Questions - OCR MEI S1 - A-Level Maths

$r$	10	20	30	40
$\mathrm { P } ( X = r )$	0.2	0.3	0.3	0.2

Explain why $\mathrm { E } ( X ) = 25$.
Calculate $\operatorname { Var } ( X )$.

OCR MEI S1 2009 June Q5

5 The frequency table below shows the distance travelled by 1200 visitors to a particular UK tourist destination in August 2008.

Distance $( d$ miles $)$	$0 \leqslant d < 50$	$50 \leqslant d < 100$	$100 \leqslant d < 200$	$200 \leqslant d < 400$
Frequency	360	400	307	133

Draw a histogram on graph paper to illustrate these data.
Calculate an estimate of the median distance.

OCR MEI S1 2009 June Q6

6 Whitefly, blight and mosaic virus are three problems which can affect tomato plants. 100 tomato plants are examined for these problems. The numbers of plants with each type of problem are shown in the Venn diagram. 47 of the plants have none of the problems.
\includegraphics[max width=\textwidth, alt={}, center]{3a5d18f5-b1fe-4513-ae4e-f37c20f172b5-3_668_812_998_664}

One of the 100 plants is selected at random. Find the probability that this plant has
(A) at most one of the problems,
(B) exactly two of the problems.
Three of the 100 plants are selected at random. Find the probability that all of them have at least one of the problems. Section B (36 marks)

OCR MEI S1 2009 June Q7

7 Laura frequently flies to business meetings and often finds that her flights are delayed. A flight may be delayed due to technical problems, weather problems or congestion problems, with probabilities $0.2,0.15$ and 0.1 respectively. The tree diagram shows this information.
\includegraphics[max width=\textwidth, alt={}, center]{3a5d18f5-b1fe-4513-ae4e-f37c20f172b5-4_608_1651_532_248}

Write down the values of the probabilities $a , b$ and $c$ shown in the tree diagram. One of Laura's flights is selected at random.
Find the probability that Laura's flight is not delayed and hence write down the probability that it is delayed.
Find the probability that Laura's flight is delayed due to just one of the three problems.
Given that Laura's flight is delayed, find the probability that the delay is due to just one of the three problems.
Given that Laura's flight has no technical problems, find the probability that it is delayed.
In a particular year, Laura has 110 flights. Find the expected number of flights that are delayed.

OCR MEI S1 2009 June Q8

8 The Department of Health 'eat five a day' advice recommends that people should eat at least five portions of fruit and vegetables per day. In a particular school, $20 \%$ of pupils eat at least five a day.

15 children are selected at random.
(A) Find the probability that exactly 3 of them eat at least five a day.
(B) Find the probability that at least 3 of them eat at least five a day.
(C) Find the expected number who eat at least five a day. A programme is introduced to encourage children to eat more portions of fruit and vegetables per day. At the end of this programme, the diets of a random sample of 15 children are analysed. A hypothesis test is carried out to examine whether the proportion of children in the school who eat at least five a day has increased.
(A) Write down suitable null and alternative hypotheses for the test.
(B) Give a reason for your choice of the alternative hypothesis.
Find the critical region for the test at the $10 \%$ significance level, showing all of your calculations. Hence complete the test, given that 7 of the 15 children eat at least five a day.

OCR MEI S1 2010 June Q1

1 A business analyst collects data about the distribution of hourly wages, in $\pounds$, of shop-floor workers at a factory. These data are illustrated in the box and whisker plot.
\includegraphics[max width=\textwidth, alt={}, center]{091d6f43-ad01-4849-9f3c-3e58349aa169-2_204_1422_484_363}

Name the type of skewness of the distribution.
Find the interquartile range and hence show that there are no outliers at the lower end of the distribution, but there is at least one outlier at the upper end.
Suggest possible reasons why this may be the case.

OCR MEI S1 2010 June Q2

2 The probability distribution of the random variable $X$ is given by the formula $$\mathrm { P } ( X = r ) = k r ( 5 - r ) \text { for } r = 1,2,3,4 .$$

Show that $k = 0.05$.
Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

OCR MEI S1 2010 June Q3

3 The lifetimes in hours of 90 components are summarised in the table.

Lifetime $( x$ hours $)$	$0 < x \leqslant 20$	$20 < x \leqslant 30$	$30 < x \leqslant 50$	$50 < x \leqslant 65$	$65 < x \leqslant 100$
Frequency	24	13	14	21	18

Draw a histogram to illustrate these data.
In which class interval does the median lie? Justify your answer.

OCR MEI S1 2010 June Q4

4 Each packet of Cruncho cereal contains one free fridge magnet. There are five different types of fridge magnet to collect. They are distributed, with equal probability, randomly and independently in the packets. Keith is about to start collecting these fridge magnets.

Find the probability that the first 2 packets that Keith buys contain the same type of fridge magnet.
Find the probability that Keith collects all five types of fridge magnet by buying just 5 packets.
Hence find the probability that Keith has to buy more than 5 packets to acquire a complete set.

OCR MEI S1 2010 June Q5

5 A retail analyst records the numbers of loaves of bread of a particular type bought by a sample of shoppers in a supermarket.

Number of loaves	0	1	2	3	4	5
Frequency	37	23	11	3	0	1

Calculate the mean and standard deviation of the numbers of loaves bought per person.
Each loaf costs $\pounds 1.04$. Calculate the mean and standard deviation of the amount spent on loaves per person.

OCR MEI S1 2010 June Q6

6 A manufacturer produces tiles. On average 10\% of the tiles produced are faulty. Faulty tiles occur randomly and independently. A random sample of 18 tiles is selected.

(A) Find the probability that there are exactly 2 faulty tiles in the sample.
(B) Find the probability that there are more than 2 faulty tiles in the sample.
(C) Find the expected number of faulty tiles in the sample. A cheaper way of producing the tiles is introduced. The manufacturer believes that this may increase the proportion of faulty tiles. In order to check this, a random sample of 18 tiles produced using the cheaper process is selected and a hypothesis test is carried out.
(A) Write down suitable null and alternative hypotheses for the test.
(B) Explain why the alternative hypothesis has the form that it does.
Find the critical region for the test at the $5 \%$ level, showing all of your calculations.
In fact there are 4 faulty tiles in the sample. Complete the test, stating your conclusion clearly.

OCR MEI S1 2010 June Q7

7 One train leaves a station each hour. The train is either on time or late. If the train is on time, the probability that the next train is on time is 0.95 . If the train is late, the probability that the next train is on time is 0.6 . On a particular day, the 0900 train is on time.

Illustrate the possible outcomes for the 1000,1100 and 1200 trains on a probability tree diagram.
Find the probability that
(A) all three of these trains are on time,
(B) just one of these three trains is on time,
(C) the 1200 train is on time.
Given that the 1200 train is on time, find the probability that the 1000 train is also on time. 3
Write any calculations on page 5.
\includegraphics[max width=\textwidth, alt={}, center]{091d6f43-ad01-4849-9f3c-3e58349aa169-4_2276_1490_324_363}

OCR MEI S1 2011 June Q1

1 In the Paris-Roubaix cycling race, there are a number of sections of cobbled road. The lengths of these sections, measured in metres, are illustrated in the histogram.
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Find the number of sections which are between 1000 and 2000 metres in length.
Name the type of skewness suggested by the histogram.
State the minimum and maximum possible values of the midrange.

OCR MEI S1 2011 June Q2

2 I have 5 books, each by a different author. The authors are Austen, Brontë, Clarke, Dickens and Eliot.

If I arrange the books in a random order on my bookshelf, find the probability that the authors are in alphabetical order with Austen on the left.
If I choose two of the books at random, find the probability that I choose the books written by Austen and Brontë.
$325 \%$ of the plants of a particular species have red flowers. A random sample of 6 plants is selected.
Find the probability that there are no plants with red flowers in the sample.
If 50 random samples of 6 plants are selected, find the expected number of samples in which there are no plants with red flowers.

OCR MEI S1 2011 June Q4

4 Two fair six-sided dice are thrown. The random variable $X$ denotes the difference between the scores on the two dice. The table shows the probability distribution of $X$.

$r$	0	1	2	3	4	5
$\mathrm { P } ( X = r )$	$\frac { 1 } { 6 }$	$\frac { 5 } { 18 }$	$\frac { 2 } { 9 }$	$\frac { 1 } { 6 }$	$\frac { 1 } { 9 }$	$\frac { 1 } { 18 }$

Draw a vertical line chart to illustrate the probability distribution.
Use a probability argument to show that
(A) $\mathrm { P } ( X = 1 ) = \frac { 5 } { 18 }$,
(B) $\mathrm { P } ( X = 0 ) = \frac { 1 } { 6 }$.
Find the mean value of $X$.

OCR MEI S1 2011 June Q6

6 The numbers of eggs laid by a sample of 70 female herring gulls are shown in the table.

Number of eggs	1	2	3	4
Frequency	10	40	15	5

Find the mean and standard deviation of the number of eggs laid per gull.
The sample did not include female herring gulls that laid no eggs. How would the mean and standard deviation change if these gulls were included?

OCR MEI S1 2011 June Q7

7 Any patient who fails to turn up for an outpatient appointment at a hospital is described as a 'no-show'. At a particular hospital, on average $15 \%$ of patients are no-shows. A random sample of 20 patients who have outpatient appointments is selected.

Find the probability that
(A) there is exactly 1 no-show in the sample,
(B) there are at least 2 no-shows in the sample. The hospital management introduces a policy of telephoning patients before appointments. It is hoped that this will reduce the proportion of no-shows. In order to check this, a random sample of $n$ patients is selected. The number of no-shows in the sample is recorded and a hypothesis test is carried out at the 5\% level.
Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis.
In the case that $n = 20$ and the number of no-shows in the sample is 1 , carry out the test.
In another case, where $n$ is large, the number of no-shows in the sample is 6 and the critical value for the test is 8 . Complete the test.
In the case that $n \leqslant 18$, explain why there is no point in carrying out the test at the $5 \%$ level.

OCR MEI S1 2011 June Q8

8 The heating quality of the coal in a sample of 50 sacks is measured in suitable units. The data are summarised below.

Heating quality $( x )$	$9.1 \leqslant x \leqslant 9.3$	$9.3 < x \leqslant 9.5$	$9.5 < x \leqslant 9.7$	$9.7 < x \leqslant 9.9$	$9.9 < x \leqslant 10.1$
Frequency	5	7	15	16	7

Draw a cumulative frequency diagram to illustrate these data.
Use the diagram to estimate the median and interquartile range of the data.
Show that there are no outliers in the sample.
Three of these 50 sacks are selected at random. Find the probability that
(A) in all three, the heating quality $x$ is more than 9.5,
$( B )$ in at least two, the heating quality $x$ is more than 9.5. 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 S1 2012 June Q1

1 At a garden centre there is a box containing 50 hyacinth bulbs. Of these, 30 will produce a blue flower and the remaining 20 will produce a red flower. Unfortunately they have become mixed together so that it is not known which of the bulbs will produce a blue flower and which will produce a red flower. Karen buys 3 of these bulbs.

Find the probability that all 3 of these bulbs will produce blue flowers.
Find the probability that Karen will have at least one flower of each colour from her 3 bulbs.

OCR MEI S1 2012 June Q2

2 An examination paper consists of two sections. Section A has 5 questions and Section B has 9 questions. Candidates are required to answer 6 questions.

In how many different ways can a candidate choose 6 questions, if 3 are from Section A and 3 are from Section B?
Another candidate randomly chooses 6 questions to answer. Find the probability that this candidate chooses 3 questions from each section.

OCR MEI S1 2012 June Q3

3 At a call centre, $85 \%$ of callers are put on hold before being connected to an operator. A random sample of 30 callers is selected.

Find the probability that exactly 29 of these callers are put on hold.
Find the probability that at least 29 of these callers are put on hold.
If 10 random samples, each of 30 callers, are selected, find the expected number of samples in which at least 29 callers are put on hold.

OCR MEI S1 2012 June Q4

4 It is known that $8 \%$ of the population of a large city use a particular web browser. A researcher wishes to interview some people from the city who use this browser. He selects people at random, one at a time.

Find the probability that the first person that he finds who uses this browser is
(A) the third person selected,
(B) the second or third person selected.
Find the probability that at least one of the first 20 people selected uses this browser.

OCR MEI S1 2012 June Q5

5 A manufacturer produces titanium bicycle frames. The bicycle frames are tested before use and on average $5 \%$ of them are found to be faulty. A cheaper manufacturing process is introduced and the manufacturer wishes to check whether the proportion of faulty bicycle frames has increased. A random sample of 18 bicycle frames is selected and it is found that 4 of them are faulty. Carry out a hypothesis test at the $5 \%$ significance level to investigate whether the proportion of faulty bicycle frames has increased.

OCR MEI S1 2012 June Q6

6 The engine sizes $x \mathrm {~cm} ^ { 3 }$ of a sample of 80 cars are summarised in the table below.

Engine size $x$	$500 \leqslant x \leqslant 1000$	$1000 < x \leqslant 1500$	$1500 < x \leqslant 2000$	$2000 < x \leqslant 3000$	$3000 < x \leqslant 5000$
Frequency	7	22	26	18	7

Draw a histogram to illustrate the distribution.
A student claims that the midrange is $2750 \mathrm {~cm} ^ { 3 }$. Discuss briefly whether he is likely to be correct.
Calculate estimates of the mean and standard deviation of the engine sizes. Explain why your answers are only estimates.
Hence investigate whether there are any outliers in the sample.
A vehicle duty of $\pounds 1000$ is proposed for all new cars with engine size greater than $2000 \mathrm {~cm} ^ { 3 }$. Assuming that this sample of cars is representative of all new cars in Britain and that there are 2.5 million new cars registered in Britain each year, calculate an estimate of the total amount of money that this vehicle duty would raise in one year.
Why in practice might your estimate in part (v) turn out to be too high?

OCR MEI S1 2012 June Q7

7 Yasmin has 5 coins. One of these coins is biased with P (heads) $= 0.6$. The other 4 coins are fair. She tosses all 5 coins once and records the number of heads, $X$.

Show that $\mathrm { P } ( X = 0 ) = 0.025$.
Show that $\mathrm { P } ( X = 1 ) = 0.1375$. The table shows the probability distribution of $X$.
$r$ 0 1 2 3 4 5
$\mathrm { P } ( X = r )$ 0.025 0.1375 0.3 0.325 0.175 0.0375
Draw a vertical line chart to illustrate the probability distribution.
Comment on the skewness of the distribution.
Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
Yasmin tosses the 5 coins three times. Find the probability that the total number of heads is 3 . \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

\(r\)	0	1	2	3	4	5
\(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 6 }\)	\(\frac { 5 } { 18 }\)	\(\frac { 2 } { 9 }\)	\(\frac { 1 } { 6 }\)	\(\frac { 1 } { 9 }\)	\(\frac { 1 } { 18 }\)

Heating quality \(( x )\)	\(9.1 \leqslant x \leqslant 9.3\)	\(9.3 < x \leqslant 9.5\)	\(9.5 < x \leqslant 9.7\)	\(9.7 < x \leqslant 9.9\)	\(9.9 < x \leqslant 10.1\)
Frequency	5	7	15	16	7

Engine size \(x\)	\(500 \leqslant x \leqslant 1000\)	\(1000 < x \leqslant 1500\)	\(1500 < x \leqslant 2000\)	\(2000 < x \leqslant 3000\)	\(3000 < x \leqslant 5000\)
Frequency	7	22	26	18	7

Distance \(( d\) miles \()\)	\(0 \leqslant d < 50\)	\(50 \leqslant d < 100\)	\(100 \leqslant d < 200\)	\(200 \leqslant d < 400\)
Frequency	360	400	307	133

Lifetime \(( x\) hours \()\)	\(0 < x \leqslant 20\)	\(20 < x \leqslant 30\)	\(30 < x \leqslant 50\)	\(50 < x \leqslant 65\)	\(65 < x \leqslant 100\)
Frequency	24	13	14	21	18

\(r\)	0	1	2	3	4	5
\(\mathrm { P } ( X = r )\)	0.025	0.1375	0.3	0.325	0.175	0.0375

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