Questions - OCR MEI S1 - A-Level Maths

Find the probability that Steve is delayed on his outward flight but not on his return flight.
Find the probability that he is delayed on at least one of the two flights.
Given that he is delayed on at least one flight, find the probability that he is delayed on both flights.

OCR MEI S1 2008 January Q4

4 A company is searching for oil reserves. The company has purchased the rights to make test drillings at four sites. It investigates these sites one at a time but, if oil is found, it does not proceed to any further sites. At each site, there is probability 0.2 of finding oil, independently of all other sites. The random variable $X$ represents the number of sites investigated. The probability distribution of $X$ is shown below.

$r$	1	2	3	4
$\mathrm { P } ( X = r )$	0.2	0.16	0.128	0.512

Find the expectation and variance of $X$.
It costs $\pounds 45000$ to investigate each site. Find the expected total cost of the investigation.
Draw a suitable diagram to illustrate the distribution of $X$.

OCR MEI S1 2008 January Q5

5 Sophie and James are having a tennis competition. The winner of the competition is the first to win 2 matches in a row. If the competition has not been decided after 5 matches, then the player who has won more matches is declared the winner of the competition. For example, the following sequences are two ways in which Sophie could win the competition. (S represents a match won by Sophie; $\mathbf { J }$ represents a match won by James.) \section*{SJSS SJSJS}

Explain why the sequence $\mathbf { S S J }$ is not possible.
Write down the other three possible sequences in which Sophie wins the competition.
The probability that Sophie wins a match is 0.7 . Find the probability that she wins the competition in no more than 4 matches.

OCR MEI S1 2008 January Q6

6 The maximum temperatures $x$ degrees Celsius recorded during each month of 2005 in Cambridge are given in the table below.

Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
9.2	7.1	10.7	14.2	16.6	21.8	22.0	22.6	21.1	17.4	10.1	7.8

These data are summarised by $n = 12 , \Sigma x = 180.6 , \Sigma x ^ { 2 } = 3107.56$.

Calculate the mean and standard deviation of the data.
Determine whether there are any outliers.
The formula $y = 1.8 x + 32$ is used to convert degrees Celsius to degrees Fahrenheit. Find the mean and standard deviation of the 2005 maximum temperatures in degrees Fahrenheit.
In New York, the monthly maximum temperatures are recorded in degrees Fahrenheit. In 2005 the mean was 63.7 and the standard deviation was 16.0 . Briefly compare the maximum monthly temperatures in Cambridge and New York in 2005. The total numbers of hours of sunshine recorded in Cambridge during the month of January for each of the last 48 years are summarised below.
Hours $h$ $70 \leqslant h < 100$ $100 \leqslant h < 110$ $110 \leqslant h < 120$ $120 \leqslant h < 150$ $150 \leqslant h < 170$ $170 \leqslant h < 190$
Number of years 6 8 10 11 10 3
Draw a cumulative frequency graph for these data.
Use your graph to estimate the 90th percentile.

OCR MEI S1 2008 January Q7

7 A particular product is made from human blood given by donors. The product is stored in bags. The production process is such that, on average, $5 \%$ of bags are faulty. Each bag is carefully tested before use.

12 bags are selected at random.
(A) Find the probability that exactly one bag is faulty.
(B) Find the probability that at least two bags are faulty.
(C) Find the expected number of faulty bags in the sample.
A random sample of $n$ bags is selected. The production manager wishes there to be a probability of one third or less of finding any faulty bags in the sample. Find the maximum possible value of $n$, showing your working clearly.
A scientist believes that a new production process will reduce the proportion of faulty bags. A random sample of 60 bags made using the new process is checked and one bag is found to be faulty. Write down suitable hypotheses and carry out a hypothesis test at the $10 \%$ level to determine whether there is evidence to suggest that the scientist is correct.

OCR MEI S1 2005 June Q1

1 At a certain stage of a football league season, the numbers of goals scored by a sample of 20 teams in the league were as follows.
$\begin{array} { l l l l l l l l l l l l l l l l l l l l l } 22 & 23 & 23 & 23 & 26 & 28 & 28 & 30 & 31 & 33 & 33 & 34 & 35 & 35 & 36 & 36 & 37 & 46 & 49 & 49 \end{array}$

Calculate the sample mean and sample variance, $s ^ { 2 }$, of these data.
The three teams with the most goals appear to be well ahead of the other teams. Determine whether or not any of these three pieces of data may be considered outliers.

OCR MEI S1 2005 June Q2

2 Answer part (i) of this question on the insert provided.
A taxi driver operates from a taxi rank at a main railway station in London. During one particular week he makes 120 journeys, the lengths of which are summarised in the table.

On the insert, draw a cumulative frequency diagram to illustrate the data.
Use your graph to estimate the median length of journey and the quartiles. Hence find the interquartile range.
State the type of skewness of the distribution of the data.

OCR MEI S1 2005 June Q3

3 Jeremy is a computing consultant who sometimes works at home. The number, $X$, of days that Jeremy works at home in any given week is modelled by the probability distribution $$\mathrm { P } ( X = r ) = \frac { 1 } { 40 } r ( r + 1 ) \quad \text { for } r = 1,2,3,4 .$$

Verify that $\mathrm { P } ( X = 4 ) = \frac { 1 } { 2 }$.
Calculate $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
Jeremy works for 45 weeks each year. Find the expected number of weeks during which he works at home for exactly 2 days.

OCR MEI S1 2005 June Q4

1 marks

4 An examination paper consists of three sections.

Section A contains 6 questions of which the candidate must answer 3
Section B contains 7 questions of which the candidate must answer 4
Section C contains 8 questions of which the candidate must answer 5
1. In how many ways can a candidate choose 3 questions from Section A?
2. In how many ways can a candidate choose 3 questions from Section A, 4 from Section B and 5 from Section C?

A candidate does not read the instructions and selects 12 questions at random.

Find the probability that they happen to be 3 from Section A, 4 from Section B and 5 from Section C.

OCR MEI S1 2005 June Q7

7 A game requires 15 identical ordinary dice to be thrown in each turn.
Assuming the dice to be fair, find the following probabilities for any given turn.

No sixes are thrown.
Exactly four sixes are thrown.
More than three sixes are thrown. David and Esme are two players who are not convinced that the dice are fair. David believes that the dice are biased against sixes, while Esme believes the dice to be biased in favour of sixes. In his next turn, David throws no sixes. In her next turn, Esme throws 5 sixes.
Writing down your hypotheses carefully in each case, decide whether
(A) David's turn provides sufficient evidence at the $10 \%$ level that the dice are biased against sixes,
(B) Esme's turn provides sufficient evidence at the $10 \%$ level that the dice are biased in favour of sixes.
Comment on your conclusions from part (iv).

OCR MEI S1 2006 June Q1

1 Every day, George attempts the quiz in a national newspaper. The quiz always consists of 7 questions. In the first 25 days of January, the numbers of questions George answers correctly each day are summarised in the table below.

Number correct	1	2	3	4	5	6	7
Frequency	1	2	3	3	4	7	5

Draw a vertical line chart to illustrate the data.
State the type of skewness shown by your diagram.
Calculate the mean and the mean squared deviation of the data.
How many correct answers would George need to average over the next 6 days if he is to achieve an average of 5 correct answers for all 31 days of January?

OCR MEI S1 2006 June Q2

2 Isobel plays football for a local team. Sometimes her parents attend matches to watch her play.

$A$ is the event that Isobel's parents watch a match.
$B$ is the event that Isobel scores in a match.

You are given that $\mathrm { P } ( B \mid A ) = \frac { 3 } { 7 }$ and $\mathrm { P } ( A ) = \frac { 7 } { 10 }$.

Calculate $\mathrm { P } ( A \cap B )$. The probability that Isobel does not score and her parents do not attend is 0.1 .
Draw a Venn diagram showing the events $A$ and $B$, and mark in the probability corresponding to each of the regions of your diagram.
Are events $A$ and $B$ independent? Give a reason for your answer.
By comparing $\mathrm { P } ( B \mid A )$ with $\mathrm { P } ( B )$, explain why Isobel should ask her parents not to attend.

OCR MEI S1 2006 June Q3

3 The score, $X$, obtained on a given throw of a biased, four-faced die is given by the probability distribution $$\mathrm { P } ( X = r ) = k r ( 8 - r ) \text { for } r = 1,2,3,4 .$$

Show that $k = \frac { 1 } { 50 }$.
Calculate $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

OCR MEI S1 2006 June Q4

4 Peter and Esther visit a restaurant for a three-course meal. On the menu there are 4 starters, 5 main courses and 3 sweets. Peter and Esther each order a starter, a main course and a sweet.

Calculate the number of ways in which Peter may choose his three-course meal.
Suppose that Peter and Esther choose different dishes from each other.
(A) Show that the number of possible combinations of starters is 6 .
(B) Calculate the number of possible combinations of 6 dishes for both meals.
Suppose instead that Peter and Esther choose their dishes independently.
(A) Write down the probability that they choose the same main course.
(B) Find the probability that they choose different dishes from each other for every course.

OCR MEI S1 2006 June Q5

5 Douglas plays darts, and the probability that he hits the number he is aiming at is 0.87 for any particular dart. Douglas aims a set of three darts at the number 20; the number of times he is successful can be modelled by $\mathrm { B } ( 3,0.87 )$.

Calculate the probability that Douglas hits 20 twice.
Douglas aims fifty sets of 3 darts at the number 20. Find the expected number of sets for which Douglas hits 20 twice.
Douglas aims four sets of 3 darts at the number 20. Calculate the probability that he hits 20 twice for two sets out of the four.

OCR MEI S1 2006 June Q6

6 It has been estimated that $90 \%$ of paintings offered for sale at a particular auction house are genuine, and that the other $10 \%$ are fakes. The auction house has a test to determine whether or not a given painting is genuine. If this test gives a positive result, it suggests that the painting is genuine. A negative result suggests that the painting is a fake. If a painting is genuine, the probability that the test result is positive is 0.95 .
If a painting is a fake, the probability that the test result is positive is 0.2 .

Copy and complete the probability tree diagram below, to illustrate the information above.
\includegraphics[max width=\textwidth, alt={}, center]{16488e7a-36fb-47f1-8dbf-dec57387f2bf-4_469_668_861_699} Calculate the probabilities of the following events.
The test gives a positive result.
The test gives a correct result.
The painting is genuine, given a positive result.
The painting is a fake, given a negative result. A second test is more accurate, but very expensive. The auction house has a policy of only using this second test on those paintings with a negative result on the original test.
Using your answers to parts (iv) and (v), explain why the auction house has this policy. The probability that the second test gives a correct result is 0.96 whether the painting is genuine or a fake.
Three paintings are independently offered for sale at the auction house. Calculate the probability that all three paintings are genuine, are judged to be fakes in the first test, but are judged to be genuine in the second test.

OCR MEI S1 2006 June Q7

7 A geologist splits rocks to look for fossils. On average $10 \%$ of the rocks selected from a particular area do in fact contain fossils. The geologist selects a random sample of 20 rocks from this area.

Find the probability that
(A) exactly one of the rocks contains fossils,
(B) at least one of the rocks contains fossils.
A random sample of $n$ rocks is selected from this area. The geologist wants to have a probability of 0.8 or greater of finding fossils in at least one of the $n$ rocks. Find the least possible value of $n$.
The geologist explores a new area in which it is claimed that less than $10 \%$ of rocks contain fossils. In order to investigate the claim, a random sample of 30 rocks from this area is selected, and the number which contain fossils is recorded. A hypothesis test is carried out at the 5\% level.
(A) Write down suitable hypotheses for the test.
(B) Show that the critical region consists only of the value 0 .
(C) In fact, 2 of the 30 rocks in the sample contain fossils. Complete the test, stating your conclusions clearly.

OCR MEI S1 2007 June Q1

1 A girl is choosing tracks from an album to play at her birthday party. The album has 8 tracks and she selects 4 of them.

In how many ways can she select the 4 tracks?
In how many different orders can she arrange the 4 tracks once she has chosen them?

OCR MEI S1 2007 June Q2

2 The histogram shows the amount of money, in pounds, spent by the customers at a supermarket on a particular day.
\includegraphics[max width=\textwidth, alt={}, center]{5e4f3310-b96e-43db-9b6d-61da3270db06-2_977_1132_808_340}
□ represents 20 customers

Express the data in the form of a grouped frequency table.
Use your table to estimate the total amount of money spent by customers on that day.

OCR MEI S1 2007 June Q3

3 The marks $x$ scored by a sample of 56 students in an examination are summarised by $$n = 56 , \quad \Sigma x = 3026 , \quad \Sigma x ^ { 2 } = 178890 .$$

Calculate the mean and standard deviation of the marks.
The highest mark scored by any of the 56 students in the examination was 93 . Show that this result may be considered to be an outlier.
The formula $y = 1.2 x - 10$ is used to scale the marks. Find the mean and standard deviation of the scaled marks.

OCR MEI S1 2007 June Q4

4 A local council has introduced a recycling scheme for aluminium, paper and kitchen waste. 50 residents are asked which of these materials they recycle. The numbers of people who recycle each type of material are shown in the Venn diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5e4f3310-b96e-43db-9b6d-61da3270db06-3_803_803_406_671} One of the residents is selected at random.

Find the probability that this resident recycles
(A) at least one of the materials,
(B) exactly one of the materials.
Given that the resident recycles aluminium, find the probability that this resident does not recycle paper. Two residents are selected at random.
Find the probability that exactly one of them recycles kitchen waste.

OCR MEI S1 2007 June Q5

5 A GCSE geography student is investigating a claim that global warming is causing summers in Britain to have more rainfall. He collects rainfall data from a local weather station for 2001 and 2006. The vertical line chart shows the number of days per week on which some rainfall was recorded during the 22 weeks of summer 2001.
\includegraphics[max width=\textwidth, alt={}, center]{5e4f3310-b96e-43db-9b6d-61da3270db06-4_720_1557_443_296} Number of days per week with rain recorded in summer 2001

Show that the median of the data is 4 , and find the interquartile range.
For summer 2006 the median is 3 and the interquartile range is also 3. The student concludes that the data demonstrate that global warming is causing summer rainfall to decrease rather than increase. Is this a valid conclusion from the data? Give two brief reasons to justify your answer.

OCR MEI S1 2007 June Q6

6 In a phone-in competition run by a local radio station, listeners are given the names of 7 local personalities and are told that 4 of them are in the studio. Competitors phone in and guess which 4 are in the studio.

Show that the probability that a randomly selected competitor guesses all 4 correctly is $\frac { 1 } { 35 }$. Let $X$ represent the number of correct guesses made by a randomly selected competitor. The probability distribution of $X$ is shown in the table.
$r$ 0 1 2 3 4
$\mathrm { P } ( X = r )$ 0 $\frac { 4 } { 35 }$ $\frac { 18 } { 35 }$ $\frac { 12 } { 35 }$ $\frac { 1 } { 35 }$
Find the expectation and variance of $X$.

OCR MEI S1 2007 June Q7

7 A screening test for a particular disease is applied to everyone in a large population. The test classifies people into three groups: 'positive', 'doubtful' and 'negative'. Of the population, $3 \%$ is classified as positive, $6 \%$ as doubtful and the rest negative. In fact, of the people who test positive, only $95 \%$ have the disease. Of the people who test doubtful, $10 \%$ have the disease. Of the people who test negative, $1 \%$ actually have the disease. People who do not have the disease are described as 'clear'.

Copy and complete the tree diagram to show this information.
\includegraphics[max width=\textwidth, alt={}, center]{5e4f3310-b96e-43db-9b6d-61da3270db06-5_830_1157_845_536}
Find the probability that a randomly selected person tests negative and is clear.
Find the probability that a randomly selected person has the disease.
Find the probability that a randomly selected person tests negative given that the person has the disease.
Comment briefly on what your answer to part (iv) indicates about the effectiveness of the screening test. Once the test has been carried out, those people who test doubtful are given a detailed medical examination. If a person has the disease the examination will correctly identify this in $98 \%$ of cases. If a person is clear, the examination will always correctly identify this.
A person is selected at random. Find the probability that this person either tests negative originally or tests doubtful and is then cleared in the detailed medical examination.

OCR MEI S1 2007 June Q8

8 A multinational accountancy firm receives a large number of job applications from graduates each year. On average $20 \%$ of applicants are successful. A researcher in the human resources department of the firm selects a random sample of 17 graduate applicants.

Find the probability that at least 4 of the 17 applicants are successful.
Find the expected number of successful applicants in the sample.
Find the most likely number of successful applicants in the sample, justifying your answer. It is suggested that mathematics graduates are more likely to be successful than those from other fields. In order to test this suggestion, the researcher decides to select a new random sample of 17 mathematics graduate applicants. The researcher then carries out a hypothesis test at the $5 \%$ significance level.
(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 $5 \%$ level, showing all of your calculations.
Explain why the critical region found in part (v) would be unaltered if a $10 \%$ significance level were used.

Hours \(h\)	\(70 \leqslant h < 100\)	\(100 \leqslant h < 110\)	\(110 \leqslant h < 120\)	\(120 \leqslant h < 150\)	\(150 \leqslant h < 170\)	\(170 \leqslant h < 190\)
Number of years	6	8	10	11	10	3

\(r\)	0	1	2	3	4
\(\mathrm { P } ( X = r )\)	0	\(\frac { 4 } { 35 }\)	\(\frac { 18 } { 35 }\)	\(\frac { 12 } { 35 }\)	\(\frac { 1 } { 35 }\)

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