Questions — OCR MEI S1 (292 questions)

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OCR MEI S1 2009 January Q6
6 The temperature of a supermarket fridge is regularly checked to ensure that it is working correctly. Over a period of three months the temperature (measured in degrees Celsius) is checked 600 times. These temperatures are displayed in the cumulative frequency diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{7b92607f-1bf9-45f0-997b-fe76c88b5fcd-4_1054_1649_539_248}
  1. Use the diagram to estimate the median and interquartile range of the data.
  2. Use your answers to part (i) to show that there are very few, if any, outliers in the sample.
  3. Suppose that an outlier is identified in these data. Discuss whether it should be excluded from any further analysis.
  4. Copy and complete the frequency table below for these data.
    Temperature
    \(( t\) degrees Celsius \()\)
    		\(3.0 \leqslant t \leqslant 3.4\)\(3.4 < t \leqslant 3.8\)\(3.8 < t \leqslant 4.2\)\(4.2 < t \leqslant 4.6\)\(4.6 < t \leqslant 5.0\)
    Frequency243157
  5. Use your table to calculate an estimate of the mean.
  6. The standard deviation of the temperatures in degrees Celsius is 0.379 . The temperatures are converted from degrees Celsius into degrees Fahrenheit using the formula \(F = 1.8 C + 32\). Hence estimate the mean and find the standard deviation of the temperatures in degrees Fahrenheit.
OCR MEI S1 2009 January Q7
7 An online shopping company takes orders through its website. On average \(80 \%\) of orders from the website are delivered within 24 hours. The quality controller selects 10 orders at random to check when they are delivered.
  1. Find the probability that
    (A) exactly 8 of these orders are delivered within 24 hours,
    (B) at least 8 of these orders are delivered within 24 hours. The company changes its delivery method. The quality controller suspects that the changes will mean that fewer than \(80 \%\) of orders will be delivered within 24 hours. A random sample of 18 orders is checked and it is found that 12 of them arrive within 24 hours.
  2. Write down suitable hypotheses and carry out a test at the \(5 \%\) significance level to determine whether there is any evidence to support the quality controller's suspicion.
  3. A statistician argues that it is possible that the new method could result in either better or worse delivery times. Therefore it would be better to carry out a 2 -tail test at the \(5 \%\) significance level. State the alternative hypothesis for this test. Assuming that the sample size is still 18, find the critical region for this test, showing all of your calculations.
OCR MEI S1 2016 June Q1
1 The stem and leaf diagram illustrates the weights in grams of 20 house sparrows.
250
26058
2779
28145
29002
3077
316
32047
3333
Key: \(\quad 27 \quad \mid \quad 7 \quad\) represents 27.7 grams
  1. Find the median and interquartile range of the data.
  2. Determine whether there are any outliers.
OCR MEI S1 2016 June Q2
2 In a hockey league, each team plays every other team 3 times. The probabilities that Team A wins, draws and loses to Team B are given below.
  • \(\mathrm { P } (\) Wins \() = 0.5\)
  • \(\mathrm { P } (\) Draws \() = 0.3\)
  • \(\mathrm { P } (\) Loses \() = 0.2\)
The outcomes of the 3 matches are independent.
  1. Find the probability that Team A does not lose in any of the 3 matches.
  2. Find the probability that Team A either wins all 3 matches or draws all 3 matches or loses all 3 matches.
  3. Find the probability that, in the 3 matches, exactly two of the outcomes, 'Wins', 'Draws' and 'Loses' occur for Team A.
OCR MEI S1 2016 June Q3
3
  1. There are 5 runners in a race. How many different finishing orders are possible? [You should assume that there are no 'dead heats', where two runners are given the same position.] For the remainder of this question you should assume that all finishing orders are equally likely.
  2. The runners are denoted by \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\). Find the probability that they either finish in the order ABCDE or in the order EDCBA.
  3. Find the probability that the first 3 runners to finish are \(\mathrm { A } , \mathrm { B }\) and C , in that order.
  4. Find the probability that the first 3 runners to finish are \(\mathrm { A } , \mathrm { B }\) and C , in any order.
OCR MEI S1 2016 June Q4
4 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = \frac { k } { r ( r - 1 ) } \text { for } r = 2,3,4,5,6 .$$
  1. Show that the value of \(k\) is 1.2 . Using this value of \(k\), show the probability distribution of \(X\) in a table.
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 2016 June Q5
5 Measurements of sunshine and rainfall are made each day at a particular weather station. For a randomly chosen day,
  • \(R\) is the event that at least 1 mm of rainfall is recorded,
  • \(S\) is the event that at least 1 hour of sunshine is recorded.
You are given that \(\mathrm { P } ( R ) = 0.28 , \mathrm { P } ( S ) = 0.87\) and \(\mathrm { P } ( R \cup S ) = 0.94\).
  1. Find \(\mathrm { P } ( R \cap S )\).
  2. Draw a Venn diagram showing the events \(R\) and \(S\), and fill in the probability corresponding to each of the four regions of your diagram.
  3. Find \(\mathrm { P } ( R \mid S )\) and state what this probability represents in this context.
OCR MEI S1 2016 June Q6
6 An online store has a total of 930 different types of women's running shoe on sale. The prices in pounds of the types of women's running shoe are summarised in the table below.
Price \(( \pounds x )\)\(10 \leqslant x \leqslant 40\)\(40 < x \leqslant 50\)\(50 < x \leqslant 60\)\(60 < x \leqslant 80\)\(80 < x \leqslant 200\)
Frequency147109182317175
  1. Calculate estimates of the mean and standard deviation of the shoe prices.
  2. Calculate an estimate of the percentage of types of shoe that cost at least \(\pounds 100\).
  3. Draw a histogram to illustrate the data. The corresponding histogram below shows the prices in pounds of the 990 types of men's running shoe on sale at the same online store.
    \includegraphics[max width=\textwidth, alt={}, center]{aff0c5b2-011b-49a0-bf05-6d905f890eba-4_643_1192_340_440}
  4. State the type of skewness shown by the histogram for men's running shoes.
  5. Martin is investigating the percentage of types of shoe on sale at the store that cost more than \(\pounds 100\). He believes that this percentage is greater for men's shoes than for women's shoes. Estimate the percentage for men's shoes and comment on whether you can be certain which percentage is higher.
  6. You are given that the mean and standard deviation of the prices of men's running shoes are \(\pounds 68.83\) and \(\pounds 42.93\) respectively. Compare the central tendency and variation of the prices of men's and women's running shoes at the store.
OCR MEI S1 2016 June Q7
7 To withdraw money from a cash machine, the user has to enter a 4-digit PIN (personal identification number). There are several thousand possible 4-digit PINs, but a survey found that \(10 \%\) of cash machine users use the PIN '1234'.
  1. 16 cash machine users are selected at random.
    (A) Find the probability that exactly 3 of them use 1234 as their PIN.
    (B) Find the probability that at least 3 of them use 1234 as their PIN.
    (C) Find the expected number of them who use 1234 as their PIN. An advertising campaign aims to reduce the number of people who use 1234 as their PIN. A hypothesis test is to be carried out to investigate whether the advertising campaign has been successful.
  2. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis.
  3. A random sample of 20 cash machine users is selected.
    (A) Explain why the test could not be carried out at the \(10 \%\) significance level.
    (B) The test is to be carried out at the \(k \%\) significance level. State the lowest integer value of \(k\) for which the test could result in the rejection of the null hypothesis.
  4. A new random sample of 60 cash machine users is selected. It is found that 2 of them use 1234 as their PIN. You are given that, if \(X \sim \mathrm {~B} ( 60,0.1 )\), then (to 4 decimal places) $$\mathrm { P } ( X = 2 ) = 0.0393 , \quad \mathrm { P } ( X < 2 ) = 0.0138 , \quad \mathrm { P } ( X \leqslant 2 ) = 0.0530 .$$ Using the same hypotheses as in part (ii), carry out the test at the \(5 \%\) significance level. \section*{END OF QUESTION PAPER}
OCR MEI S1 Q3
3 The Venn diagram illustrates the occurrence of two events \(A\) and \(B\).
\includegraphics[max width=\textwidth, alt={}, center]{1ad9c390-b42f-47d8-86c5-f73a42d97721-02_513_826_1713_658} You are given that \(\mathrm { P } ( A \cap B ) = 0.3\) and that the probability that neither \(A\) nor \(B\) occurs is 0.1 . You are also given that \(\mathrm { P } ( A ) = 2 \mathrm { P } ( B )\). Find \(\mathrm { P } ( B )\).
OCR MEI S1 Q7
7 The cumulative frequency graph below illustrates the distances that 176 children live from their primary school. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Distance from school} \includegraphics[alt={},max width=\textwidth]{1ad9c390-b42f-47d8-86c5-f73a42d97721-04_1073_1571_580_340}
\end{figure}
  1. Use the graph to estimate, to the nearest 10 metres,
    (A) the median distance from school,
    (B) the lower quartile, upper quartile and interquartile range.
  2. Draw a box and whisker plot to illustrate the data. The graph on page 4 used the following grouped data.
    Distance (metres)20040060080010001200
    Cumulative frequency2064118150169176
  3. Copy and complete the grouped frequency table below describing the same data.
    Distance ( \(d\) metres)Frequency
    \(0 < d \leqslant 200\)20
    \(200 < d \leqslant 400\)
  4. Hence estimate the mean distance these children live from school. It is subsequently found that none of the 176 children lives within 100 metres of the school.
  5. Calculate the revised estimate of the mean distance.
  6. Describe what change needs to be made to the cumulative frequency graph.
OCR MEI S1 2005 June Q5
  1. On the insert, complete the table giving the lowest common multiples of all pairs of integers between 1 and 6 .
    [0pt] [1]
    \multirow{2}{*}{}Second integer
    123456
    \multirow{6}{*}{First integer}1123456
    22264106
    336312156
    4441212
    551015
    666612
    Two fair dice are thrown and the lowest common multiple of the two scores is found.
  2. Use the table to find the probabilities of the following events.
    (A) The lowest common multiple is greater than 6 .
    (B) The lowest common multiple is a multiple of 5 .
    (C) The lowest common multiple is both greater than 6 and a multiple of 5 .
  3. Use your answers to part (ii) to show that the events "the lowest common multiple is greater than 6 " and "the lowest common multiple is a multiple of 5 " are not independent.
OCR MEI S1 Q3
  1. On die insert, complete the lable giving due lowest common multiples of all pairs of integers between 1 and 6 .
    Second integer
    \cline { 2 - 8 } \multicolumn{2}{|c|}{}123456
    \multirow{5}{*}{
    First
    integer
    }    		1123456
    \cline { 2 - 8 }22264106
    \cline { 2 - 8 }336312156
    \cline { 2 - 8 }4441212
    \cline { 2 - 8 }551015
    \cline { 2 - 8 }666612
    Two fair dice are thrown and the lowest common multiple of the two scores is found.
  2. Use the table to find the probabilities of the following events.
    (A) The lowest common multiple is greater than 6 .
    (B) The lowest common multiple is a multiple of 5 .
    (C) The lowest common multiple is both greater than 6 and a multiple of 5.
  3. Use your answers to part (ii) to show that the events "the lowest common multiple is greater than 6 " and "the lowest common multiple is a multiple of 5 " are not independent.
OCR MEI S1 2005 June Q6
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
    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
    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
    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.