Questions - OCR MEI S2 - A-Level Maths

\multirow{2}{*}{}		Sex		\multirow{2}{*}{Row totals}
		Male	Female
\multirow{5}{*}{Type of car}	Hatchback	96	36	132
	Saloon	77	35	112
	People carrier	38	44	82
	4WD	19	8	27
	Sports car	22	25	47
Column totals		252	148	400

Carry out a test at the $5 \%$ significance level to examine whether there is any association between type of car and sex of driver. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.
For each type of car, comment briefly on how the number of drivers of each sex compares with what would be expected if there were no association. OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations, is given to all schools that receive assessment material and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1PB.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

OCR MEI S2 2010 June Q1

1 Two celebrities judge a talent contest. Each celebrity gives a score out of 20 to each of a random sample of 8 contestants. The scores, $x$ and $y$, given by the celebrities to each contestant are shown below.

Contestant	A	B	C	D	E	F	G	H
$x$	6	17	9	20	13	15	11	14
$y$	6	13	10	11	9	7	12	15

Calculate the value of Spearman's rank correlation coefficient.
Carry out a hypothesis test at the $5 \%$ significance level to determine whether there is positive association between the scores allocated by the two celebrities.
State the distributional assumption required for a test based on the product moment correlation coefficient. Sketch a scatter diagram of the scores above, and discuss whether it appears that the assumption is likely to be valid.

OCR MEI S2 2010 June Q2

2 A radioactive source is decaying at a mean rate of 3.4 counts per 5 seconds.

State conditions for a Poisson distribution to be a suitable model for the rate of decay of the source. You may assume that a Poisson distribution with a mean rate of 3.4 counts per 5 seconds is a suitable model.
State the variance of this Poisson distribution.
Find the probability of
(A) exactly 3 counts in a 5 -second period,
(B) at least 3 counts in a 5 -second period.
Find the probability of exactly 40 counts in a period of 60 seconds.
Use a suitable approximating distribution to find the probability of at least 40 counts in a period of 60 seconds.
The background radiation rate also, independently, follows a Poisson distribution and produces a mean count of 1.4 per 5 seconds. Find the probability that the radiation source together with the background radiation give a total count of at least 8 in a 5 -second period.

OCR MEI S2 2010 June Q3

3 In a men's cycling time trial, the times are modelled by the random variable $X$ minutes which is Normally distributed with mean 63 and standard deviation 5.2.

Find $$\begin{aligned} & \text { (A) } \mathrm { P } ( X < 65 ) \text {, }
& \text { (B) } \mathrm { P } ( 60 < X < 65 ) \text {. } \end{aligned}$$
Find the probability that 5 riders selected at random all record times between 60 and 65 minutes.
A competitor aims to be in the fastest $5 \%$ of entrants (i.e. those with the lowest times). Find the maximum time that he can take. It is suggested that holding the time trial on a new course may result in lower times. To investigate this, a random sample of 15 competitors is selected. These 15 competitors do the time trial on the new course. The mean time taken by these riders is 61.7 minutes. You may assume that times are Normally distributed and the standard deviation is still 5.2 minutes. A hypothesis test is carried out to investigate whether times on the new course are lower.
Write down suitable null and alternative hypotheses for the test. Carry out the test at the 5\% significance level.

OCR MEI S2 2010 June Q4

4 In a survey a random sample of 63 runners is selected. The category of runner and the type of running are classified as follows.

\multirow{2}{*}{}		Category of runner			\multirow{2}{*}{Row totals}
		Junior	Senior	Veteran
\multirow{3}{*}{Type of running}	Track	9	8	2	19
	Road	4	8	12	24
	Both	4	10	6	20
Column totals		17	26	20	63

Carry out a test at the $5 \%$ significance level to examine whether there is any association between category of runner and the type of running. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.
For each category of runner, comment briefly on how the type of running compares with what would be expected if there were no association.

OCR MEI S2 2011 June Q1

1 An experiment is performed to determine the response of maize to nitrogen fertilizer. Data for the amount of nitrogen fertilizer applied, $x \mathrm {~kg} / \mathrm { hectare }$, and the average yield of maize, $y$ tonnes/hectare, in 5 experimental plots are given in the table below.

$x$	0	30	60	90	120
$y$	0.5	2.5	4.7	6.2	7.4

Draw a scatter diagram to illustrate these data.
Calculate the equation of the regression line of $y$ on $x$.
Draw your regression line on your scatter diagram and comment briefly on its fit.
Calculate the value of the residual for the data point where $x = 30$ and $y = 2.5$.
Use the equation of the regression line to calculate estimates of average yield with nitrogen fertilizer applications of
(A) $45 \mathrm {~kg} / \mathrm { hectare }$,
(B) $150 \mathrm {~kg} /$ hectare.
In a plot where $150 \mathrm {~kg} /$ hectare of nitrogen fertilizer is applied, the average yield of maize is 8.7 tonnes/hectare. Comment on this result.

OCR MEI S2 2011 June Q2

2 At a drive-through fast food takeaway, cars arrive independently, randomly and at a uniform average rate. The numbers of cars arriving per minute may be modelled by a Poisson distribution with mean 0.62.

Briefly explain the meaning of each of the three terms 'independently', 'randomly' and 'at a uniform average rate', in the context of cars arriving at a fast food takeaway.
Find the probability of at most 1 car arriving in a period of 1 minute.
Find the probability of more than 5 cars arriving in a period of 10 minutes.
State the exact distribution of the number of cars arriving in a period of 1 hour.
Use a suitable approximating distribution to find the probability that at least 40 cars arrive in a period of 1 hour.

OCR MEI S2 2011 June Q3

3 The weights of Braeburn apples on display in a supermarket, measured in grams, are Normally distributed with mean 210.5 and standard deviation 15.2.

Find the probability that a randomly selected apple weighs at least 220 grams.
These apples are sold in packs of 3. You may assume that the weights of apples in each pack are independent. Find the probability that all 3 of the apples in a randomly selected pack weigh at least 220 grams.
100 packs are selected at random.
(A) State the exact distribution of the number of these 100 packs in which all 3 apples weigh at least 220 grams.
(B) Use a suitable approximating distribution to find the probability that in at most one of these packs all 3 apples weigh at least 220 grams.
(C) Explain why this approximating distribution is suitable.
The supermarket also sells Cox's Orange Pippin apples. The weights of these apples, measured in grams, are Normally distributed with mean 185 and standard deviation $\sigma$.
(A) Given that $10 \%$ of randomly selected Cox's Orange Pippin apples weigh less than 170 grams, calculate the value of $\sigma$.
(B) Sketch the distributions of the weights of both types of apple on a single diagram.

OCR MEI S2 2011 June Q4

In a survey on internet usage, a random sample of 200 people is selected. The people are asked how much they have spent on internet shopping during the last three months. The results, classified by amount spent and sex, are shown in the table.

\multirow{2}{*}{}		Sex		\multirow{2}{*}{Row totals}
		Male	Female
\multirow{5}{*}{Amount spent}	Nothing	28	34	62
	Less than £50	17	21	38
	£50 up to £200	22	26	48
	£200 up to £1000	23	16	39
	£1000 or more	8	5	13
Column totals		98	102	200

Write down null and alternative hypotheses for a test to examine whether there is any association between amount spent and sex of person. The contributions to the test statistic for the usual $\chi ^ { 2 }$ test are shown in the table below.
\multirow{2}{*}{} Sex
Male Female
\multirow{5}{*}{Amount spent} Nothing 0.1865 0.1791
Less than £50 0.1409 0.1354
£50 up to £200 0.0982 0.0944
£200 up to £1000 0.7918 0.7608
£1000 or more 0.4171 0.4007
The sum of these contributions, correct to 3 decimal places, is 3.205.
Calculate the expected frequency for females spending nothing. Verify the corresponding contribution, 0.1791 , to the test statistic.
Carry out the test at the $5 \%$ level of significance, stating your conclusion clearly.

A bakery sells loaves specified as having a mean weight of 400 grams. It is known that the weights of these loaves are Normally distributed and that the standard deviation is 5.7 grams. An inspector suspects that the true mean weight may be less than 400 grams. In order to test this, the inspector takes a random sample of 6 loaves. Carry out a suitable test at the $5 \%$ level, given that the weights, in grams, of the 6 loaves are as follows.
$\begin{array} { l l l l l l } 392.1 & 405.8 & 401.3 & 387.4 & 391.8 & 400.6 \end{array}$ RECOGNISING ACHIEVEMENT

OCR MEI S2 2012 June Q1

1 The times, in seconds, taken by ten randomly selected competitors for the first and last sections of an Olympic bobsleigh run are denoted by $x$ and $y$ respectively. Summary statistics for these data are as follows. $$\Sigma x = 113.69 \quad \Sigma y = 52.81 \quad \Sigma x ^ { 2 } = 1292.56 \quad \Sigma y ^ { 2 } = 278.91 \quad \Sigma x y = 600.41 \quad n = 10$$

Calculate the sample product moment correlation coefficient.
Carry out a hypothesis test at the $10 \%$ significance level to investigate whether there is any correlation between times taken for the first and last sections of the bobsleigh run.
State the distributional assumption which is necessary for this test to be valid. Explain briefly how a scatter diagram may be used to check whether this assumption is likely to be valid.
A commentator says that in order to have a fast time on the last section, you must have a fast time on the first section. Comment briefly on this suggestion.
(A) Would your conclusion in part (ii) have been different if you had carried out the hypothesis test at the $1 \%$ level rather than the $10 \%$ level? Explain your answer.
(B) State one advantage and one disadvantage of using a $1 \%$ significance level rather than a $10 \%$ significance level in a hypothesis test.

OCR MEI S2 2012 June Q2

2 A particular genetic mutation occurs in one in every 300 births on average. A random sample of 1200 births is selected.

State the exact distribution of $X$, the number of births in the sample which have the mutation.
Explain why $X$ has, approximately, a Poisson distribution.
Use a Poisson approximating distribution to find
(A) $\mathrm { P } ( X = 1 )$,
(B) $\mathrm { P } ( X > 4 )$.
Twenty independent samples, each of 1200 births, are selected. State the mean and variance of a Normal approximating distribution suitable for modelling the total number of births with the mutation in the twenty samples.
Use this Normal approximating distribution to
(A) find the probability that there are at least 90 births which have the mutation,
( $B$ ) find the least value of $k$ such that the probability that there are at most $k$ births with this mutation is greater than 5\%.

OCR MEI S2 2012 June Q3

3 At a vineyard, the process used to fill bottles with wine is subject to variation. The contents of bottles are independently Normally distributed with mean $\mu = 751.4 \mathrm { ml }$ and standard deviation $\sigma = 2.5 \mathrm { ml }$.

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

9 marks

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
[0pt] [9] OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

OCR MEI S2 2013 June Q1

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

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

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

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

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

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

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

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

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

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

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

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}

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

Contestant	A	B	C	D	E	F	G	H
\(x\)	6	17	9	20	13	15	11	14
\(y\)	6	13	10	11	9	7	12	15

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