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OCR MEI Further Statistics A AS 2023 June Q4
10 marks Standard +0.3
4 At a parcel delivery company it is known that the probability that a parcel is delivered to the wrong address is 0.0005 . On a particular day, 15000 parcels are delivered. The number of parcels delivered to the wrong address is denoted by the random variable \(X\).
  1. Explain why the binomial distribution and the Poisson distribution could both be suitable models for the distribution of \(X\).
  2. Use a Poisson distribution to find each of the following.
    • \(\mathrm { P } ( X = 5 )\)
    • \(\mathrm { P } ( X \geqslant 8 )\)
    You are given that 15000 parcels are delivered each day in a 5-day working week.
    1. Determine the probability that at least 40 parcels are delivered to the wrong address during the week.
    2. Determine the probability that at least 8 parcels are delivered to the wrong address on each of the 5 days in the week.
OCR MEI Further Statistics A AS 2023 June Q5
10 marks Standard +0.3
5 Two practice GCSE examinations in mathematics are given to all of the students in a large year group. A teacher wants to check whether there is a positive relationship between the marks obtained by the students in the two examinations. She selects a random sample of 20 students. Summary data for the marks obtained in the first and second practice examinations, \(x\) and \(y\) respectively, are as follows. $$\sum x = 565 \quad \sum y = 724 \quad \sum x ^ { 2 } = 17103 \quad \sum y ^ { 2 } = 29286 \quad \sum x y = 21635$$ The teacher decides to carry out a hypothesis test based on Pearson's product moment correlation coefficient.
  1. In this question you must show detailed reasoning. Calculate the value of Pearson's product moment correlation coefficient.
  2. Carry out the test at the \(5 \%\) significance level.
  3. Given that the teacher did not draw a scatter diagram before carrying out the test, comment on the validity of the test.
OCR MEI Further Statistics A AS 2023 June Q6
15 marks Standard +0.3
6 An eight-sided dice has its faces numbered \(1,2 , \ldots , 8\).
  1. In this part of the question you should assume that the dice is fair.
    1. State the probability that, when the dice is rolled once, the score is at least 6 .
    2. Show that the probability that the score is within 2 standard deviations of its mean is 1 .
  2. A student thinks that the dice may be biased. To investigate this, the student decides to roll the dice 80 times and then carry out a \(\chi ^ { 2 }\) goodness of fit test of a uniform distribution. The spreadsheet below shows the data for the test, where some of the values have been deliberately omitted.
    \multirow[b]{2}{*}{1}ABCD
    ScoreObserved frequencyExpected frequencyChi-squared contribution
    2114101.6
    324103.6
    4310100
    541510
    656101.6
    7611100.1
    877100.9
    98100.9
    1. Explain why all of the expected frequencies are equal to 10 .
    2. Determine the missing values in each of the following cells.
      • B9
  3. D5
    (iii) In this question you must show detailed reasoning.
    4. Carry out the \(\chi ^ { 2 }\) test at the \(5 \%\) significance level.
OCR MEI Further Statistics A AS 2024 June Q1
7 marks Easy -1.2
1 The probability distribution for a discrete random variable \(X\) is given in the table below.
\(x\)0123
\(\mathrm { P } ( \mathrm { X } = \mathrm { x } )\)\(2 c\)\(3 c\)\(0.5 - c\)\(c\)
  1. Find the value of \(c\).
  2. Find the value of each of the following.
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\)
    The random variable \(Y\) is defined by \(Y = 2 X - 3\).
  3. Find the value of each of the following.
    • E(Y)
    • \(\operatorname { Var } ( Y )\)
OCR MEI Further Statistics A AS 2024 June Q2
9 marks Standard +0.3
2 In a game of chance there are 32 slots, numbered 1 to 32, and on each turn a ball lands in one of them. You may assume that the process is completely random. You are given that \(X\) is the random variable denoting the number of the slot that the ball lands in on a given turn.
  1. Suggest a suitable distribution to model \(X\). You should state the value(s) of any parameter(s).
  2. Write down \(\mathrm { P } ( X = 7 )\). Players of the game start with a score of 0 . On each turn a player may choose to play the game by selecting a number. If the ball lands in the slot with that number then 15 is added to the player's score. Otherwise, the player's score is reduced by 1 . A player's score may become negative. A player decides to play the game, selecting the number 7 on each turn, until the ball lands in the slot numbered 7. You are given that \(Y\) is the random variable denoting the number of turns up to and including the turn in which the ball lands in the slot numbered 7.
  3. Determine \(\mathrm { P } ( Y \leqslant 15 )\).
  4. Determine the player's expected final score.
OCR MEI Further Statistics A AS 2024 June Q3
14 marks Standard +0.3
3 A glassware factory produces a large number of ornaments each week. Just before they leave the factory, all the ornaments are checked and some may be found to be defective. The Quality Assurance Manager of the factory wishes to model the number of defective ornaments that are found each week using a Poisson distribution. The numbers of defective ornaments found each week in a period of 40 weeks are shown in Table 3.1. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Table 3.1}
No. of defective ornaments in a week, \(r\)0123456\(\geqslant 7\)
No. of weeks with \(r\) defective ornaments, \(f\)2141353120
\end{table} You are given that summary statistics for the data are \(\sum f = 40 , \sum \mathrm { rf } = 84\) and \(\sum \mathrm { r } ^ { 2 } \mathrm { f } = 256\).
  1. By using the summary statistics to determine estimates for the mean and variance of the number of defective ornaments produced by the factory each week, explain how the data support the suggestion that the number of defective ornaments produced each week can be modelled using a Poisson distribution. The Quality Assurance Manager is asked by the head office to carry out a chi-squared hypothesis test for goodness of fit based on a \(\operatorname { Po } ( 2 )\) distribution.
  2. Table 3.2, which is incomplete, gives observed frequency, probability, expected frequency and chi-squared contribution. \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Table 3.2}
    No. of defective ornaments in a week, \(r\)Observed frequencyProbabilityExpected frequencyChi-squared contribution
    020.135345.41342.15232
    114
    2130.270670.43620
    357.2179
    \(\geqslant 4\)60.142880.01421
    \end{table}
    1. Complete the copy of the table in the Printed Answer Booklet.
    2. Carry out the test at the \(10 \%\) significance level.
  3. On one occasion a fork-lift truck in the factory drops a crate containing eight ornaments and all of them are subsequently found to be defective. Explain why the Poisson model cannot model defects occurring in this manner.
OCR MEI Further Statistics A AS 2024 June Q4
10 marks Standard +0.3
4 A chemist is conducting an experiment in which the concentration of a certain chemical, A , is supposed to be recorded at the start of the experiment and then every 30 seconds after the start. The time after the start is denoted by \(t \mathrm {~s}\) and the concentration by \(\mathrm { z } \mathrm { mg } \mathrm { cm } ^ { - 3 }\). The collected data are shown in the table below. Note that the concentration at \(t = 90\) was not recorded.
Time, \(t\)03060120150
Concentration of A, \(z\)40.031.327.512.811.4
The chemist wishes to plot the data on a graph.
  1. Explain why \(t\) should be plotted on the horizontal axis. You are given that the summary statistics for the data are as follows. \(n = 5 \quad \sum t = 360 \quad \sum z = 123.0 \quad \sum t ^ { 2 } = 41400 \quad \sum z ^ { 2 } = 3629.74 \quad \sum \mathrm { t } = 5835\) The regression line of \(z\) on \(t\) is given by \(\mathbf { z = a + b t }\) and is used to model the concentration of chemical A for \(t \geqslant 0\).
    1. Use the summary statistics to determine the value of \(a\) and the value of \(b\).
    2. Find the value of the residual at each of the following values of \(t\).
      • \(t = 60\)
  3. \(t = 120\)
    1. Use the equation of the regression line to estimate the value of the concentration at 90 seconds.
    2. With reference to your answers to part (b)(ii), comment on the reliability of your answer to part (c)(i).
    5. Further experiments indicate that the model is reasonably reliable for times greater than 150 seconds up to about 200 seconds.
  5. Show that the model cannot be valid beyond a time of about 200 seconds.
OCR MEI Further Statistics A AS 2024 June Q5
10 marks Moderate -0.3
5 A student is investigating possible association between the amount of coffee that an adult drinks each day and the number of hours that they remain awake each day. In an initial investigation, a random sample of 8 adults is selected. The student obtains the following information from each of these adults: the amount of coffee that they drink each day and the number of hours that they remain awake each day. The student analyses the data and finds that the associated product moment correlation coefficient is 0.6030 .
  1. State one assumption that must be made for a hypothesis test based on the product moment correlation coefficient to be carried out. For the remainder of this question you may assume that this assumption is true.
  2. Carry out a test at the \(5 \%\) significance level to investigate whether there is any correlation between amount of coffee drunk and number of hours awake. The student conducts a second investigation which is similar to the first but this time based on a random sample of 30 adults. The product moment correlation coefficient for the new data is 0.5487 . The student carries out an equivalent hypothesis test to the one carried out in part (b), again using a 5\% significance level.
  3. Identify any differences between the two tests and their results. You do not need to restate the hypotheses or explain the conclusion in context.
  4. You may assume the following guidelines for considering effect size.
    Product moment
    correlation coefficient
    		Effect size
    0.1Small
    0.3Medium
    0.5Large
    Explain briefly why the results of the student's second investigation are likely to be more reliable than the results of the initial investigation.
OCR MEI Further Statistics A AS 2024 June Q6
10 marks Moderate -0.8
6 A bank monitors the amounts of cash withdrawn from a cash machine. It categorises any withdrawal of an amount of \(\pounds 50\) or less as 'small' and any withdrawal of an amount greater than \(\pounds 50\) as 'large'. Over a long period of time the bank finds that the proportion of withdrawals that are small is 0.43 .
The bank wishes to model a sample of 10 withdrawals to examine the number of small withdrawals.
    1. State a suitable probability distribution for such a model, justifying your answer.
    2. State one assumption needed for the model to be valid.
    1. Find the probability that exactly 4 of the 10 withdrawals are small.
    2. Find the probability that exactly 4 of the 10 withdrawals are large.
    3. Find the probability that no more than 4 of the 10 withdrawals are large.
  3. Find the probability that, in the 10 withdrawals, the 7th withdrawal is large and there are exactly 3 that are small.
OCR MEI Further Statistics A AS 2020 November Q1
12 marks Moderate -0.3
1 The random variable \(X\) represents the number of cars arriving at a car wash per 10-minute period. From observations over a number of days, an estimate was made of the probability distribution of \(X\). Table 1 shows this estimated probability distribution. \begin{table}[h]
\(r\)01234\(> 4\)
\(\mathrm { P } ( X = r )\)0.300.380.190.080.050
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
  1. In this question you must show detailed reasoning. Use Table 1 to calculate estimates of each of the following.
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\)
    • Explain how your answers to part (a) indicate that a Poisson distribution may be a suitable model for \(X\).
    You should now assume that \(X\) can be modelled by a Poisson distribution with mean equal to the value which you calculated in part (a).
  2. Find each of the following.
    • \(\mathrm { P } ( X = 2 )\)
    • \(\mathrm { P } ( X > 3 )\)
    • Given that the probability that there is at least 1 car arriving in a period of \(k\) minutes is at least 0.99 , find the least possible value of \(k\).
OCR MEI Further Statistics A AS 2020 November Q2
12 marks Standard +0.3
2 A researcher is investigating the concentration of bacteria and fungi in the air in buildings. The researcher selects a random sample of 12 buildings and measures the concentrations of bacteria, \(x\), and fungi, \(y\), in the air in each building. Both concentrations are measured in the same standard units. Fig. 2 illustrates the data collected. The researcher wishes to test for a relationship between \(x\) and \(y\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ba3fcd3c-6834-4116-be0e-d5b27aed0a7e-3_595_844_513_255} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. Explain why a test based on the product moment correlation coefficient is likely to be appropriate for these data. Summary statistics for the data are as follows. \(n = 12 \quad \sum x = 18030 \quad \sum y = 15550 \quad \sum x ^ { 2 } = 31458700 \quad \sum y ^ { 2 } = 21980500 \quad \sum x y = 25626800\)
  2. In this question you must show detailed reasoning. Calculate the product moment correlation coefficient between \(x\) and \(y\).
  3. Carry out a test at the \(5 \%\) significance level based on the product moment correlation coefficient to investigate whether there is any correlation between concentrations of bacteria and fungi.
  4. Explain why, in order for proper inference to be undertaken, the sample should be chosen randomly.
OCR MEI Further Statistics A AS 2020 November Q3
8 marks Moderate -0.3
3 A child is trying to draw court cards from an ordinary pack of 52 cards (court cards are Kings, Queens and Jacks; there are 12 in a pack). She draws cards, one at a time, with replacement, from the pack. Find the probabilities of the following events.
  1. She draws a court card for the first time on the sixth try.
  2. She draws a court card at least once in the first six tries.
  3. She draws a court card for the second time on the sixth try.
  4. She draws at least two court cards in the first six tries.
OCR MEI Further Statistics A AS 2020 November Q4
8 marks Easy -1.2
4 A fair 8 -sided dice has faces labelled \(1,2 , \ldots , 8\). The random variable \(X\) represents the score when the dice is rolled once.
  1. State the distribution of \(X\).
  2. Find \(\mathrm { P } ( X < 4 )\).
  3. Find each of the following.
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\)
    • The random variable \(Y\) is defined by \(Y = 10 X + 5\). Find each of the following.
    • \(\mathrm { E } ( Y )\)
    • \(\operatorname { Var } ( Y )\)
OCR MEI Further Statistics A AS 2020 November Q5
8 marks Moderate -0.3
5 A doctor is investigating the relationship between the levels in the blood of a particular hormone and of calcium in healthy adults. The levels of the hormone and of calcium, each measured in suitable units, are denoted by \(x\) and \(y\) respectively. The doctor selects a random sample of 14 adults and measures the hormone and calcium levels in each of them. The spreadsheet in Fig. 5 shows the values obtained, together with a scatter diagram which illustrates the data. The equation of the regression line of \(y\) on \(x\) is shown on the scatter diagram, together with the value of the square of the product moment correlation coefficient. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ba3fcd3c-6834-4116-be0e-d5b27aed0a7e-5_801_1644_646_255} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Use the equation of the regression line to estimate the mean calcium level of people with the following hormone levels.
    • 150
    • 250
    • Explain which of your two estimates is likely to be more reliable.
    • Comment on the goodness of fit of the regression line.
    • Explain whether it would be appropriate to plot the scatter diagram the other way around with calcium level on the horizontal axis and hormone level on the vertical axis.
    • Calculate the equation of a regression line which would be suitable for estimating the mean hormone level of people with a known calcium level.
OCR MEI Further Statistics A AS 2020 November Q6
12 marks Standard +0.3
6 A researcher is investigating whether there is any relationship between whether a cyclist wears a helmet and the distance, \(x \mathrm {~m}\), the cyclist is from the kerb (the edge of the road). Data are collected at a particular location for a random sample of 250 cyclists. The researcher carries out a chi-squared test. Fig. 6 is a screenshot showing part of a spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted. \begin{table}[h]
ABCDEFG
1\multirow{2}{*}{}Observed frequency
2\(\boldsymbol { x } \boldsymbol { \leq } \mathbf { 0 . 3 }\)\(0.3 < x \leq 0.5\)\(0.5 < x \leq 0.8\)x > 0.8Totals
3\multirow[t]{2}{*}{Wears helmet}Yes26272346122
4No45312131128
5\multirow{2}{*}{}Totals71584477250
6
7Expected frequency
8\(\boldsymbol { x } \boldsymbol { \leq } \mathbf { 0 . 3 }\)\(0.3 < x \leq 0.5\)\(0.5 < x \leq 0.8\)\(\boldsymbol { x } \boldsymbol { > } \mathbf { 0 . 8 }\)
9\multirow[t]{2}{*}{Wears helmet}Yes34.648037.5760
10No36.352039.4240
11
12\multirow{2}{*}{}Contribution to the test statistic
13\(\boldsymbol { x } \boldsymbol { \leq } \mathbf { 0 . 3 }\)\(0.3 < x \leq 0.5\)\(0.5 < x \leq 0.8\)\(\boldsymbol { x } \boldsymbol { > } \mathbf { 0 . 8 }\)
14\multirow[t]{2}{*}{Wears helmet}Yes2.15850.06010.10871.8885
15No2.05730.05731.8000
16
\captionsetup{labelformat=empty} \caption{Fig. 6}
\end{table}
  1. Showing your calculations, find the missing values in each of the following cells.
    • E10
    • E15
    • In this question you must show detailed reasoning.
    Carry out a hypothesis test at the \(10 \%\) significance level to investigate whether there is any association between helmet wearing and distance from the kerb.
  2. Discuss briefly what the data suggest about helmet wearing for different distances from the kerb.
OCR MEI Further Statistics A AS 2021 November Q1
4 marks Easy -1.3
1 The random variable \(X\) represents the clutch size (the number of eggs laid) by female birds of a particular species. The probability distribution of \(X\) is given in the table.
\(r\)234567
\(\mathrm { P } ( X = r )\)0.030.070.270.490.130.01
  1. Find each of the following.
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\)
    On average 65\% of eggs laid result in a young bird successfully leaving the nest.
    1. Find the mean number of young birds that successfully leave the nest.
    2. Find the standard deviation of the number of young birds that successfully leave the nest.
OCR MEI Further Statistics A AS 2021 November Q2
10 marks Moderate -0.3
2 A football player is practising taking penalties. On each attempt the player has a \(70 \%\) chance of scoring a goal. The random variable \(X\) represents the number of attempts that it takes for the player to score a goal.
  1. Determine \(\mathrm { P } ( X = 4 )\).
  2. Find each of the following.
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\)
    • Determine the probability that the player needs exactly 4 attempts to score 2 goals.
    • The player has \(n\) attempts to score a goal.
      1. Determine the least value of \(n\) for which the probability that the player first scores a goal on the \(n\)th attempt is less than 0.001 .
      2. Determine the least value of \(n\) for which the probability that the player scores at least one goal in \(n\) attempts is at least 0.999.
OCR MEI Further Statistics A AS 2021 November Q3
9 marks Standard +0.3
3 A student is investigating the link between temperature (in degrees Celsius) and electricity consumption (in Gigawatt-hours) in the country in which he lives. The student has read that there is strong negative correlation between daily mean temperature over the whole country and daily electricity consumption during a year. He wonders if this applies to an individual season. He therefore obtains data on the mean temperature and electricity consumption on ten randomly selected days in the summer. The spreadsheet output below shows the data, together with a scatter diagram to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{5be067ff-4668-48d6-8ed2-b8dfa3e678f7-3_798_1593_639_251}
  1. Calculate Pearson's product moment correlation coefficient between daily mean temperature and daily electricity consumption. The student decides to carry out a hypothesis test to investigate whether there is negative correlation between daily mean temperature and daily electricity consumption during the summer.
  2. Explain why the student decides to carry out a test based on Pearson's product moment correlation coefficient.
  3. Show that the test at the \(5 \%\) significance level does not result in the null hypothesis being rejected.
  4. The student concludes that there is no correlation between the variables in the summer months. Comment on the student's conclusion.
OCR MEI Further Statistics A AS 2021 November Q4
6 marks Standard +0.3
4 It is known that in an electronic circuit, the number of electrons passing per nanosecond can be modelled by a Poisson distribution. In a particular electronic circuit, the mean number of electrons passing per nanosecond is 12 .
    1. Determine the probability that there are more than 15 electrons passing in a randomly selected nanosecond.
    2. Determine the probability that there are fewer than 50 electrons passing in a randomly selected period of 5 nanoseconds.
  2. Explain what you can deduce about the electrons passing in the circuit from the fact that a Poisson distribution is a suitable model.
OCR MEI Further Statistics A AS 2021 November Q5
7 marks Moderate -0.3
5 A fair spinner has five faces, labelled 0, 1, 2, 3, 4.
  1. State the distribution of the score when the spinner is spun once.
  2. Determine the probability that, when the spinner is spun twice, one of the scores is less than 2 and the other is at least 2.
  3. Find the variance of the total score when the spinner is spun 5 times.
OCR MEI Further Statistics A AS 2021 November Q6
11 marks Moderate -0.3
6 A health researcher is investigating the relationship between age and maximum heart rate. A commonly quoted formula states that 'maximum heart rate \(= 220\) - age in years'. The researcher wants to check if this formula is a satisfactory model for people who work in the large hospital where she is employed. The researcher selects a random sample of 20 people who work in her hospital, and measures their maximum heart rates.
  1. Explain why the researcher selects a sample, rather than using all of the people who work in the hospital. The ages, \(x\) years, and maximum heart rates, \(y\) beats per minute, of the people in the researcher's sample are summarised as follows. \(n = 20 \quad \sum x = 922 \quad \sum y = 3638 \quad \sum x ^ { 2 } = 47250 \quad \sum y ^ { 2 } = 664610 \quad \sum x y = 164998\) These data are illustrated below. \includegraphics[max width=\textwidth, alt={}, center]{5be067ff-4668-48d6-8ed2-b8dfa3e678f7-5_758_1246_1027_244}
    1. Draw the line which represents the formula 'maximum heart rate \(= 220 -\) age in years' on the copy of the scatter diagram in the Printed Answer Booklet.
    2. Comment on how well this model fits the data.
  3. Determine the equation of the regression line of maximum heart rate on age.
  4. Use the equation of the regression line to predict the values of the maximum heart rate for each of the following ages.
    • 40 years
    • 5 years
    • Comment on the reliability of your predictions in part (d).
OCR MEI Further Statistics A AS 2021 November Q7
13 marks Standard +0.3
7 A biologist is investigating migrating butterflies. Fig. 7.1 shows the numbers of migrating butterflies passing her location in 100 randomly chosen one-minute periods. \begin{table}[h]
Number of butterflies01234567\(\geqslant 8\)
Frequency6918261316930
\captionsetup{labelformat=empty} \caption{Fig. 7.1}
\end{table}
    1. Use the data to show that a suitable estimate for the mean number of butterflies passing her location per minute is 3.3.
    2. Explain how the value of the variance estimate calculated from the sample supports the suggestion that a Poisson distribution may be a suitable model for these data. The biologist decides to carry out a test to investigate whether a Poisson distribution may be a suitable model for these data.
  2. In this question you must show detailed reasoning. Complete the copy of Fig. 7.2 of expected frequencies and contributions for a chi-squared test in the Printed Answer Booklet. \begin{table}[h]
    Number of butterfliesFrequencyProbabilityExpected frequencyChi-squared contribution
    060.03693.68831.4489
    190.121712.17140.8264
    2180.2160
    3260.6916
    4130.182318.22521.4981
    5160.120312.0286
    690.06626.61580.8593
    \(\geqslant 7\)30.05105.09660.8625
    \captionsetup{labelformat=empty} \caption{Fig. 7.2}
    \end{table}
  3. Complete the chi-squared test at the \(5 \%\) significance level.
OCR MEI Further Statistics A AS Specimen Q1
6 marks Moderate -0.8
1 The number of failures of a machine each week at a factory is modelled by a Poisson distribution with mean 0.45.
  1. Write down the variance of the distribution.
  2. Find the probability that there are exactly 2 failures in a week.
  3. State a distribution which can be used to model the number of failures in a period of 4 weeks.
  4. Find the probability that there are at least 2 failures in a period of 4 weeks.
OCR MEI Further Statistics A AS Specimen Q2
6 marks Moderate -0.8
2 The discrete random variable \(Y\) is uniformly distributed over the values \(\{ 12,13 , \ldots , 20 \}\).
  1. Write down \(\mathrm { P } ( Y < 15 )\).
  2. Two independent observations of \(Y\) are taken. Find the probability that one of these values is less than 15 and the other is greater than 15 .
  3. Find \(\mathrm { P } ( Y > \mathrm { E } ( Y ) )\).
OCR MEI Further Statistics A AS Specimen Q4
18 marks Moderate -0.3
4 The discrete random variable \(X\) has probability distribution defined by $$\mathrm { P } ( X = r ) = k ( 2 r - 1 ) \quad \text { for } r = 1,2,3,4,5,6 \text {, where } k \text { is a constant. }$$
  1. Complete the table in the Printed Answer Booklet giving the probabilities in terms of \(k\).
    \(r\)123456
    \(\mathrm { P } ( X = r )\)
  2. Show that the value of \(k\) is \(\frac { 1 } { 36 }\).
  3. Draw a graph to illustrate the distribution.
  4. In this question you must show detailed reasoning. Find
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\).
    A game consists of a player throwing two fair dice. The score is the maximum of the two values showing on the dice.
  5. Show that the probability of a score of 3 is \(\frac { 5 } { 36 }\).
  6. Show that the probability distribution for the score in the game is the same as the probability distribution of the random variable \(X\).
  7. The game is played three times. Find
    • the mean of the total of the three scores.
    • the variance of the total of the three scores.