ) = \begin{cases} k x ^ { 2 } & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where $a$ and $k$ are constants.

Sketch the graph of $y = \mathrm { f } ( x )$ and explain in non-technical language what this tells you about $X$.
Given that $\mathrm { E } ( X ) = 4.5$, find
(a) the value of $a$,
(b) $\operatorname { Var } ( X )$.

OCR S2 2012 June Q8

12 marks Standard +0.8

8 The random variable $X$ has the distribution $\mathrm { N } \left( \mu , 8 ^ { 2 } \right)$. A test is carried out, at the $5 \%$ significance level, of $\mathrm { H } _ { 0 } : \mu = 30$ against $\mathrm { H } _ { 1 } : \mu > 30$, based on a random sample of size 18 .

Find the critical region for the test.
If $\mu = 30$ and the outcome of the test is that $\mathrm { H } _ { 0 }$ is rejected, state the type of error that is made. On a particular day this test is carried out independently a total of 20 times, and for 4 of these tests the outcome is that $\mathrm { H } _ { 0 }$ is rejected. It is known that the value of $\mu$ remains the same throughout these 20 tests.
Find the probability that $\mathrm { H } _ { 0 }$ is rejected at least 4 times if $\mu = 30$. Hence state whether you think that $\mu = 30$, giving a reason.
Given that the probability of making an error of the type different from that stated in part (ii) is 0.4 , calculate the actual value of $\mu$, giving your answer correct to 4 significant figures. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

OCR S2 2013 June Q1

4 marks Moderate -0.8

1 It is required to select a random sample of 30 pupils from a school with 853 pupils. A student suggests the following method.
"Give each pupil sequentially a three-digit number from 001 to 853 . Use a calculator to generate random three-digit numbers from 0.000 to 0.999 inclusive, multiply the answer by 853 , add 1 and round off to the nearest whole number. Select the corresponding pupil, and repeat as necessary".

Determine which pupil would be picked for each of the following calculator outputs: $$0.103 , \quad 0.104 , \quad 0.105 , \quad 0.106 , \quad 0.107$$
Use your answers to part (i) to show that this method is biased, and suggest an improvement.

OCR S2 2013 June Q2

4 marks Standard +0.3

2 The number of neutrinos that pass through a certain region in one second is a random variable with the distribution $\operatorname { Po } \left( 5 \times 10 ^ { 4 } \right)$. Use a suitable approximation to calculate the probability that the number of neutrinos passing through the region in 40 seconds is less than $1.999 \times 10 ^ { 6 }$.

OCR S2 2013 June Q3

9 marks Challenging +1.2

3 The mean of a sample of 80 independent observations of a continuous random variable $Y$ is denoted by $\bar { Y }$. It is given that $\mathrm { P } ( \bar { Y } \leqslant 157.18 ) = 0.1$ and $\mathrm { P } ( \bar { Y } \geqslant 164.76 ) = 0.7$.

Calculate $\mathrm { E } ( Y )$ and the standard deviation of $Y$.
State
(a) where in your calculations you have used the Central Limit Theorem,
(b) why it was necessary to use the Central Limit Theorem,
(c) why it was possible to use the Central Limit Theorem.

OCR S2 2013 June Q4

7 marks Standard +0.3

4 The number of floods in a certain river plain is known to have a Poisson distribution. It is known that up until 10 years ago the mean number of floods per year was 0.32 . During the last 10 years there were 6 floods. Test at the $1 \%$ significance level whether there is evidence of an increase in the mean number of floods per year.

OCR S2 2013 June Q5

10 marks Moderate -0.3

5 Two random variables $S$ and $T$ have probability density functions given by $$\begin{aligned} & f _ { S } ( x ) = \begin{cases} \frac { 3 } { a ^ { 3 } } ( x - a ) ^ { 2 } & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases} \\ & f _ { T } ( x ) = \begin{cases} c & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases} \end{aligned}$$ where $a$ and $c$ are constants.

On a single diagram sketch both probability density functions.
Calculate the mean of $S$, in terms of $a$.
Use your diagram to explain which of $S$ or $T$ has the bigger variance. (Answers obtained by calculation will score no marks.)

OCR S2 2013 June Q6

11 marks Standard +0.3

6 The random variable $X$ denotes the yield, in kilograms per acre, of a certain crop. Under the standard treatment it is known that $\mathrm { E } ( X ) = 38.4$. Under a new treatment, the yields of 50 randomly chosen regions can be summarised as $$n = 50 , \quad \sum x = 1834.0 , \quad \sum x ^ { 2 } = 70027.37 .$$ Test at the $1 \%$ level whether there has been a change in the mean crop yield.

OCR S2 2013 June Q7

11 marks Standard +0.3

7 Past experience shows that $35 \%$ of the senior pupils in a large school know the regulations about bringing cars to school. The head teacher addresses this subject in an assembly, and afterwards a random sample of 120 senior pupils is selected. In this sample it is found that 50 of these pupils know the regulations. Use a suitable approximation to test, at the $10 \%$ significance level, whether there is evidence that the proportion of senior pupils who know the regulations has increased. Justify your approximation.

OCR S2 2013 June Q8

6 marks Challenging +1.2

8 The random variable $R$ has the distribution $\mathrm { B } ( 14 , p )$. A test is carried out at the $\alpha \%$ significance level of the null hypothesis $\mathrm { H } _ { 0 } : p = 0.25$, against $\mathrm { H } _ { 1 } : p > 0.25$.

Given that $\alpha$ is as close to 5 as possible, find the probability of a Type II error when the true value of $p$ is 0.4 .
State what happens to the probability of a Type II error as
(a) $p$ increases from 0.4,
(b) $\alpha$ increases, giving a reason.

OCR S2 2013 June Q9

10 marks Standard +0.3

9 The managers of a car breakdown recovery service are discussing whether the number of breakdowns per day can be modelled by a Poisson distribution. They agree that breakdowns occur randomly. Manager $A$ says, "it must be assumed that breakdowns occur at a constant rate throughout the day".

Give an improved version of Manager $A$ 's statement, and explain why the improvement is necessary.
Explain whether you think your improved statement is likely to hold in this context. Assume now that the number $B$ of breakdowns per day can be modelled by the distribution $\operatorname { Po } ( \lambda )$.
Given that $\lambda = 9.0$ and $\mathrm { P } \left( B > B _ { 0 } \right) < 0.1$, use tables to find the smallest possible value of $B _ { 0 }$, and state the corresponding value of $\mathrm { P } \left( B > B _ { 0 } \right)$.
Given that $\mathrm { P } ( B = 2 ) = 0.0072$, show that $\lambda$ satisfies an equation of the form $\lambda = 0.12 \mathrm { e } ^ { k \lambda }$, for a value of $k$ to be stated. Evaluate the expression $0.12 \mathrm { e } ^ { k \lambda }$ for $\lambda = 8.5$ and $\lambda = 8.6$, giving your answers correct to 4 decimal places. What can be deduced about a possible value of $\lambda$ ?

OCR MEI S2 2009 January Q1

20 marks Moderate -0.3

1 A researcher is investigating whether there is a relationship between the population size of cities and the average walking speed of pedestrians in the city centres. Data for the population size, $x$ thousands, and the average walking speed of pedestrians, $y \mathrm {~m} \mathrm {~s} ^ { - 1 }$, of eight randomly selected cities are given in the table below.

$x$	18	43	52	94	98	206	784	1530
$y$	1.15	0.97	1.26	1.35	1.28	1.42	1.32	1.64

Calculate the value of Spearman's rank correlation coefficient.
Carry out a hypothesis test at the $5 \%$ significance level to determine whether there is any association between population size and average walking speed. In another investigation, the researcher selects a random sample of six adult males of particular ages and measures their maximum walking speeds. The data are shown in the table below, where $t$ years is the age of the adult and $w \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the maximum walking speed. Also shown are summary statistics and a scatter diagram on which the regression line of $w$ on $t$ is drawn.
$t$ 20 30 40 50 60 70
$w$ 2.49 2.41 2.38 2.14 1.97 2.03
$$n = 6 \quad \Sigma t = 270 \quad \Sigma w = 13.42 \quad \Sigma t ^ { 2 } = 13900 \quad \Sigma w ^ { 2 } = 30.254 \quad \Sigma t w = 584.6$$ \includegraphics[max width=\textwidth, alt={}, center]{77b97142-afb6-41d6-8fec-e982b7a7501b-2_728_1091_1379_529}
Calculate the equation of the regression line of $w$ on $t$.
(A) Use this equation to calculate an estimate of maximum walking speed of an 80 -year-old male.
(B) Explain why it might not be appropriate to use the equation to calculate an estimate of maximum walking speed of a 10 -year-old male.

OCR MEI S2 2009 January Q2

18 marks Moderate -0.3

2 Clover stems usually have three leaves. Occasionally a clover stem has four leaves. This is considered by some to be lucky and is known as a four-leaf clover. On average 1 in 10000 clover stems is a four-leaf clover. You may assume that four-leaf clovers occur randomly and independently. A random sample of 5000 clover stems is selected.

State the exact distribution of $X$, the number of four-leaf clovers in the sample.
Explain why $X$ may be approximated by a Poisson distribution. Write down the mean of this Poisson distribution.
Use this Poisson distribution to find the probability that the sample contains at least one four-leaf clover.
Find the probability that in 20 samples, each of 5000 clover stems, there are exactly 9 samples which contain at least one four-leaf clover.
Find the expected number of these 20 samples which contain at least one four-leaf clover. The table shows the numbers of four-leaf clovers in these 20 samples.
Number of four-leaf clovers 0 1 2 $> 2$
Number of samples 11 7 2 0
Calculate the mean and variance of the data in the table.
Briefly comment on whether your answers to parts (v) and (vi) support the use of the Poisson approximating distribution in part (iii).

OCR MEI S2 2009 January Q3

17 marks Moderate -0.3

3 The number of minutes, $X$, for which a particular model of laptop computer will run on battery power is Normally distributed with mean 115.3 and standard deviation 21.9.

(A) Find $\mathrm { P } ( X < 120 )$.
(B) Find $\mathrm { P } ( 100 < X < 110 )$.
(C) Find the value of $k$ for which $\mathrm { P } ( X > k ) = 0.9$. The number of minutes, $Y$, for which a different model of laptop computer will run on battery power is known to be Normally distributed with mean $\mu$ and standard deviation $\sigma$.
Given that $\mathrm { P } ( Y < 180 ) = 0.7$ and $\mathrm { P } ( Y < 140 ) = 0.15$, find the values of $\mu$ and $\sigma$.
Find values of $a$ and $b$ for which $\mathrm { P } ( a < Y < b ) = 0.95$.

OCR MEI S2 2009 January Q4

17 marks Standard +0.3

4 A gardening research organisation is running a trial to examine the growth and the size of flowers of various plants.

In the trial, seeds of three types of plant are sown. The growth of each plant is classified as good, average or poor. The results are shown in the table.
\multirow{2}{*}{} Growth \multirow[t]{2}{*}{Row totals}
Good Average Poor
\multirow{3}{*}{Type of plant} Coriander 12 28 15 55
Aster 7 18 23 48
Fennel 14 22 11 47
Column totals 33 68 49 150
Carry out a test at the $5 \%$ significance level to examine whether there is any association between growth and type of plant. State carefully your null and alternative hypotheses. Include a table of the contributions of each cell to the test statistic.
It is known that the diameter of marigold flowers is Normally distributed with mean 47 mm and standard deviation 8.5 mm . A certain fertiliser is expected to cause flowers to have a larger mean diameter, but without affecting the standard deviation. A large number of marigolds are grown using this fertiliser. The diameters of a random sample of 50 of the flowers are measured and the mean diameter is found to be 49.2 mm . Carry out a hypothesis test at the $1 \%$ significance level to check whether flowers grown with this fertiliser appear to be larger on average. Use hypotheses $\mathrm { H } _ { 0 } : \mu = 47 , \mathrm { H } _ { 1 } : \mu > 47$, where $\mu \mathrm { mm }$ represents the mean diameter of all marigold flowers grown with this fertiliser.

OCR MEI S2 2010 January Q1

19 marks Moderate -0.3

1 A pilot records the take-off distance for his light aircraft on runways at various altitudes. The data are shown in the table below, where $a$ metres is the altitude and $t$ metres is the take-off distance. Also shown are summary statistics for these data.

$a$	0	300	600	900	1200	1500	1800
$t$	635	704	776	836	923	1008	1105

$$n = 7 \quad \Sigma a = 6300 \quad \Sigma t = 5987 \quad \Sigma a ^ { 2 } = 8190000 \quad \Sigma t ^ { 2 } = 5288931 \quad \Sigma a t = 6037800$$

Draw a scatter diagram to illustrate these data.
State which of the two variables $a$ and $t$ is the independent variable and which is the dependent variable. Briefly explain your answer.
Calculate the equation of the regression line of $t$ on $a$.
Use the equation of the regression line to calculate estimates of the take-off distance for altitudes
(A) 800 metres,
(B) 2500 metres. Comment on the reliability of each of these estimates.
Calculate the value of the residual for the data point where $a = 1200$ and $t = 923$, and comment on its sign.

OCR MEI S2 2010 January Q2

18 marks Moderate -0.5

2 On average 2\% of a particular model of laptop computer are faulty. Faults occur independently and randomly.

Find the probability that exactly 1 of a batch of 10 laptops is faulty.
State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
A school buys a batch of 150 of these laptops. Use a Poisson approximating distribution to find the probability that
(A) there are no faulty laptops in the batch,
(B) there are more than the expected number of faulty laptops in the batch.
A large company buys a batch of 2000 of these laptops for its staff.
(A) State the exact distribution of the number of faulty laptops in this batch.
(B) Use a suitable approximating distribution to find the probability that there are at most 50 faulty laptops in this batch.

OCR MEI S2 2010 January Q3

17 marks Standard +0.3

3 In an English language test for 12-year-old children, the raw scores, $X$, are Normally distributed with mean 45.3 and standard deviation 11.5.

Find
(A) $\mathrm { P } ( X < 50 )$,
(B) $\mathrm { P } ( 45.3 < X < 50 )$.
Find the least raw score which would be obtained by the highest scoring $10 \%$ of children.
The raw score is then scaled so that the scaled score is Normally distributed with mean 100 and standard deviation 15. This scaled score is then rounded to the nearest integer. Find the probability that a randomly selected child gets a rounded score of exactly 111 .
In a Mathematics test for 12-year-old children, the raw scores, $Y$, are Normally distributed with mean $\mu$ and standard deviation $\sigma$. Given that $\mathrm { P } ( Y < 15 ) = 0.3$ and $\mathrm { P } ( Y < 22 ) = 0.8$, find the values of $\mu$ and $\sigma$.

OCR MEI S2 2010 January Q4

18 marks Moderate -0.3

4 A council provides waste paper recycling services for local businesses. Some businesses use the standard service for recycling paper, others use a special service for dealing with confidential documents, and others use both. Businesses are classified as small or large. A survey of a random sample of 285 businesses gives the following data for size of business and recycling service.

Recycling Service

\cline { 3 - 5 } \multicolumn{2}{|c|}{}

Write down null and alternative hypotheses for a test to examine whether there is any association between size of business and recycling service used. The contributions to the test statistic for the usual $\chi ^ { 2 }$ test are shown in the table below.
Recycling Service
\cline { 3 - 5 } \multicolumn{2}{|c|}{} Standard Special Both
\multirow{2}{*}{
Size of
business
} Small 0.1023 0.2607 0.0186
Large 0.0597 0.1520 0.0108
The sum of these contributions is 0.6041 .
Calculate the expected frequency for large businesses using the special service. Verify the corresponding contribution 0.1520 to the test statistic.
Carry out the test at the $5 \%$ level of significance, stating your conclusion clearly. The council is also investigating the weight of rubbish in domestic dustbins. In 2008 the average weight of rubbish in bins was 32.8 kg . The council has now started a recycling initiative and wishes to determine whether there has been a reduction in the weight of rubbish in bins. A random sample of 50 domestic dustbins is selected and it is found that the mean weight of rubbish per bin is now 30.9 kg , and the standard deviation is 3.4 kg .
Carry out a test at the $5 \%$ level to investigate whether the mean weight of rubbish has been reduced in comparison with 2008 . State carefully your null and alternative hypotheses. {www.ocr.org.uk}) after the live examination series.
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OCR MEI S2 2011 January Q1

17 marks Standard +0.3

1 The scatter diagram below shows the birth rates $x$, and death rates $y$, measured in standard units, in a random sample of 14 countries in a particular year. Summary statistics for the data are as follows. $$\Sigma x = 139.8 \quad \Sigma y = 140.4 \quad \Sigma x ^ { 2 } = 1411.66 \quad \Sigma y ^ { 2 } = 1417.88 \quad \Sigma x y = 1398.56 \quad n = 14$$ \includegraphics[max width=\textwidth, alt={}, center]{cd1a8f39-dd3c-44c9-90b0-6a919361d593-2_643_1047_488_550}

Calculate the sample product moment correlation coefficient.
Carry out a hypothesis test at the $5 \%$ significance level to determine whether there is any correlation between birth rates and death rates.
State the distributional assumption which is necessary for this test to be valid. Explain briefly in the light of the scatter diagram why it appears that the assumption may be valid.
The values of $x$ and $y$ for another country in that year are 14.4 and 7.8 respectively. If these values are included, the value of the sample product moment correlation coefficient is - 0.5694 . Explain why this one observation causes such a large change to the value of the sample product moment correlation coefficient. Discuss whether this brings the validity of the test into question.

\(x\)	18	43	52	94	98	206	784	1530
\(y\)	1.15	0.97	1.26	1.35	1.28	1.42	1.32	1.64

\multirow{2}{*}{}		Growth			\multirow[t]{2}{*}{Row totals}
		Good	Average	Poor
\multirow{3}{*}{Type of plant}	Coriander	12	28	15	55
	Aster	7	18	23	48
	Fennel	14	22	11	47
Column totals		33	68	49	150

\(t\)	20	30	40	50	60	70
\(w\)	2.49	2.41	2.38	2.14	1.97	2.03

Number of four-leaf clovers	0	1	2	\(> 2\)
Number of samples	11	7	2	0

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