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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6 A bank monitors the amounts of cash withdrawn from a cash machine.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.
OCR MEI Further Statistics A AS Specimen Q5
8 marks Moderate -0.8
5 In a recent report, it was stated that \(40 \%\) of working people have a degree. For the whole of this question, you should assume that this is true. A researcher wishes to interview a working person who has a degree. He asks working people at random whether they have a degree and counts the number of people he has to ask until he finds one with a degree.
  1. Find the probability that he has to ask 5 people.
  2. Find the mean number of people the researcher has to ask. Subsequently, the researcher decides to take a random sample from the population of working people.
  3. A random sample of 5 working people is chosen. What is the probability that at least one of them has a degree?
  4. How large a random sample of working people would the researcher need to take to ensure that the probability that at least one person has a degree is 0.99 or more?
OCR MEI Further Statistics A AS Specimen Q6
12 marks Standard +0.3
6 A motorist decides to check the fuel consumption, \(y\) miles per gallon, of her car at particular speeds, \(x \mathrm { mph }\), on flat roads. She carries out the check on a suitable stretch of motorway. Fig. 6 shows her results. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{880026ad-1cd3-40bb-bc87-8dcc94bd9bbd-4_707_1091_1320_477} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Explain why it would not be appropriate to carry out a hypothesis test for correlation based on the product moment correlation coefficient.
  2. (A) One of the results is an outlier. Circle the outlier on the copy of Fig. 6 in the Printed Answer Booklet.
    (B) Suggest one possible reason for the outlier in part (ii) (A) not being used in any analysis. The motorist decides to remove this item of data from any analysis. The table below shows part of a spreadsheet that was used to analyse the 14 remaining data items (with the outlier removed). Some rows of the spreadsheet have been deliberately omitted.
    Data item\(x\)\(y\)\(x ^ { 2 }\)\(y ^ { 2 }\)\(x y\)
    15053.625002872.962680
    25053.325002840.892665
    137044.849002007.043136
    147044.249001953.643094
    Sum8406865115033779.740812
  3. Calculate the equation of the regression line of \(y\) on \(x\).
  4. Use the equation of the regression line to predict the fuel consumption of the car at
    (A) 58 mph ,
    (B) 30 mph .
  5. Comment on the reliability of your predictions in part (iv). OCR is committed to seeking permission to reproduce all third-party content that it uses in the assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
    For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
    OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }\section*{}
OCR MEI Further Statistics B AS 2018 June Q1
6 marks Moderate -0.8
1 The birth weights, in kilograms, of a random sample of 9 captive-bred elephants are as follows. $$\begin{array} { l l l l l l l l l } 94 & 138 & 130 & 118 & 146 & 165 & 82 & 115 & 69 \end{array}$$ A researcher uses software to produce a \(99 \%\) confidence interval for the mean birth weight of captive-bred elephants. The output from the software is shown in Fig. 1. \begin{table}[h]
Result
T Estimate of a Mean
Mean
s
SE
N
df
Lower limit
Upper limit
Interval
\captionsetup{labelformat=empty} \caption{Fig. 1}
\end{table}
  1. State an assumption about the distribution of the population from which these weights come that is necessary in order to produce this interval.
  2. State the confidence interval which the software gives, in the form \(a < \mu < b\).
  3. Explain
    • what the label df means,
    • how the value of df is calculated for a confidence interval produced using the \(t\) distribution.
    • State two ways in which the researcher could have obtained a narrower confidence interval.
OCR MEI Further Statistics B AS 2018 June Q2
9 marks Standard +0.3
2 A supermarket sells oranges. Their weights are modelled by the random variable \(X\) which has a Normal distribution with mean 345 grams and standard deviation 15 grams. When the oranges have been peeled, their weights in grams, \(Y\), are modelled by \(Y = 0.7 X\).
  1. Find the probability that a randomly chosen peeled orange weighs less than 250 grams. I randomly choose 5 oranges to buy.
  2. Find the probability that the total weight of the 5 unpeeled oranges is at least 1800 grams.
  3. I peel three of the oranges and leave the remaining two unpeeled. Find the probability that the total weight of the two unpeeled oranges is greater than the total weight of the three peeled ones.
OCR MEI Further Statistics B AS 2018 June Q3
10 marks Standard +0.3
3 The probability density function of the continuous random variable \(X\) is given by $$\mathrm { f } ( x ) = \begin{cases} c + x & - c \leqslant x \leqslant 0 \\ c - x & 0 \leqslant x \leqslant c \\ 0 & \text { otherwise } \end{cases}$$ where \(c\) is a positive constant.
  1. (A) Sketch the graph of the probability density function.
    (B) Show that \(c = 1\).
  2. Find \(\mathrm { P } \left( X < \frac { 1 } { 4 } \right)\).
  3. Find
    • the mean of \(X\),
    • the standard deviation of \(X\).
OCR MEI Further Statistics B AS 2018 June Q4
15 marks Easy -1.2
4 The random variable \(X\) has a continuous uniform distribution on [ 0,10 ].
  1. Find \(\mathrm { P } ( 3 < X < 6 )\).
  2. Find each of the following.
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\)
    Marisa is investigating the sample mean, \(Y\), of 8 independent values of \(X\). She designs a simulation shown in the spreadsheet in Fig. 4.1. Each of the 25 rows below the heading row consists of 8 values of \(X\) together with the value of \(Y\). All of the values in the spreadsheet have been rounded to 2 decimal places. \begin{table}[h]
    1ABCDEFGHIJ
    1\(X _ { 1 }\)\(X _ { 2 }\)\(X _ { 3 }\)\(X _ { 4 }\)\(X _ { 5 }\)\(X _ { 6 }\)\(X _ { 7 }\)\(X _ { 8 }\)\(Y\)
    26.312.453.273.064.161.530.437.993.65
    31.701.527.108.936.442.709.967.835.77
    49.150.524.956.996.523.150.815.354.68
    50.652.717.929.650.504.876.462.674.43
    63.096.113.960.090.184.670.676.203.12
    77.065.841.973.609.361.974.483.474.72
    81.461.575.450.373.767.568.489.124.72
    99.421.854.911.611.948.001.775.344.36
    102.985.322.914.129.161.769.976.885.39
    112.833.443.287.851.000.938.774.034.01
    124.510.595.849.878.653.947.180.235.10
    134.490.693.658.784.968.963.771.434.59
    146.578.084.856.757.920.279.694.046.02
    158.351.098.638.047.232.122.579.595.95
    165.249.536.088.213.617.076.657.636.75
    177.895.503.090.716.475.496.474.955.07
    188.367.272.359.040.582.263.017.905.10
    193.761.019.619.657.899.986.284.346.56
    209.946.843.385.530.268.535.725.125.66
    217.259.100.342.884.662.656.377.635.11
    227.187.145.380.044.096.474.964.234.94
    238.695.044.902.942.004.234.130.974.11
    243.466.330.489.350.231.187.976.374.42
    252.377.267.161.245.262.803.553.844.19
    262.168.307.173.322.961.309.110.314.33
    27
    \captionsetup{labelformat=empty} \caption{Fig. 4.1}
    \end{table}
  3. Use the spreadsheet to estimate \(\mathrm { P } ( 3 < Y < 6 )\).
  4. Explain why it is not surprising that this estimated probability is substantially greater than the value which you calculated in part (i). Marisa wonders whether, even though the sample size is only 8, use of the Central Limit Theorem will provide a good approximation to \(\mathrm { P } ( 3 < Y < 6 )\).
  5. Calculate an estimate of \(\mathrm { P } ( 3 < Y < 6 )\) using the Central Limit Theorem. A Normal probability plot of the 25 simulated values of \(Y\) is shown in Fig. 4.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0c58d4d7-10e9-473a-888a-b407ec90bf08-5_800_1291_306_386} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
    \end{figure}
  6. Explain what the Normal probability plot suggests about the use of the Central Limit Theorem to approximate \(\mathrm { P } ( 3 < Y < 6 )\). Marisa now decides to use a spreadsheet with 1000 rows below the heading row, rather than the 25 which she used in the initial simulation shown in Fig. 4.1. She uses a counter to count the number of values of \(Y\) between 3 and 6. This value is 808.
  7. Explain whether the value 808 supports the suggestion that the Central Limit Theorem provides a good approximation to \(\mathrm { P } ( 3 < Y < 6 )\). Marisa decides to repeat each of her two simulations many times in order to investigate how variable the probability estimates are in each case.
  8. Explain whether you would expect there to be more, the same or less variability in the probability estimates based on 1000 rows than in the probability estimates based on 25 rows.
OCR MEI Further Statistics B AS 2018 June Q5
10 marks Moderate -0.3
5 The flight time between two airports is known to be Normally distributed with mean 3.75 hours and standard deviation 0.21 hours. A new airline starts flying the same route. The flight times for a random sample of 12 flights with the new airline are shown in the spreadsheet (Fig. 5), together with the sample mean. \begin{table}[h]
ABCDEFGHIJKL
13.5953.7233.5843.6433.6693.6973.5503.6743.9243.5633.3303.706
2
3Mean3.638
\captionsetup{labelformat=empty} \caption{Fig. 5}
\end{table} \section*{(i) In this question you must show detailed reasoning.} You should assume that:
  • the flight times for the new airline are Normally distributed,
  • the standard deviation of the flight times is still 0.21 hours.
Carry out a test at the \(5 \%\) significance level to investigate whether the mean flight time for the new airline is less than 3.75 hours.
(ii) If both of the assumptions in part (i) were false, name an alternative test that you could carry out to investigate average flight times, stating any assumption necessary for this test.
(iii) If instead the flight times were still Normally distributed but the standard deviation was not known to be 0.21 hours, name another test that you could carry out.
OCR MEI Further Statistics B AS 2018 June Q6
10 marks Standard +0.3
6 A company has a large fleet of cars. It is claimed that use of a fuel additive will reduce fuel consumption. In order to test this claim a researcher at the company randomly selects 40 of the cars. The fuel consumption of each of the cars is measured, both with and without the fuel additive. The researcher then calculates the difference \(d\) litres per kilometre between the two figures for each car, where \(d\) is the fuel consumption without the additive minus the fuel consumption with the additive. The sample mean of \(d\) is 0.29 and the sample standard deviation is 1.64 .
  1. Showing your working, find a 95\% confidence interval for the population mean difference.
  2. Explain whether the confidence interval suggests that, on average, the fuel additive does reduce fuel consumption.
  3. Explain why you can construct the interval in part (i) despite not having any information about the distribution of the population of differences.
  4. Explain why the sample used was random.