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OCR MEI Further Statistics B AS 2022 June Q6
9 marks Standard +0.3
6 The length \(L\) of a particular type of fence panel is Normally distributed with mean 179.2 cm and standard deviation 0.8 cm . You should assume that the lengths of individual fence panels are independent of each other.
  1. Find the probability that the length of a randomly chosen fence panel is at least 180 cm .
  2. Find the probability that the total length of 5 randomly chosen fence panels is less than 895 cm . The width \(W\) of a fence post is Normally distributed with mean 9.8 cm and standard deviation 0.3 cm . A straight fence is constructed using 6 posts and 5 panels with no gaps between them. Fig. 6 shows a view from above of the first two posts, the first panel and the start of the second panel. You should assume that the lengths of fence panels and widths of fence posts are independent. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4caa7409-cb32-41da-ad64-012a45753296-6_213_1522_934_244} \captionsetup{labelformat=empty} \caption{Fig. 6}
    \end{figure}
  3. Determine the probability that the total length of the fence, including the posts, is less than 9.5 m .
  4. State another assumption that is necessary for the calculation of the probability in part (c) to be valid.
OCR MEI Further Statistics B AS 2022 June Q7
9 marks Standard +0.3
7 Many cars have pollen filters to try to remove as much pollen as possible from the passenger compartment. In a test car, the amount of pollen is regularly monitored. The amount of pollen is measured using a scale from 0 to 1 , and is modelled by the continuous random variable \(X\) with probability density function given by \(f ( x ) = \begin{cases} k \left( 5 x ^ { 4 } - 16 x ^ { 2 } + 11 x \right) & 0 \leqslant x \leqslant 1 , \\ 0 & \text { otherwise } , \end{cases}\) where \(k\) is a positive constant.
  1. Show that \(k = \frac { 6 } { 7 }\).
  2. Determine \(\mathrm { P } ( X < \mathrm { E } ( X ) )\).
  3. Verify that the median amount of pollen according to the model lies between 0.417 and 0.418.
OCR MEI Further Statistics B AS 2021 November Q1
9 marks Moderate -0.8
1 Over time LED light bulbs gradually lose brightness. For a particular type of LED bulb, it is known that the mean reduction in brightness after 10000 hours is \(2.6 \%\). A manufacturer produces a new version of this bulb, which costs less to make, but is claimed to have the same reduction in brightness after 10000 hours as the previous version. In order to check this claim, a random sample of 10 bulbs is selected. For each bulb, the original brightness and the brightness after 10000 hours are measured, in suitable units. A spreadsheet is used to produce a \(95 \%\) confidence interval for the mean percentage reduction in brightness. A screenshot of the spreadsheet is shown in Fig. 1. \begin{table}[h]
ABCDEFGH1JK
1Original brightness1075112111061095110111091114112311081115
2After 10000 hours1042108410761065107010791081109110801082
3Percentage reduction3.073.302.712.742.822.712.962.852.532.96
4
5
6Sample mean (\%)2.8650
7Sample sd (\%)0.2179
8SE0.0689
9DF9
10tvalue2.262
11Lower limit2.709
12Upper limit3.021
1.3
\captionsetup{labelformat=empty} \caption{Fig. 1}
\end{table}
  1. State the confidence interval in the form \(a < \mu < b\).
  2. Explain whether the confidence interval suggests that the mean percentage reduction in brightness after 10000 hours is different from 2.6\%.
  3. Explain how the value in cell B8 was calculated.
  4. State an assumption necessary for this confidence interval to be calculated.
  5. Explain the advantage of using the same bulbs for both measurements.
OCR MEI Further Statistics B AS 2021 November Q2
10 marks Moderate -0.3
2 Natasha is investigating binomial distributions. She constructs the spreadsheet in Fig. 2 which shows the first 3 and last 4 rows of a simulation involving two independent variables, \(X\) and \(Y\), with distributions \(\mathrm { B } ( 10,0.3 )\) and \(\mathrm { B } ( 50,0.3 )\) respectively. The spreadsheet also shows the corresponding value of the random variable \(Z\), defined by \(Z = 5 X - Y\), for each pair of values of \(X\) and \(Y\). There are 100 simulated values of each of \(X , Y\) and \(Z\). The spreadsheet also shows whether each value of \(Z\) is greater than 6, and cells D103 and D104 show the number of values of \(Z\) which are greater than 6 and not greater than 6 respectively. \begin{table}[h]
1ABCDE
1XY\(\mathbf { Z } = \mathbf { 5 } \mathbf { X } - \mathbf { Y }\)\(\mathbf { Z } > \mathbf { 6 }\)
24137Y
34173N
4321-6N
5
6
98114-9N
9951213Y
100318-3N
1013150N
102
103Number of Y19
104Number of N81
105
\captionsetup{labelformat=empty} \caption{Fig. 2}
\end{table}
  1. Use the information in the spreadsheet to write down an estimate of \(\mathrm { P } ( Z > 6 )\).
  2. Explain how a more reliable estimate of \(\mathrm { P } ( Z > 6 )\) could be obtained.
    1. State the greatest possible value of \(Z\).
    2. Explain why it is very unlikely that \(Z\) would have this value.
  4. Use the Central Limit Theorem to calculate an estimate of the probability that the mean of 20 independent values of \(Z\) is greater than 2 .
OCR MEI Further Statistics B AS 2021 November Q3
12 marks Standard +0.3
3 The weights in kg of male otters in a large river system are known to be Normally distributed with mean 8.3 and standard deviation 1.8. A researcher believes that weights of male otters in another river are higher because of what he suspects is better availability of food. The researcher records the weights of a random sample of 9 male otters in this other river. The sum of these 9 weights is 83.79 kg .
  1. In this question you must show detailed reasoning. You should assume that:
    Show that a test at the \(5 \%\) significance level provides sufficient evidence to conclude that the mean weight of male otters in the other river is greater than 8.3 kg .
  2. Explain whether the result of the test suggests that the weights are higher due to better availability of food.
  3. If the standard deviation of the weights of otters in the other river could not be assumed to be 1.8 kg , name an alternative test that the researcher could carry out to investigate otter weights.
  4. Explain why, even if a test at the \(5 \%\) significance level results in the rejection of the null hypothesis, you cannot be sure that the alternative hypothesis is true.
OCR MEI Further Statistics B AS 2021 November Q4
9 marks Standard +0.3
4 John regularly downloads podcasts onto his mobile phone. From past experience he knows that the average time to download one 30 -minute podcast is 12.7 s . He believes that this time has recently increased. At each of 12 randomly chosen times, he downloads a 30-minute podcast. The times in seconds to download the 12 podcasts are as follows. \(\begin{array} { l l l l l l l l l l l } 12.63 & 13.24 & 11.73 & 14.91 & 13.17 & 13.53 & 12.33 & 14.27 & 11.48 & 13.51 & 13.05 \end{array} 13.83\)
  1. Given that it is not known whether the times are Normally distributed, carry out a suitable test at the \(5 \%\) significance level to investigate whether the average download time has increased.
  2. What assumption is required to carry out the test in part (a)?
OCR MEI Further Statistics B AS 2021 November Q5
9 marks Standard +0.3
5 A food company makes mini apple pies. The weight of pastry in a pie is Normally distributed with mean 75 g and standard deviation 4 g . The weight of filling in a pie is Normally distributed with mean 130 g and standard deviation 8 g . You should assume that the weights of pastry and filling in a pie are independent.
  1. Find the probability that the weight of pastry in a randomly chosen pie is between 70 g and 80 g .
  2. Find the probability that the mean weight of filling in 10 randomly chosen pies is at least 125 g. The pies are sold in packs of 4 . The weight of the packaging is Normally distributed with mean 165 g and standard deviation 6 g .
  3. In order to find the probability that the total weight of a pack of 4 pies is less than 1 kg , you must assume that the weight of the packaging is independent of the weight of the pies.
    1. State another necessary assumption.
    2. Given that the assumptions are valid, calculate this probability.
OCR MEI Further Statistics B AS 2021 November Q6
11 marks Standard +0.3
6 The probability density function of the continuous random variable \(X\) is given by \(f ( x ) = \begin{cases} 2 ( 1 + a x ) & 0 \leqslant x \leqslant 1 , \\ 0 & \text { otherwise } , \end{cases}\) where \(a\) is a constant.
  1. Show that \(a = - 1\).
  2. Find the cumulative distribution function of \(X\).
  3. Find \(\mathrm { P } ( X < 0.5 )\).
  4. Show that \(\mathrm { E } ( X )\) is greater than the median of \(X\).
OCR MEI Further Statistics B AS Specimen Q1
9 marks Easy -1.2
1 Abby runs a stall at a charity event. Visitors to the stall pay to play a game in which six fair dice are rolled. If the difference between the highest and lowest scores is less than 3 then the player wins \(\pounds 5\). Otherwise the player wins nothing. Abby designs the spreadsheet shown in Fig. 1 to estimate the probability of a player winning, by simulating 20 goes at the game. Cell C5, highlighted, shows that the 2nd dice in simulated game 4 scores 5 . Cells H5 and I5 show the highest and lowest scores, respectively, in game 4, and cell J5 gives the difference between them. \begin{table}[h]
C5\(\times \vee f _ { x }\)=RANDBETWEEN(1,6)
ABCDEFGHJ
1dice 1dice 2dice 3dice 4dice 5dice 6High scoreLow scoreDifference
2game 1224233422
3game 2263212615
4game 3315346615
5game 4652563624
6game 5633532624
7game 6563514615
8game 7231264615
9game 8666615615
10game 9362541615
11game 10511461615
12game 11256165615
13game 12256666624
14game 13222244422
15game 14166635615
16game 15223351514
17game 16123433413
18game 17524216615
19game 18615215615
20game 19135135514
21game 20543251
\captionsetup{labelformat=empty} \caption{Fig. 1}
\end{table}
  1. (A) Write down the numbers in columns H , I and J for game 20 .
    (B) Use the spreadsheet to estimate the probability of a player winning a game.
  2. State how the estimate of probability in (i) (B) could be improved.
  3. Give one advantage and one disadvantage of using this simulation technique compared with working out the theoretical probability. All profit made by the stall is given to charity. Abby has to decide how much to charge players to play.
  4. If Abby charges \(\pounds 1\) per game, estimate the total profit when 50 players each play the game once.
OCR MEI Further Statistics B AS Specimen Q2
7 marks Standard +0.3
2 The cumulative distribution function of the continuous random variable, \(Y\), is given below. $$\mathrm { F } ( y ) = \left\{ \begin{array} { c c } 0 & y < 0 \\ \frac { y ^ { 3 } - y ^ { 2 } } { 4 } & 1 \leq y \leq 2 \\ 1 & y > 2 \end{array} \right.$$
  1. Find \(\mathrm { P } ( Y \leq 1.5 )\)
  2. Verify that the median of \(Y\) lies between 1.6 and 1.7.
  3. Find the probability density function of \(Y\).
OCR MEI Further Statistics B AS Specimen Q3
11 marks Standard +0.3
3 At a factory, flour is packed into bags. A model for the mass in grams of flour packed into each bag is \(1500 + X\), where \(X\) is a continuous random variable with probability density function $$f ( x ) = \left\{ \begin{array} { c c } k x ( 6 - x ) & 0 \leq x \leq 6 \\ 0 & \text { elsewhere, } \end{array} \right.$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 36 }\).
  2. Find the probability that a randomly selected bag of flour contains 1505 grams of flour or more.
  3. Find
OCR MEI Further Statistics B AS Specimen Q4
8 marks Standard +0.3
4 An online encyclopedia claims that the average mass of an adult European hedgehog is 720 g . In an investigation to check this average figure, the masses in grams of twelve randomly chosen adult European hedgehogs are measured and shown below.
705730720691718680
731723745708724736
  1. What assumption is required to carry out a Wilcoxon test in this situation?
  2. Given that this assumption is met, carry out a 2 -tail Wilcoxon test at the \(5 \%\) level to test whether the median mass is 720 g . You should state your hypotheses and complete the table of calculations in the Printed Answer Booklet.
OCR MEI Further Statistics B AS Specimen Q5
11 marks Standard +0.3
5 A particular alloy of bronze is specified as containing \(11.5 \%\) copper on average. A researcher takes a random sample of 14 specimens of this bronze and undertakes an analysis of each of them. The percentages of copper are found to be as follows.
11.1211.2911.4211.4311.2011.2511.65
11.3311.5611.3411.4411.2411.6011.52
The researcher uses software to draw a Normal probability plot for these data and to conduct a Kolmogorov-Smirnov test for Normality. The output is shown in Fig 5.1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0de8222f-7df5-4e17-ab68-0f9d84fc615d-4_428_1550_1434_299} \captionsetup{labelformat=empty} \caption{Fig 5.1}
\end{figure}
  1. Comment on what the Normal probability plot and the \(p\)-value of the test suggest about the data. The researcher uses software to produce a \(99 \%\) confidence interval for the mean percentage of copper in the alloy, based on the \(t\) distribution. The output from the software is shown in Fig 5.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0de8222f-7df5-4e17-ab68-0f9d84fc615d-5_1058_615_434_726} \captionsetup{labelformat=empty} \caption{Fig 5.2}
    \end{figure}
  2. State the confidence interval which the software gives, in the form \(a < \mu < b\).
  3. (A) State an assumption necessary for the use of the \(t\) distribution in the construction of this confidence interval.
    (B) State whether the assumption in part (iii) (A) seems reasonable.
  4. Does the confidence interval suggest that the copper content is different from \(11.5 \%\), on average? Explain your answer.
  5. In the output from the software shown in Fig 5.2, SE stands for 'standard error'.
    (A) Explain what a standard error is.
    (B) Show how the standard error was calculated in this case.
  6. Suggest a way in which the researcher could produce a narrower confidence interval.
OCR MEI Further Statistics B AS Specimen Q6
8 marks Standard +0.3
6 The table below shows the mean and variance of the test scores of a random samples of 70 girls who are starting an A level Mathematics course.
Sample meanSample variance
118.8686.57
  1. Showing your working, find a \(95 \%\) confidence interval for the population mean.
  2. Explain why you can construct the interval in part (i) despite no information about the distribution of the parent population being given.
  3. The same random sample of girls repeats the test. The mean improvement in score is 0.9 . The \(95 \%\) confidence interval for the improvement is \([ - 1.5,3.3 ]\). What is the sample variance for the improvement in score?
OCR MEI Further Statistics B AS Specimen Q7
6 marks Standard +0.3
7 Two flatmates work at the same location. One of them takes the bus to work and the other one cycles. Journey times, measured in minutes, are distributed as follows.
  • By bus: Normally distributed with mean 23 and standard deviation 6
  • By bicycle: Normally distributed with mean 21 and standard deviation 2
You should assume that all journey times are independent.
  1. One morning the two flatmates set out at the same time. Find the probability that the person who takes the bus arrives before the cyclist.
  2. Find the probability that the total time taken for 5 bus journeys is less than 2 hours.
  3. Comment on the assumption that all journey times are independent. \section*{END OF QUESTION PAPER} }{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.
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OCR MEI Further Pure Core 2019 June Q1
4 marks Easy -1.2
1 Find \(\sum _ { r = 1 } ^ { n } \left( 2 r ^ { 2 } - 1 \right)\), expressing your answer in fully factorised form.
OCR MEI Further Pure Core 2019 June Q2
3 marks Moderate -0.5
2 The plane \(x + 2 y + c z = 4\) is perpendicular to the plane \(2 x - c y + 6 z = 9\), where \(c\) is a constant. Find the value of \(c\).
OCR MEI Further Pure Core 2019 June Q3
7 marks Moderate -0.8
3 Matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by \(\mathbf { A } = \left( \begin{array} { l l } 3 & 1 \\ 2 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } k & 1 \\ 2 & 0 \end{array} \right)\), where \(k\) is a constant.
  1. Verify the result \(( \mathbf { A B } ) ^ { - 1 } = \mathbf { B } ^ { - 1 } \mathbf { A } ^ { - 1 }\) in this case.
  2. Investigate whether \(\mathbf { A }\) and \(\mathbf { B }\) are commutative under matrix multiplication.
OCR MEI Further Pure Core 2019 June Q5
5 marks Standard +0.3
5 Using the Maclaurin series for \(\cos 2 x\), show that, for small values of \(x\), \(\sin ^ { 2 } x \approx a x ^ { 2 } + b x ^ { 4 } + c x ^ { 6 }\),
where the values of \(a , b\) and \(c\) are to be given in exact form.
OCR MEI Further Pure Core 2019 June Q6
4 marks Standard +0.8
6 In this question you must show detailed reasoning.
Find \(\int _ { 2 } ^ { \infty } \frac { 1 } { 4 + x ^ { 2 } } \mathrm {~d} x\).
OCR MEI Further Pure Core 2019 June Q7
8 marks Standard +0.3
7 A curve has cartesian equation \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2 c ^ { 2 } x y\), where \(c\) is a positive constant.
  1. Show that the polar equation of the curve is \(r ^ { 2 } = c ^ { 2 } \sin 2 \theta\).
  2. Sketch the curves \(r = c \sqrt { \sin 2 \theta }\) and \(r = - c \sqrt { \sin 2 \theta }\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  3. Find the area of the region enclosed by one of the loops in part (b). Section B (110 marks)
    Answer all the questions.
OCR MEI Further Pure Core 2019 June Q9
7 marks Moderate -0.3
9 Prove by induction that \(5 ^ { n } + 2 \times 11 ^ { n }\) is divisible by 3 for all positive integers \(n\).
OCR MEI Further Pure Core 2019 June Q11
12 marks Standard +0.3
11
  1. Specify fully the transformations represented by the following matrices.
OCR MEI Further Pure Core 2019 June Q12
9 marks Challenging +1.2
12 Three intersecting lines \(L _ { 1 } , L _ { 2 }\) and \(L _ { 3 }\) have equations \(L _ { 1 } : \frac { x } { 2 } = \frac { y } { 3 } = \frac { z } { 1 } , \quad L _ { 2 } : \frac { x } { 1 } = \frac { y } { 2 } = \frac { z } { - 4 } \quad\) and \(\quad L _ { 3 } : \frac { x - 1 } { 1 } = \frac { y - 2 } { 1 } = \frac { z + 4 } { 5 }\).
Find the area of the triangle enclosed by these lines.
OCR MEI Further Pure Core 2019 June Q13
11 marks Standard +0.8
13
  1. Using the logarithmic form of \(\operatorname { arcosh } x\), prove that the derivative of \(\operatorname { arcosh } x\) is \(\frac { 1 } { \sqrt { x ^ { 2 } - 1 } }\).
  2. Hence find \(\int _ { 1 } ^ { 2 } \operatorname { arcosh } x \mathrm {~d} x\), giving your answer in exact logarithmic form.
  3. Ali tries to evaluate \(\int _ { 0 } ^ { 1 } \operatorname { arcosh } x \mathrm {~d} x\) using his calculator, and gets an 'error'. Explain why.