Questions — OCR MEI (4301 questions)

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OCR MEI Further Statistics Minor 2021 November Q3
3 A student wants to know whether there is any association between age and whether or not people smoke. The student takes a sample of 120 adults and asks each of them whether or not they smoke. Below is a screenshot showing part of a spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted.
ABCDE
1\multirow{3}{*}{}Observed frequency
2Age
316-3435-5960 and over
4\multirow{2}{*}{Smoking status}Smoker1373
5Non-smoker284326
6
7Expected frequency
87.8583
933.1417
10
11Contributions to the test statistic
123.36420.69641.1775
130.16510.2792
11
  1. The student wants to carry out a chi-squared test to analyse the data. State a requirement of the sample if the test is to be valid. For the rest of this question, you should assume that this requirement is met.
  2. Determine the missing values in each of the following cells.
    • E8
    • C13
    • In this question you must show detailed reasoning.
    Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether there is any association between age and smoking status.
  3. Discuss what the data suggest about the smoking status for each different age group.
OCR MEI Further Statistics Minor 2021 November Q4
4 A scientist is investigating sea salinity (the level of salt in the sea) in a particular area. She wishes to check whether satellite measurements, \(y\), of salinity are similar to those directly measured, \(x\). Both variables are measured in parts per thousand in suitable units. The scientist obtains a random sample of 10 values of \(x\) and the related values of \(y\). Below is a screenshot of a scatter diagram to illustrate the data. She decides to carry out a hypothesis test to check if there is any correlation between direct measurement, \(x\), and satellite measurement, \(y\).
\includegraphics[max width=\textwidth, alt={}, center]{691e8b55-e9a1-4fff-b9ee-a71ff1f73ead-5_830_837_589_246}
  1. Explain why the scientist might decide to carry out a test based on the product moment correlation coefficient. Summary statistics for \(x\) and \(y\) are as follows.
    \(n = 10 \quad \sum x = 351.9 \quad \sum y = 350.0 \quad \sum x ^ { 2 } = 12384.5 \quad \sum y ^ { 2 } = 12251.2 \quad \sum \mathrm { xy } = 12317.2\)
  2. In this question you must show detailed reasoning. Calculate the product moment correlation coefficient.
  3. Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether there is positive correlation between directly measured and satellite measured salinity levels.
  4. Explain why it would be preferable to use a larger sample. The scientist is also interested in whether there is any correlation between salinity and numbers of a particular species of shrimp in the water. She takes a large sample and finds that the product moment correlation coefficient for this sample is 0.165 . The result of a test based on this sample is to reject the null hypothesis and conclude that there is correlation between salinity and numbers of shrimp.
  5. Comment on the outcome of the hypothesis test with reference to the effect size of 0.165 .
OCR MEI Further Statistics Minor 2021 November Q5
5 Biological cell membranes have receptor molecules which perform various functions. It is known that the number of receptor molecules of a particular type can be modelled by a Poisson distribution with mean 6 per area of 1 square unit.
    1. Determine the probability that there are at least 10 of these receptor molecules in an area of 1 square unit.
    2. Determine the probability that there are fewer than 50 of these receptor molecules in an area of 10 square units.
  2. A scientist is looking at areas of 1 square unit of cell membrane in order to find one which has at least 10 receptor molecules. Find the probability that she has to look at more than 20 to find such an area. It is known that the number of receptor molecules of another type in an area of 1 square unit can be modelled by the random variable \(X\) which has a Poisson distribution with mean \(\mu\). It is given that \(\mathrm { E } \left( X ^ { 2 } \right) = 12\).
  3. Determine \(\mathrm { P } ( X < 5 )\).
OCR MEI Further Statistics Minor 2021 November Q6
6 A lottery has tickets numbered 1 to \(n\) inclusive, where \(n\) is a positive integer. The random variable \(X\) denotes the number on a ticket drawn at random.
  1. Determine \(\mathrm { P } \left( \mathrm { X } \leqslant \frac { 1 } { 4 } \mathrm { n } \right)\) in each of the following cases.
    1. \(n\) is a multiple of 4 .
    2. \(n\) is of the form \(4 k + 1\), where \(k\) is a positive integer. Give your answer as a single fraction in terms of \(n\).
  2. Given that \(n = 101\), find the probability that \(X\) is within one standard deviation of the mean.
OCR MEI Further Statistics Minor Specimen Q1
1 A darts player is trying to hit the bullseye on a dart board. On each throw the probability that she hits it is 0.05 , independently of any other throw.
  1. Find the probability that she hits the bullseye for the first time on her 10th throw.
  2. Find the probability that she does not hit the bullseye in her first 10 throws.
  3. Write down the expected number of throws which it takes her to hit the bullseye for the first time.
  4. Write down the expected number of throws which it takes her to hit the bullseye for the first time.
OCR MEI Further Statistics Minor Specimen Q2
2 The number of televisions of a particular model sold per week at a retail store can be modelled by a random variable \(X\) with the probability function shown in the table.
\(x\)01234
\(\mathrm { P } ( X = x )\)0.050.20.50.20.05
  1. (A) Explain why \(\mathrm { E } ( X ) = 2\).
    (B) Find \(\operatorname { Var } ( X )\).
  2. The profit, measured in pounds made in a week, on the sales of this model of television is given by \(Y\), where \(Y = 250 X - 80\).
    Find
    • \(\mathrm { E } ( Y )\) and
    • \(\operatorname { Var } ( Y )\).
    The remote controls for the televisions are quality tested by the manufacturer to see how long they last before they fail.
  3. Explain why it would be inappropriate to test all the remote controls in this way.
  4. State an advantage of using random sampling in this context. A website awards a random number of loyalty points each time a shopper buys from it. The shopper gets a whole number of points between 0 and 10 (inclusive). Each possibility is equally likely, each time the shopper buys from the website. Awards of points are independent of each other.
  5. Let \(X\) be the number of points gained after shopping once. Find
    • the mean of \(X\)
    • the variance of \(X\).
    • Let \(Y\) be the number of points gained after shopping twice.
    Find
    • the mean of \(Y\)
    • the variance of \(Y\).
    • Find the probability of the most likely number of points gained after shopping twice. Justify your answer.
    • State the conditions under which the Poisson distribution is an appropriate model for the number of emails received by one person in a day.
    Jane records the number of junk emails which she receives each day. During working hours (9am to 5pm, Monday to Friday) the mean number of junk emails is 7.4 per day. Outside working hours ( 5 pm to 9am), the mean number of junk emails is 0.3 per hour. For the remainder of this question, you should assume that Poisson models are appropriate for the number of junk emails received during each of "working hours" and "outside working hours".
  6. Find the probability that the number of junk emails which she receives between 9am and 5pm on a Monday is
    (A) exactly 10 ,
    (B) at least 10 .
  7. (A) What assumption must you make to calculate the probability that the number of junk emails which she receives from 9am Monday to 9am Tuesday is at most 20?
    (B) Find the probability.
OCR MEI Further Statistics Minor Specimen Q5
5 Each contestant in a talent competition is given a score out of 20 by a judge. The organisers suspect that the judge's scores are associated with the age of the contestant. Table 5.1 and the scatter diagram in Fig. 5.2 show the scores and ages of a random sample of 7 contestants. \begin{table}[h]
ContestantABCDEFG
Age6651392992214
Score1211151716189
\captionsetup{labelformat=empty} \caption{Table 5.1}
\end{table} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4109d98-1009-4929-a0d2-2ba12234894b-4_638_1079_772_502} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
\end{figure} Contestant G did not finish her performance, so it is decided to remove her data.
  1. Spearman's rank correlation coefficient between age and score, including all 7 contestants, is - 0.25 . Explain why Spearman's rank correlation coefficient becomes more negative when the data for contestant G is removed.
  2. Calculate Spearman's rank correlation coefficient for the 6 remaining contestants.
  3. Using this value of Spearman's rank correlation coefficient, carry out a hypothesis test at the \(5 \%\) level to investigate whether there is any association between age and score.
  4. Briefly explain why it may be inappropriate to carry out a hypothesis test based on Pearson's product moment correlation coefficient using these data.
OCR MEI Further Statistics Minor Specimen Q6
6 At a bird feeding station, birds are captured and ringed. If a bird is recaptured, the ring enables it to be identified. The table below shows the number of recaptures, \(x\), during a period of a month, for each bird of a particular species in a random sample of 40 birds.
Number of
recaptures, \(x\)
012345678910
Frequency255910431010
  1. The sample mean of \(x\) is 3.4. Calculate the sample variance of \(x\).
  2. Briefly comment on whether the results of part (i) support a suggestion that a Poisson model might be a good fit to the data. The screenshot below shows part of a spreadsheet for a \(\chi ^ { 2 }\) test to assess the goodness of fit of a Poisson model. The sample mean of 3.4 has been used as an estimate of the Poisson parameter. Some values in the spreadsheet have been deliberately omitted.
    ABCDE
    1Number of recapturesObserved frequencyPoisson probabilityExpected frequencyChi-squared contribution
    20 or 170.14685.87370.2160
    3250.9560
    4390.21868.74470.0075
    54100.18587.43300.8865
    6540.12645.0544
    7\(\geq 6\)50.12955.17830.0061
  3. State the null and alternative hypotheses for the test.
  4. Calculate the missing values in cells
    • C3,
    • D3 and
    • E6.
    • Complete the test at the \(10 \%\) significance level.
    • The screenshot below shows part of a spreadsheet for a \(\chi ^ { 2 }\) test for a different species of bird. Find the value of the Poisson parameter used.
    ABCDE
    1Number of recapturesObserved frequencyPoisson probabilityExpected frequencyChi-squared contribution
    31100.2571612.85790.6352
    4270.2700213.50083.1302
    53150.189019.45063.2587
    6\(\geq 4\)110.161368.06791.0656
OCR MEI Further Statistics Minor Specimen Q7
4 marks
7 A fair coin has + 1 written on the heads side and - 1 on the tails side. The coin is tossed 100 times. The sum of the numbers showing on the 100 tosses is the random variable \(Y\). Show that the variance of \(Y\) is 100 . [4] \section*{END OF QUESTION PAPER} 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.
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OCR MEI Further Statistics Major 2019 June Q1
1 A fair six-sided dice is rolled three times.
The random variable \(X\) represents the lowest of the three scores.
The probability distribution of \(X\) is given by the formula
\(\mathrm { P } ( X = r ) = k \left( 127 - 39 r + 3 r ^ { 2 } \right)\) for \(r = 1,2,3,4,5,6\).
  1. Complete the copy of the table in the Printed Answer Booklet.
    \(r\)123456
    \(\mathrm { P } ( X = r )\)\(91 k\)\(61 k\)\(37 k\)
  2. Show that \(k = \frac { 1 } { 216 }\).
  3. Draw a graph to illustrate the distribution.
  4. Comment briefly on the shape of the distribution.
  5. In this question you must show detailed reasoning. Find each of the following.
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\)
OCR MEI Further Statistics Major 2019 June Q2
2 A special railway coach detects faults in the railway track before they become dangerous.
  1. Write down the conditions required for the numbers of faults in the track to be modelled by a Poisson distribution. You should now assume that these conditions do apply, and that the mean number of faults in a 5 km length of track is 1.6 .
  2. Find the probability that there are at least 2 faults in a randomly chosen 5 km length of track.
  3. Find the probability that there are at most 10 faults in a randomly chosen 25 km length of track.
  4. On a particular day the coach is used to check 10 randomly chosen 1 km lengths of track. Find the probability that exactly 1 fault, in total, is found.
OCR MEI Further Statistics Major 2019 June Q3
3 The weights of bananas sold by a supermarket are modelled by a Normal distribution with mean 205 g and standard deviation 11 g .
  1. Find the probability that the total weight of 5 randomly selected bananas is at least 1 kg . When a banana is peeled the change in its weight is modelled as being a reduction of \(35 \%\).
  2. Find the probability that the weight of a randomly selected peeled banana is at most 150 g Andy makes smoothies. Each smoothie is made using 2 peeled bananas and 20 strawberries from the supermarket, all the items being randomly chosen. The weight of a strawberry is modelled by a Normal distribution with mean 22.5 g and standard deviation 2.7 g .
  3. Find the probability that the total weight of a smoothie is less than 700 g .
OCR MEI Further Statistics Major 2019 June Q4
4 Shellfish in the sea near nuclear power stations are regularly monitored for levels of radioactivity. On a particular occasion, the levels of caesium-137 (a radioactive isotope) in a random sample of 8 cockles, measured in becquerels per kilogram, were as follows.
\(\begin{array} { l l l l l l l l } 2.36 & 2.97 & 2.69 & 3.00 & 2.51 & 2.45 & 2.21 & 2.63 \end{array}\) Software is used to produce a 95\% confidence interval for the level of caesium-137 in the cockles. The output from the software is shown in Fig. 4. The value for 'SE' has been deliberately omitted. T Estimate of a Mean
Confidence Level 0.95 Sample
Mean 2.6025
s 0.2793
□
0.2793 N □ 8 Result T Estimate of a Mean \begin{table}[h]
Mean2.6025
s0.2793
SE
N8
df7
Interval\(2.6025 \pm 0.2335\)
\captionsetup{labelformat=empty} \caption{Fig. 4}
\end{table}
  1. State an assumption necessary for the use of the \(t\) distribution in the construction of this confidence interval.
  2. State the confidence interval which the software gives in the form \(a < \mu < b\).
  3. In the software output shown in Fig. 4, SE stands for standard error. Find the standard error in this case.
  4. Show how the value of 0.2335 in the confidence interval was calculated.
  5. State how, using this sample, a wider confidence interval could be produced.
OCR MEI Further Statistics Major 2019 June Q5
5 In an investigation into the possible relationship between smoking and weight in adults in a particular country, a researcher selected a random sample of 500 adults.
The adults in the sample were classified according to smoking status (non-smoker, light smoker or heavy smoker, where light smoker indicates less than 10 cigarettes per day) and body weight (underweight, normal weight or overweight). Fig. 5 is a screenshot showing part of the spreadsheet used to calculate the contributions for a chisquared test. Some values in the spreadsheet have been deliberately omitted. \begin{table}[h]
ABCDEF
1Observed frequencies
2UnderweightNormalOverweightTotals
3Non-smoker852178238
4Light smoker104068118
5Heavy smoker54792144
6Totals23139338500
7
8Expected frequencies
9Non-smoker10.948066.1640160.8880
10Light smoker5.428079.7680
11Heavy smoker40.032097.3440
12
13
14Non-smoker0.79381.8200
15Light smoker3.85101.57851.7361
16Heavy smoker0.39821.21290.2934
17
\captionsetup{labelformat=empty} \caption{Fig. 5}
\end{table}
  1. Showing your calculations, find the missing values in each of the following cells.
    • B11
    • C10
    • C14
    • Complete the hypothesis test at the \(1 \%\) level of significance.
    • For each smoking status, give a brief interpretation of the largest of the three contributions to the test statistic.
OCR MEI Further Statistics Major 2019 June Q6
6
  1. A researcher is investigating the date of the 'start of spring' at different locations around the country.
    A suitable date (measured in days from the start of the year) can be identified by checking, for example, when buds first appear for certain species of trees and plants, but this is time-consuming and expensive. Satellite data, measuring microwave emissions, can alternatively be used to estimate the date that land-based measurements would give. The researcher chooses a random sample of 12 locations, and obtains land-based measurements for the start of spring date at each location, together with relevant satellite measurements. The scatter diagram in Fig. 6.1 shows the results; the land-based measurements are denoted by \(x\) days and the corresponding values derived from satellite measurements by \(y\) days. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3a89edc4-ac93-4691-ade8-4d4665b55202-06_732_1342_781_333} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
    \end{figure} Fig. 6.2 shows part of a spreadsheet used to analyse the data. Some rows of the spreadsheet have been deliberately omitted. \begin{table}[h]
    1ABCDEF
    1x\(\boldsymbol { y }\)\(\boldsymbol { x } ^ { \mathbf { 2 } }\)\(\boldsymbol { y } ^ { \mathbf { 2 } }\)xy
    2901028100104049180
    3
    10
    11
    129497883694099118
    13991019801102019999
    14Sum11311227107783126725116724
    15
    \captionsetup{labelformat=empty} \caption{Fig. 6.2}
    \end{table}
    1. Calculate the equation of a regression line suitable for estimating the land-based date of the start of spring from satellite measurements.
    2. Using this equation, estimate the land-based date of the start of spring for the following dates from satellite measurements.
      • 95 days
  2. 60 days
    (iii) Comment on the reliability of each of your estimates.
  3. The researcher is also investigating whether there is any correlation between the average temperature during a month in spring and the total rainfall during that month at a particular location. The average temperatures in degrees Celsius and total rainfall in mm for a random selection, over several years, of 10 spring months at this location are as follows.
    4. Temperature4.27.15.63.58.66.52.75.96.74.1
    Rainfall18264276154384536636
    The researcher plots the scatter diagram shown in Fig. 6.3 to check which type of test to carry out. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3a89edc4-ac93-4691-ade8-4d4665b55202-07_693_880_1174_338} \captionsetup{labelformat=empty} \caption{Fig. 6.3}
    \end{figure} (i) Explain why the researcher might come to the conclusion that a test based on Pearson's product moment correlation coefficient may be valid.
    (ii) Find the value of Pearson's product moment correlation coefficient.
    (iii) Carry out a test at the \(5 \%\) significance level to investigate whether there is any correlation between temperature and rainfall.
OCR MEI Further Statistics Major 2019 June Q7
7 A swimming coach believes that times recorded by people using stopwatches are on average 0.2 seconds faster than those recorded by an electronic timing system. In order to test this, the coach takes a random sample of 40 competitors' times recorded by both methods, and finds the differences between the times recorded by the two methods. The mean difference in the times (electronic time minus stopwatch time) is 0.1442 s and the standard deviation of the differences is 0.2580 s .
  1. Find a 95\% confidence interval for the mean difference between electronic and stopwatch times.
  2. Explain whether there is evidence to suggest that the coach’s belief is correct.
  3. Explain how you can calculate the confidence interval in part (a) even though you do not know the distribution of the parent population of differences.
  4. If the coach wanted to produce a \(95 \%\) confidence interval of width no more than 0.12 s , what is the minimum sample size that would be needed, assuming that the standard deviation remains the same?
OCR MEI Further Statistics Major 2019 June Q8
8 A student doing a school project wants to test a claim which she read in a newspaper that drinking a cup of tea will improve a person's arithmetic skills.
She chooses 13 students from her school and gets each of them to drink a cup of tea. She then gives each of them an arithmetic test. She knows that the average score for this test in students of the same age group as those she has chosen is 33.5.
The scores of the students she tests, arranged in ascending order, are as follows.
\(\begin{array} { l l l l l l l l l l l l l } 26 & 28 & 29 & 30 & 31 & 32 & 34 & 42 & 49 & 54 & 55 & 56 & 61 \end{array}\) The student decides to use software to draw a Normal probability plot for these data, and to carry out a Normality test as shown in Fig. 8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3a89edc4-ac93-4691-ade8-4d4665b55202-09_536_1234_792_244} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. The student uses the output from the software to help in deciding on a suitable hypothesis test to use for investigating the claim about drinking tea.
    Explain what the student should conclude.
  2. The student's teacher agrees with the student's choice of hypothesis test, but says that even this test may not be valid as there may be some unsatisfactory features in the student's project. Give three features that the teacher might identify as unsatisfactory.
  3. Assuming that the student's procedures can be justified, carry out an appropriate test at the \(5 \%\) significance level to investigate the claim about drinking tea.
OCR MEI Further Statistics Major 2019 June Q9
3 marks
9 Every weekday Jonathan takes an underground train to work. On any weekday the time in minutes that he has to wait at the station for a train is modelled by the continuous uniform distribution over \([ 0,5 ]\).
  1. Find the probability that Jonathan has to wait at least 3 minutes for a train. The total time that Jonathan has to wait on two days is modelled by the continuous random variable \(X\) with probability density function given by
    \(\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 25 } x & 0 \leqslant x \leqslant 5 ,
    \frac { 1 } { 25 } ( 10 - x ) & 5 < x \leqslant 10 ,
    0 & \text { otherwise } . \end{cases}\)
  2. Find the probability that Jonathan has to wait a total of at most 6 minutes on two days. Jonathan's friend suggests that the total waiting time for 5 days, \(T\) minutes, will almost certainly be less than 18 minutes. In order to investigate this suggestion, Jonathan constructs the simulation shown in Fig. 9. All of the numbers in the simulation have been rounded to 2 decimal places. \begin{table}[h]
    ABCDEF
    1MonTueWedThuFriTotal T
    21.784.362.743.884.6417.41
    30.951.304.834.291.8113.18
    44.274.904.571.413.6618.81
    50.800.063.201.760.356.17
    60.034.821.263.530.139.77
    73.884.731.193.751.2914.84
    84.113.544.330.774.5017.25
    93.540.113.852.861.5811.94
    101.871.823.003.531.8312.05
    114.002.984.591.731.7615.06
    121.913.852.081.722.8212.38
    130.104.862.510.522.1710.15
    141.244.260.951.331.789.57
    152.990.693.853.412.4213.36
    164.671.762.133.483.1015.14
    171.941.070.910.633.347.89
    180.112.290.714.210.868.18
    190.434.584.891.862.8414.60
    204.230.882.714.884.2016.91
    213.724.583.114.893.1819.49
    \captionsetup{labelformat=empty} \caption{Fig. 9}
    \end{table}
  3. Use the simulation to estimate \(\mathrm { P } ( T > 18 )\).
  4. Explain how Jonathan could obtain a better estimate. Jonathan thinks that he can use the Central Limit Theorem to provide a very good approximation to the distribution of \(T\).
  5. Find each of the following.
    • \(\mathrm { E } ( T )\)
    • \(\operatorname { Var } ( T )\)
    • Use the Central Limit Theorem to estimate \(\mathrm { P } ( T > 18 )\).
    • Comment briefly on the use of the Central Limit Theorem in this case.
    Jonathan travels to work on 200 days in a year.
  6. Find the probability that the total waiting time for Jonathan in a year is more than 510 minutes.
    [0pt] [3]
OCR MEI Further Statistics Major 2019 June Q10
10 The probability density function of the continuous random variable \(X\) is given by
\(f ( x ) = \begin{cases} k x ^ { m } & 0 \leqslant x \leqslant a ,
0 & \text { otherwise, } \end{cases}\)
where \(a , k\) and \(m\) are positive constants.
  1. Show that \(k = \frac { m + 1 } { a ^ { m + 1 } }\).
  2. Find the cumulative distribution function of \(X\) in terms of \(x , a\) and \(m\).
  3. Given that \(\mathrm { P } \left( \frac { 1 } { 4 } a < X < \frac { 1 } { 2 } a \right) = \frac { 1 } { 10 }\),
    1. show that \(2 p ^ { 2 } - 10 p + 5 = 0\), where \(p = 2 ^ { m }\),
    2. find the value of \(m\). \section*{END OF QUESTION PAPER}
OCR MEI Further Statistics Major 2022 June Q1
1 During a meteor shower, the number of meteors that can be seen at a particular location can be modelled by a Poisson distribution with mean 1.2 per minute.
  1. Find the probability that exactly 2 meteors are seen in a period of 1 minute.
  2. Find the probability that more than 3 meteors are seen in a period of 1 minute.
  3. Find the probability that no more than 8 meteors are seen in a period of 10 minutes.
  4. Explain what the fact that the number of meteors seen can be modelled by a Poisson distribution tells you about the occurrence of meteors.
OCR MEI Further Statistics Major 2022 June Q2
2 A manufacturer is testing how long coloured LED lights will last before the battery runs out, using two different battery types. The times in hours before the battery runs out are modelled by independent Normal distributions with means and standard deviations as shown in the table.
\cline { 2 - 3 } \multicolumn{1}{c|}{}Time
TypeMean
Standard
deviation
A232.8
B353.6
  1. In a particular test, a battery of type A is used and the time taken for it to run out is recorded. This process is repeated until a total of 5 randomly selected batteries have been used. Determine the probability that the total time the 5 batteries last is at least 120 hours.
  2. In a similar test, 3 randomly selected batteries of type A are used, one after the other. Then 2 randomly selected batteries of type B are used, one after the other. Determine the probability that the 3 type A batteries last longer in total than the 2 type B batteries.
  3. Explain why it is necessary that the Normal distributions are independent in order to be able to find the probability in part (b).
OCR MEI Further Statistics Major 2022 June Q3
3 The table shows the probability distribution of the random variable \(X\), where \(a\) and \(b\) are constants.
\(r\)01234
\(\mathrm { P } ( X = r )\)\(a\)\(b\)0.240.32\(b ^ { 2 }\)
  1. Given that \(\mathrm { E } ( X ) = 1.8\), determine the values of \(a\) and \(b\). The random variable \(Y\) is given by \(Y = 10 - 3 X\).
  2. Using the values of \(a\) and \(b\) which you found in part (a), find each of the following.
    • \(\mathrm { E } ( Y )\)
    • \(\operatorname { Var } ( Y )\)
OCR MEI Further Statistics Major 2022 June Q4
4 A pack of \(k\) cards is labelled \(1,2 , \ldots , k\). A card is drawn at random from the pack. The random variable \(X\) represents the number on the card.
  1. Given that \(k > 10\), find \(\mathrm { P } ( X \geqslant 10 )\). You are now given that \(k = 20\).
  2. A card is drawn at random from the pack and the number on it is noted. The card is then returned to the pack. This process is repeated until the second occasion on which the number noted is less than 9 . Find the probability that no more than 4 cards have to be drawn. Answer all the questions. Section B (95 marks)
OCR MEI Further Statistics Major 2022 June Q5
5 A motorist is investigating the relationship between tyre pressure and temperature. As the temperature increases during a hot day, she records the pressure (measured in bars) of one of her car tyres at specific temperatures of \(20 ^ { \circ } \mathrm { C } , 22 ^ { \circ } \mathrm { C } , \ldots , 36 ^ { \circ } \mathrm { C }\). The results are shown in Table 5.1. \begin{table}[h]
Temperature \(\left( t ^ { \circ } \mathrm { C } \right)\)202224262830323436
Tyre pressure \(( P\) bar \()\)2.0122.0362.0652.0742.1142.1402.1492.1762.192
\captionsetup{labelformat=empty} \caption{Table 5.1}
\end{table}
  1. Calculate the equation of the regression line of pressure on temperature. Give your answer in the form \(P = a t + b\), giving the values of \(a\) and \(b\) to \(\mathbf { 4 }\) significant figures.
  2. Table 5.2 shows the residuals for most of the data values. Complete the copy of the table in the Printed Answer Booklet. \begin{table}[h]
    Temperature202224262830323436
    Residual tyre
    pressure
    		- 0.003- 0.0020.004- 0.0100.011- 0.0030.001
    \captionsetup{labelformat=empty} \caption{Table 5.2}
    \end{table}
  3. With reference to the values of the residuals, comment on the goodness of fit of the regression line.
  4. Use your answer to part (a) to calculate an estimate of the pressure in the tyre at each of the following temperatures, giving your answers to \(\mathbf { 3 }\) decimal places.
    • \(25 ^ { \circ } \mathrm { C }\)
    • \(10 ^ { \circ } \mathrm { C }\)
    • Comment on the reliability of each of your estimates.
OCR MEI Further Statistics Major 2022 June Q7
7 Amir is trying to thread a needle. On each attempt the probability that he is successful is 0.3 , independently of any other attempt. The random variable \(X\) represents the number of attempts that he takes to thread the needle.
  1. Find \(\mathrm { P } ( X = 5 )\).
  2. During the course of a day, Amir has to thread 6 needles. Determine the probability that it takes him more than 3 attempts to be successful for at least 4 of the 6 needles.
  3. Amaya is also trying to thread a needle. On each attempt the probability that she is successful is \(p\), independently of any other attempt. The probability that Amaya takes 2 attempts to thread a particular needle is \(\frac { 28 } { 121 }\). Determine the possible values of \(p\).