Questions — OCR MEI S1 (300 questions)

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OCR MEI S1 2005 June Q6
15 marks Standard +0.3
6 Answer part (i) of this question on the insert provided. Mancaster Hockey Club invite prospective new players to take part in a series of three trial games. At the end of each game the performance of each player is assessed as pass or fail. Players who achieve a pass in all three games are invited to join the first team squad. Players who achieve a pass in two games are invited to join the second team squad. Players who fail in two games are asked to leave. This may happen after two games.
  • The probability of passing the first game is 0.9
  • Players who pass any game have probability 0.9 of passing the next game
  • Players who fail any game have probability 0.5 of failing the next game
    1. On the insert, complete the tree diagram which illustrates the information above. \includegraphics[max width=\textwidth, alt={}, center]{668963b4-994d-475a-a1c8-c3e3a252e4e6-4_691_1329_978_397}
    2. Find the probability that a randomly selected player
      (A) is invited to join the first team squad,
      (B) is invited to join the second team squad.
    3. Hence write down the probability that a randomly selected player is asked to leave.
    4. Find the probability that a randomly selected player is asked to leave after two games, given that the player is asked to leave.
Angela, Bryony and Shareen attend the trials at the same time. Assuming their performances are independent, find the probability that
  • at least one of the three is asked to leave,
  • they pass a total of 7 games between them.
    •
    OCR MEI S1 Q3
    8 marks Easy -1.3
    3 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 Q4
    18 marks Standard +0.8
    4 Answer part (i) of this question on the insert provided. Mancaster Hockey Club invite prospective new players to take part in a series of three trial games. At the end of each game the performance of each player is assessed as pass or fail. Players who achieve a pass in all three games are invited to join the first team squad. Players who achieve a pass in two games are invited to join the second team squad. Players who fail in two games are asked to leave. This may happen after two games.
    • The probability of passing the first game is 0.9
    • Players who pass any game have probability 0.9 of passing the next game
    • Players who fail any game have probability 0.5 of failing the next game
      1. On the insert, complete the tree diagram which illustrates the information above. \includegraphics[max width=\textwidth, alt={}, center]{64f25a40-d3bf-4212-b92e-655f980c702b-4_643_1239_942_417}
      2. Find the probability that a randomly selected player
        (A) is invited to join the first team squad,
        (B) is invited to join the second team squad.
      3. Hence write down the probability that a randomly selected player is asked to leave.
      4. Find the probability that a randomly selected player is asked to leave after two games, given that the player is asked to leave.
    Angela, Bryony and Shareen attend the trials at the same time. Assuming their performances are independent, find the probability that
  • at least one of the three is asked to leave,
  • they pass a total of 7 games between them.
    •
    OCR MEI S1 Q4
    9 marks Easy -1.8
    4 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 Q2
    Easy -1.2
    2 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 correct123
    Frequency123
    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 2010 January Q1
    8 marks Easy -1.3
    A camera records the speeds in miles per hour of 15 vehicles on a motorway. The speeds are given below. $$73 \quad 67 \quad 75 \quad 64 \quad 52 \quad 63 \quad 75 \quad 81 \quad 77 \quad 72 \quad 68 \quad 74 \quad 79 \quad 72 \quad 71$$
    1. Construct a sorted stem and leaf diagram to represent these data, taking stem values of 50, 60, ... . [4]
    2. Write down the median and midrange of the data. [2]
    3. Which of the median and midrange would you recommend to measure the central tendency of the data? Briefly explain your answer. [2]
    OCR MEI S1 2010 January Q2
    8 marks Moderate -0.8
    In her purse, Katharine has two £5 notes, two £10 notes and one £20 note. She decides to select two of these notes at random to donate to a charity. The total value of these two notes is denoted by the random variable \(£X\).
      1. Show that P(X = 10) = 0.1. [1]
      2. Show that P(X = 30) = 0.2. [2]
      The table shows the probability distribution of X.
      \(r\)1015202530
      P(X = r)0.10.40.10.20.2
    2. Find E(X) and Var(X). [5]
    OCR MEI S1 2010 January Q3
    8 marks Easy -1.2
    In a survey, a large number of young people are asked about their exercise habits. One of these people is selected at random. • \(G\) is the event that this person goes to the gym. • \(R\) is the event that this person goes running. You are given that P(G) = 0.24, P(R) = 0.13 and P(G ∩ R) = 0.06.
    1. Draw a Venn diagram, showing the events \(G\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
    2. Determine whether the events \(G\) and \(R\) are independent. [2]
    3. Find P(R | G). [3]
    OCR MEI S1 2010 January Q4
    5 marks Moderate -0.8
    In a multiple-choice test there are 30 questions. For each question, there is a 60% chance that a randomly selected student answers correctly, independently of all other questions.
    1. Find the probability that a randomly selected student gets a total of exactly 20 questions correct. [3]
    2. If 100 randomly selected students take the test, find the expected number of students who get exactly 20 questions correct. [2]
    OCR MEI S1 2010 January Q5
    3 marks Easy -1.2
    My credit card has a 4-digit code called a PIN. You should assume that any 4-digit number from 0000 to 9999 can be a PIN.
    1. If I cannot remember any digits and guess my number, find the probability that I guess it correctly. [1]
    In fact my PIN consists of four different digits. I can remember all four digits, but cannot remember the correct order.
    1. If I now guess my number, find the probability that I guess it correctly. [2]
    OCR MEI S1 2010 January Q6
    4 marks Easy -1.2
    Three prizes, one for English, one for French and one for Spanish, are to be awarded in a class of 20 students. Find the number of different ways in which the three prizes can be awarded if
    1. no student may win more than 1 prize, [2]
    2. no student may win all 3 prizes. [2]
    OCR MEI S1 2010 January Q7
    19 marks Moderate -0.8
    A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm, of these pears. \includegraphics{figure_7}
    1. Calculate the number of pears which are between 90 and 100 mm long. [2]
    2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate. [4]
    3. Calculate an estimate of the standard deviation. [3]
    4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers. [4]
    5. Name the type of skewness of the distribution. [1]
    6. Illustrate the data using a cumulative frequency diagram. [5]
    OCR MEI S1 2010 January Q8
    17 marks Standard +0.3
    An environmental health officer monitors the air pollution level in a city street. Each day the level of pollution is classified as low, medium or high. The probabilities of each level of pollution on a randomly chosen day are as given in the table.
    Pollution levelLowMediumHigh
    Probability0.50.350.15
    1. Three days are chosen at random. Find the probability that the pollution level is
      1. low on all 3 days, [2]
      2. low on at least one day, [2]
      3. low on one day, medium on another day, and high on the other day. [3]
    2. Ten days are chosen at random. Find the probability that
      1. there are no days when the pollution level is high, [2]
      2. there is exactly one day when the pollution level is high. [3]
    The environmental health officer believes that pollution levels will be low more frequently in a different street. On 20 randomly selected days she monitors the pollution level in this street and finds that it is low on 15 occasions.
    1. Carry out a test at the 5% level to determine if there is evidence to suggest that she is correct. Use hypotheses \(H_0: p = 0.5\), \(H_1: p > 0.5\), where \(p\) represents the probability that the pollution level in this street is low. Explain why \(H_1\) has this form. [5]
    OCR MEI S1 2011 January Q1
    3 marks Easy -1.8
    The stem and leaf diagram shows the weights, rounded to the nearest 10 grams, of 25 female iguanas. \begin{align} 8 &| 3 \quad 9
    9 &| 3 \quad 5 \quad 6 \quad 6 \quad 6 \quad 8 \quad 9 \quad 9
    10 &| 0 \quad 2 \quad 2 \quad 3 \quad 4 \quad 6 \quad 9
    11 &| 2 \quad 4 \quad 7 \quad 8
    12 &| 3 \quad 4 \quad 5
    13 &| 2 \end{align} Key: \(11|2\) represents 1120 grams
    1. Find the mode and the median of the data. [2]
    2. Identify the type of skewness of the distribution. [1]
    OCR MEI S1 2011 January Q2
    4 marks Moderate -0.8
    The table shows all the possible products of the scores on two fair four-sided dice.
    Score on second die
    1234
    \multirow{4}{*}{\rotatebox{90}{Score on first die}} 11234
    \cline{2-5} 22468
    \cline{2-5} 336912
    \cline{2-5} 4481216
    1. Find the probability that the product of the two scores is less than 10. [1]
    2. Show that the events 'the score on the first die is even' and 'the product of the scores on the two dice is less than 10' are not independent. [3]
    OCR MEI S1 2011 January Q3
    6 marks Moderate -0.8
    There are 13 men and 10 women in a running club. A committee of 3 men and 3 women is to be selected.
    1. In how many different ways can the three men be selected? [2]
    2. In how many different ways can the whole committee be selected? [2]
    3. A random sample of 6 people is selected from the running club. Find the probability that this sample consists of 3 men and 3 women. [2]
    OCR MEI S1 2011 January Q4
    7 marks Standard +0.3
    The probability distribution of the random variable \(X\) is given by the formula $$\text{P}(X = r) = kr(r + 1) \quad \text{for } r = 1, 2, 3, 4, 5.$$
    1. Show that \(k = \frac{1}{70}\). [2]
    2. Find E\((X)\) and Var\((X)\). [5]
    OCR MEI S1 2011 January Q5
    8 marks Moderate -0.8
    Andy can walk to work, travel by bike or travel by bus. The tree diagram shows the probabilities of any day being dry or wet and the corresponding probabilities for each of Andy's methods of travel. \includegraphics{figure_5} A day is selected at random. Find the probability that
    1. the weather is wet and Andy travels by bus, [2]
    2. Andy walks or travels by bike, [3]
    3. the weather is dry given that Andy walks or travels by bike. [3]
    OCR MEI S1 2011 January Q6
    8 marks Moderate -0.8
    A survey is being carried out into the carbon footprint of individual citizens. As part of the survey, 100 citizens are asked whether they have attempted to reduce their carbon footprint by any of the following methods.
    • Reducing car use
    • Insulating their homes
    • Avoiding air travel
    The numbers of citizens who have used each of these methods are shown in the Venn diagram. \includegraphics{figure_6} One of the citizens is selected at random.
    1. Find the probability that this citizen
      1. has avoided air travel, [1]
      2. has used at least two of the three methods. [2]
    2. Given that the citizen has avoided air travel, find the probability that this citizen has reduced car use. [2]
    Three of the citizens are selected at random.
    1. Find the probability that none of them have avoided air travel. [3]
    OCR MEI S1 2011 January Q7
    19 marks Moderate -0.3
    The incomes of a sample of 918 households on an island are given in the table below.
    Income (x thousand pounds)\(0 \leqslant x \leqslant 20\)\(20 < x \leqslant 40\)\(40 < x \leqslant 60\)\(60 < x \leqslant 100\)\(100 < x \leqslant 200\)
    Frequency23836514212845
    1. Draw a histogram to illustrate the data. [5]
    2. Calculate an estimate of the mean income. [3]
    3. Calculate an estimate of the standard deviation of the incomes. [4]
    4. Use your answers to parts (ii) and (iii) to show there are almost certainly some outliers in the sample. Explain whether or not it would be appropriate to exclude the outliers from the calculation of the mean and the standard deviation. [4]
    5. The incomes were converted into another currency using the formula \(y = 1.15x\). Calculate estimates of the mean and variance of the incomes in the new currency. [3]
    OCR MEI S1 2011 January Q8
    17 marks Standard +0.3
    Mark is playing solitaire on his computer. The probability that he wins a game is 0.2, independently of all other games that he plays.
    1. Find the expected number of wins in 12 games. [2]
    2. Find the probability that
      1. he wins exactly 2 out of the next 12 games that he plays, [3]
      2. he wins at least 2 out of the next 12 games that he plays. [3]
    3. Mark's friend Ali also plays solitaire. Ali claims that he is better at winning games than Mark. In a random sample of 20 games played by Ali, he wins 7 of them. Write down suitable hypotheses for a test at the 5\% level to investigate whether Ali is correct. Give a reason for your choice of alternative hypothesis. Carry out the test. [9]
    OCR MEI S1 2011 June Q1
    5 marks Easy -1.3
    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{figure_1}
    1. Find the number of sections which are between 1000 and 2000 metres in length. [2]
    2. Name the type of skewness suggested by the histogram. [1]
    3. State the minimum and maximum possible values of the midrange. [2]
    OCR MEI S1 2011 June Q2
    5 marks Easy -1.3
    I have 5 books, each by a different author. The authors are Austen, Brontë, Clarke, Dickens and Eliot.
    1. 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. [2]
    2. If I choose two of the books at random, find the probability that I choose the books written by Austen and Brontë. [3]
    OCR MEI S1 2011 June Q3
    4 marks Easy -1.2
    25% of the plants of a particular species have red flowers. A random sample of 6 plants is selected.
    1. Find the probability that there are no plants with red flowers in the sample. [2]
    2. If 50 random samples of 6 plants are selected, find the expected number of samples in which there are no plants with red flowers. [2]
    OCR MEI S1 2011 June Q4
    7 marks Moderate -0.8
    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 \(X\).
    \(r\)012345
    P(X = r)\(\frac{1}{6}\)\(\frac{5}{18}\)\(\frac{2}{9}\)\(\frac{1}{6}\)\(\frac{1}{9}\)\(\frac{1}{18}\)
    1. Draw a vertical line chart to illustrate the probability distribution. [2]
    2. Use a probability argument to show that
      1. P(X = 1) = \(\frac{5}{18}\). [2]
      2. P(X = 0) = \(\frac{1}{6}\). [1]
    3. Find the mean value of \(X\). [2]