Questions — OCR MEI (4301 questions)

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OCR MEI S1 2007 January Q4
4 A fair six-sided die is rolled twice. The random variable \(X\) represents the higher of the two scores. The probability distribution of \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k ( 2 r - 1 ) \text { for } r = 1,2,3,4,5,6 .$$
  1. Copy and complete the following probability table and hence find the exact value of \(k\), giving your answer as a fraction in its simplest form.
    \(r\)123456
    \(\mathrm { P } ( X = r )\)\(k\)\(11 k\)
  2. Find the mean of \(X\). A fair six-sided die is rolled three times.
  3. Find the probability that the total score is 16 .
OCR MEI S1 2007 January Q5
5 Each day the probability that Ashwin wears a tie is 0.2 . The probability that he wears a jacket is 0.4 . If he wears a jacket, the probability that he wears a tie is 0.3 .
  1. Find the probability that, on a randomly selected day, Ashwin wears a jacket and a tie.
  2. Draw a Venn diagram, using one circle for the event 'wears a jacket' and one circle for the event 'wears a tie'. Your diagram should include the probability for each region.
  3. Using your Venn diagram, or otherwise, find the probability that, on a randomly selected day, Ashwin
    (A) wears either a jacket or a tie (or both),
    (B) wears no tie or no jacket (or wears neither).
OCR MEI S1 2007 January Q6
6 The birth weights in grams of a random sample of 1000 babies are displayed in the cumulative frequency diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{05b96db3-93c7-4921-a1c6-c7b2f8952a8f-4_1264_1553_486_296}
  1. Use the diagram to estimate the median and interquartile range of the data.
  2. Use your answers to part (i) to estimate the number of outliers in the sample.
  3. Should these outliers be excluded from any further analysis? Briefly explain your answer.
  4. Any baby whose weight is below the 10th percentile is selected for careful monitoring. Use the diagram to determine the range of weights of the babies who are selected.
    \(12 \%\) of new-born babies require some form of special care. A maternity unit has 17 new-born babies. You may assume that these 17 babies form an independent random sample.
  5. Find the probability that
    (A) exactly 2 of these 17 babies require special care,
    (B) more than 2 of the 17 babies require special care.
  6. On 100 independent occasions the unit has 17 babies. Find the expected number of occasions on which there would be more than 2 babies who require special care.
OCR MEI S1 2007 January Q7
7 When onion seeds are sown outdoors, on average two-thirds of them germinate. A gardener sows seeds in pairs, in the hope that at least one will germinate.
  1. Assuming that germination of one of the seeds in a pair is independent of germination of the other seed, find the probability that, if a pair of seeds is selected at random,
    (A) both seeds germinate,
    (B) just one seed germinates,
    (C) neither seed germinates.
  2. Explain why the assumption of independence is necessary in order to calculate the above probabilities. Comment on whether the assumption is likely to be valid.
  3. A pair of seeds is sown. Find the expectation and variance of the number of seeds in the pair which germinate.
  4. The gardener plants 200 pairs of seeds. If both seeds in a pair germinate, the gardener destroys one of the two plants so that only one is left to grow. Of the plants that remain after this, only \(85 \%\) successfully grow to form an onion. Find the expected number of onions grown from the 200 pairs of seeds. If the seeds are sown in a greenhouse, the germination rate is higher. The seed manufacturing company claims that the germination rate is \(90 \%\). The gardener suspects that the rate will not be as high as this, and carries out a trial to investigate. 18 randomly selected seeds are sown in the greenhouse and it is found that 14 germinate.
  5. Write down suitable hypotheses and carry out a test at the \(5 \%\) level to determine whether there is any evidence to support the gardener's suspicions.
OCR MEI S1 2008 January Q1
1 Alice carries out a survey of the 28 students in her class to find how many text messages each sent on the previous day. Her results are shown in the stem and leaf diagram.
000113577788
1012334469
201337
357
4
58
Key: 2 | 3 represents 23
  1. Find the mode and median of the number of text messages.
  2. Identify the type of skewness of the distribution.
  3. Alice is considering whether to use the mean or the median as a measure of central tendency for these data.
    (A) In view of the skewness of the distribution, state whether Alice should choose the mean or the median.
    (B) What other feature of the distribution confirms Alice's choice?
  4. The mean number of text messages is 14.75 . If each message costs 10 pence, find the total cost of all of these messages.
OCR MEI S1 2008 January Q2
2 Codes of three letters are made up using only the letters A, C, T, G. Find how many different codes are possible
  1. if all three letters used must be different,
  2. if letters may be repeated.
OCR MEI S1 2008 January Q3
3 Steve is going on holiday. The probability that he is delayed on his outward flight is 0.3 . The probability that he is delayed on his return flight is 0.2 , independently of whether or not he is delayed on the outward flight.
  1. Find the probability that Steve is delayed on his outward flight but not on his return flight.
  2. Find the probability that he is delayed on at least one of the two flights.
  3. 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.
\(r\)1234
\(\mathrm { P } ( X = r )\)0.20.160.1280.512
  1. Find the expectation and variance of \(X\).
  2. It costs \(\pounds 45000\) to investigate each site. Find the expected total cost of the investigation.
  3. 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}
  1. Explain why the sequence \(\mathbf { S S J }\) is not possible.
  2. Write down the other three possible sequences in which Sophie wins the competition.
  3. 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.
JanFebMarAprMayJunJulAugSepOctNovDec
9.27.110.714.216.621.822.022.621.117.410.17.8
These data are summarised by \(n = 12 , \Sigma x = 180.6 , \Sigma x ^ { 2 } = 3107.56\).
  1. Calculate the mean and standard deviation of the data.
  2. Determine whether there are any outliers.
  3. 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.
  4. 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 years681011103
  5. Draw a cumulative frequency graph for these data.
  6. 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.
  1. 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.
  2. 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.
  3. 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.
\(\begin{array} { l l l l l l l l l l l l l l l l l l l l l } 22 & 23 & 23 & 23 & 26 & 28 & 28 & 30 & 31 & 33 & 33 & 34 & 35 & 35 & 36 & 36 & 37 & 46 & 49 & 49 \end{array}\)
  1. Calculate the sample mean and sample variance, \(s ^ { 2 }\), of these data.
  2. 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.
Length
\(( x\) miles \()\)
\(0 < x \leqslant 1\)\(1 < x \leqslant 2\)\(2 < x \leqslant 3\)\(3 < x \leqslant 4\)\(4 < x \leqslant 6\)\(6 < x \leqslant 10\)
Number of
journeys
3830211498
  1. On the insert, draw a cumulative frequency diagram to illustrate the data.
  2. Use your graph to estimate the median length of journey and the quartiles. Hence find the interquartile range.
  3. 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 $$\mathrm { P } ( X = r ) = \frac { 1 } { 40 } r ( r + 1 ) \quad \text { for } r = 1,2,3,4 .$$
  1. Verify that \(\mathrm { P } ( X = 4 ) = \frac { 1 } { 2 }\).
  2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  3. 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.
    1. No sixes are thrown.
    2. Exactly four sixes are thrown.
    3. 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.
    4. 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.
    5. 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 correct1234567
    Frequency1233475
    1. Draw a vertical line chart to illustrate the data.
    2. State the type of skewness shown by your diagram.
    3. Calculate the mean and the mean squared deviation of the data.
    4. 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 \(\mathrm { P } ( A ) = \frac { 7 } { 10 }\).
    1. Calculate \(\mathrm { P } ( A \cap B )\). The probability that Isobel does not score and her parents do not attend is 0.1 .
    2. Draw a Venn diagram showing the events \(A\) and \(B\), and mark in the probability corresponding to each of the regions of your diagram.
    3. Are events \(A\) and \(B\) independent? Give a reason for your answer.
    4. 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 $$\mathrm { P } ( X = r ) = k r ( 8 - r ) \text { for } r = 1,2,3,4 .$$
    1. Show that \(k = \frac { 1 } { 50 }\).
    2. 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.
    1. Calculate the number of ways in which Peter may choose his three-course meal.
    2. 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.
    3. 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 \(\mathrm { B } ( 3,0.87 )\).
    1. Calculate the probability that Douglas hits 20 twice.
    2. Douglas aims fifty sets of 3 darts at the number 20. Find the expected number of sets for which Douglas hits 20 twice.
    3. 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 .
    1. 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.
    2. The test gives a positive result.
    3. The test gives a correct result.
    4. The painting is genuine, given a positive result.
    5. 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.
    6. 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.
    7. 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.
    1. Find the probability that
      (A) exactly one of the rocks contains fossils,
      (B) at least one of the rocks contains fossils.
    2. 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 \(n\).
    3. 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.
    1. In how many ways can she select the 4 tracks?
    2. 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
    1. Express the data in the form of a grouped frequency table.
    2. Use your table to estimate the total amount of money spent by customers on that day.