Questions S1 - A-Level Maths

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

8 marks Moderate -0.8

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

18 marks Standard +0.3

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

18 marks Moderate -0.3

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

5 marks Easy -1.2

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.
\includegraphics[max width=\textwidth, alt={}, center]{854cb8fb-d75d-4854-b3ec-d7edbb21ea7e-2_899_1397_477_372}

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

5 marks Easy -1.2

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

7 marks Easy -1.3

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

7 marks Easy -1.3

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

18 marks Standard +0.3

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

18 marks Easy -1.3

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.
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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

6 marks Moderate -0.8

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

6 marks Moderate -0.8

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

8 marks Moderate -0.8

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

8 marks Moderate -0.8

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

8 marks Moderate -0.3

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

18 marks Moderate -0.3

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

18 marks Standard +0.3

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.}

OCR MEI S1 2013 June Q1

6 marks Moderate -0.8

1 The weights, $x$ grams, of 100 potatoes are summarised as follows. $$n = 100 \quad \sum x = 24940 \quad \sum x ^ { 2 } = 6240780$$

Calculate the mean and standard deviation of $x$.
The weights, $y$ grams, of the potatoes after they have been peeled are given by the formula $y = 0.9 x - 15$. Deduce the mean and standard deviation of the weights of the potatoes after they have been peeled.

OCR MEI S1 2013 June Q2

8 marks Standard +0.3

2 Every evening, 5 men and 5 women are chosen to take part in a phone-in competition. Of these 10 people, exactly 3 will win a prize. These 3 prize-winners are chosen at random.

Find the probability that, on a particular evening, 2 of the prize-winners are women and the other is a man. Give your answer as a fraction in its simplest form.
Four evenings are selected at random. Find the probability that, on at least three of the four evenings, 2 of the prize-winners are women and the other is a man.

OCR MEI S1 2013 June Q3

7 marks Standard +0.3

3 The weights of bags of a particular brand of flour are quoted as 1.5 kg . In fact, on average $10 \%$ of bags are underweight.

Find the probability that, in a random sample of 50 bags, there are exactly 5 bags which are underweight.
Bags are randomly chosen and packed into boxes of 20 . Find the probability that there is at least one underweight bag in a box.
A crate contains 48 boxes. Find the expected number of boxes in the crate which contain at least one underweight bag.

OCR MEI S1 2013 June Q4

7 marks Moderate -0.8

4 Martin has won a competition. For his prize he is given six sealed envelopes, of which he is allowed to open exactly three and keep their contents. Three of the envelopes each contain $\pounds 5$ and the other three each contain $\pounds 1000$. Since the envelopes are identical on the outside, he chooses three of them at random. Let $\pounds X$ be the total amount of money that he receives in prize money.

Show that $\mathrm { P } ( X = 15 ) = 0.05$. The probability distribution of $X$ is given in the table below.
$r$ 15 1010 2005 3000
$\mathrm { P } ( X = r )$ 0.05 0.45 0.45 0.05
Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

OCR MEI S1 2013 June Q5

8 marks Moderate -0.3

5 A researcher is investigating whether people can identify whether a glass of water they are given is bottled water or tap water. She suspects that people do no better than they would by guessing. Twenty people are selected at random; thirteen make a correct identification. She carries out a hypothesis test.

Explain why the null hypothesis should be $p = 0.5$, where $p$ represents the probability that a randomly selected person makes a correct identification.
Briefly explain why she uses an alternative hypothesis of $p > 0.5$.
Complete the test at the $5 \%$ significance level.

OCR MEI S1 2013 June Q6

18 marks Easy -1.2

6 The birth weights in kilograms of 25 female babies are shown below, in ascending order.

1.39	2.50	2.68	2.76	2.82	2.82	2.84	3.03	3.06	3.16	3.16	3.24	3.32
3.36	3.40	3.54	3.56	3.56	3.70	3.72	3.72	3.84	4.02	4.24	4.34

Find the median and interquartile range of these data.
Draw a box and whisker plot to illustrate the data.
Show that there is exactly one outlier. Discuss whether this outlier should be removed from the data. The cumulative frequency curve below illustrates the birth weights of 200 male babies.
\includegraphics[max width=\textwidth, alt={}, center]{6b886da6-3fb8-4b4c-b572-f4b770ae5a4c-3_929_1569_1450_248}
Find the median and interquartile range of the birth weights of the male babies.
Compare the weights of the female and male babies.
Two of these male babies are chosen at random. Calculate an estimate of the probability that both of these babies weigh more than any of the female babies.

OCR MEI S1 2013 June Q7

18 marks Standard +0.3

7 Jenny has six darts. She throws darts, one at a time, aiming each at the bull's-eye. The probability that she hits the bull's-eye with her first dart is 0.1 . For any subsequent throw, the probability of hitting the bull's-eye is 0.2 if the previous dart hit the bull's-eye and 0.05 otherwise.

Illustrate the possible outcomes for her first, second and third darts on a probability tree diagram.
Find the probability that
(A) she hits the bull's-eye with at least one of her first three darts,
(B) she hits the bull's-eye with exactly one of her first three darts.
Given that she hits the bull's-eye with at least one of her first three darts, find the probability that she hits the bull's-eye with exactly one of them. Jenny decides that, if she hits the bull's-eye with any of her first three darts, she will stop after throwing three darts. Otherwise she will throw all six darts.
Find the probability that she hits the bull's-eye three times in total.

OCR MEI S1 2014 June Q1

8 marks Easy -1.3

1 The ages, $x$ years, of the senior members of a running club are summarised in the table below.

Age $( x )$	$20 \leqslant x < 30$	$30 \leqslant x < 40$	$40 \leqslant x < 50$	$50 \leqslant x < 60$	$60 \leqslant x < 70$	$70 \leqslant x < 80$	$80 \leqslant x < 90$
Frequency	10	30	42	23	9	5	1

Draw a cumulative frequency diagram to illustrate the data.
Use your diagram to estimate the median and interquartile range of the data.

\(r\)	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

Age \(( x )\)	\(20 \leqslant x < 30\)	\(30 \leqslant x < 40\)	\(40 \leqslant x < 50\)	\(50 \leqslant x < 60\)	\(60 \leqslant x < 70\)	\(70 \leqslant x < 80\)	\(80 \leqslant x < 90\)
Frequency	10	30	42	23	9	5	1

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

\(r\)	15	1010	2005	3000
\(\mathrm { P } ( X = r )\)	0.05	0.45	0.45	0.05

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