Questions — OCR MEI Further Statistics Minor (42 questions)

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OCR MEI Further Statistics Minor 2020 November Q1
1 A quiz team of 4 students is to be selected from a group of 7 girls and 5 boys. The team is selected at random from the students in the group. The number of girls in the team is denoted by the random variable \(X\).
  1. Show that \(\mathrm { P } ( X = 4 ) = \frac { 7 } { 99 }\). Table 1 shows the probability distribution of \(X\). \begin{table}[h]
    \(r\)01234
    \(\mathrm { P } ( X = r )\)\(\frac { 1 } { 99 }\)\(\frac { 14 } { 99 }\)\(\frac { 42 } { 99 }\)\(\frac { 35 } { 99 }\)\(\frac { 7 } { 99 }\)
    \captionsetup{labelformat=empty} \caption{Table 1}
    \end{table}
  2. Find each of the following.
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\)
    It is decided that the quiz team must have at least 1 girl and at least 1 boy, but the team is still otherwise selected at random.
  3. Explain whether \(\mathrm { E } ( X )\) would be smaller than, equal to or larger than the value which you found in part (b).
OCR MEI Further Statistics Minor 2020 November Q2
2 On computer monitor screens there are often one or more tiny dots which are permanently dark and do not display any of the image. Such dots are known as 'dead pixels'. Dead pixels occur on screens randomly and independently of each other. A company manufactures three types of monitor, Types A, B and C. For a monitor of Type A, the screen has a total of 2304000 pixels. For this type of monitor, the probability of a randomly chosen pixel being dead is 1 in 500000 . Let \(X\) represent the number of dead pixels on a monitor screen of this type.
  1. Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of \(X\).
  2. Use a Poisson distribution to calculate estimates of each of the following probabilities.
    • \(\mathrm { P } ( X = 4 )\)
    • \(\mathrm { P } ( X > 4 )\)
    • In this question you must show detailed reasoning.
    For a monitor of Type B, the probability of a randomly chosen pixel being dead is also 1 in 500 000. The screen of a monitor of Type B has a total of \(n\) pixels. Use a binomial distribution to find the least value of \(n\) for which the probability of finding at least 1 dead pixel is greater than 0.99 . Give your answer in millions correct to 3 significant figures. For a monitor of Type C, the number of dead pixels on the screen is modelled by a Poisson distribution with mean \(\lambda\).
  3. Given that the probability of finding at least one dead pixel is 0.8 , find \(\lambda\).
OCR MEI Further Statistics Minor 2020 November Q4
4 Cards are drawn at random from a standard pack of 52 cards, one at a time, until one of the 4 aces is drawn. After each card is drawn, it is replaced in the pack before the next one is drawn. The random variable \(X\) represents the number of draws required to draw the first ace.
  1. State fully the distribution of \(X\).
  2. Find \(\mathrm { P } ( X = 10 )\).
  3. Find each of the following.
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\)
    A further \(k\) aces are added to the full pack and the process described above is repeated. The random variable \(Y\) represents the number of draws required to draw the first ace.
  4. In this question you must show detailed reasoning. Given that \(\mathrm { P } ( Y = 2 ) = \frac { 8 } { 81 }\), find the two possible values of \(k\).
OCR MEI Further Statistics Minor 2020 November Q5
5 A student is investigating immunisation. He wonders if there is any relationship between the percentage of young children who have been given measles vaccine and the percentage who have been given BCG vaccine in various countries. He takes a random sample of 8 countries and finds the data for the two variables. The spreadsheet in Fig. 5.1 shows the values obtained, together with a scatter diagram which illustrates the data. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{882f9f3c-40d8-4abb-822a-49bd505a33ea-5_910_1653_541_246} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
\end{figure}
  1. The student decides that a test based on Pearson's product moment correlation coefficient is not valid. Explain why he comes to this conclusion. The student carries out a test based on Spearman’s rank correlation coefficient.
  2. Calculate the value of Spearman’s rank correlation coefficient.
  3. Carry out a test based on this coefficient at the \(5 \%\) significance level to investigate whether there is any association between measles and BCG vaccination levels. The student then decides to investigate the relationship between number of doctors per 1000 people in a country and unemployment rate in that country (unemployment rate is the percentage of the working age population who are not in paid work). He selects a random sample of 6 countries. The spreadsheet in Fig. 5.2 shows the values obtained, together with a scatter diagram which illustrates the data. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{882f9f3c-40d8-4abb-822a-49bd505a33ea-6_776_1649_495_248} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
    \end{figure}
  4. Use your calculator to write down the equation of the regression line of unemployment rate on doctors per 1000.
  5. Use the regression line to estimate the unemployment rate for a country with 2.00 doctors per 1000.
  6. Comment briefly on the reliability of your answer to part (e). The student decides to add the data for another country with 3.99 doctors per 1000 and unemployment rate 11.42 to his diagram.
  7. Add this point to the scatter diagram in the Printed Answer Booklet.
  8. Without doing any further calculations, comment on what difference, if any, including this extra data point would make to the usefulness of a regression line of unemployment rate on doctors per 1000.
OCR MEI Further Statistics Minor 2020 November Q6
6
  1. The random variable \(X\) has a uniform distribution over the values \(\{ 1,2 , \ldots , n \}\). Show that \(\operatorname { Var } ( X )\) is given by \(\frac { 1 } { 12 } \left( n ^ { 2 } - 1 \right)\).
  2. The random variable \(Y\) has a uniform distribution over the values \(\{ 1,3,5 , \ldots , 2 n - 1 \}\). Using the result in part (a) or otherwise, show that \(\operatorname { Var } ( Y )\) is given by \(\frac { 1 } { 3 } \left( n ^ { 2 } - 1 \right)\).
  3. Given that \(n = 100\), find the least value of \(k\) for which \(\mathrm { P } ( \mu - k \sigma \leqslant Y \leqslant \mu + k \sigma ) = 1\), where the mean and standard deviation of \(Y\) are represented by \(\mu\) and \(\sigma\) respectively.
OCR MEI Further Statistics Minor 2021 November Q1
1 The probability distribution of a discrete random variable \(X\) is given by the formula \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { k } \left( ( \mathrm { r } - 1 ) ^ { 2 } + 1 \right)\) for \(r = 1,2,3,4,5\).
  1. Show that \(k = \frac { 1 } { 35 }\). The distribution of \(X\) is shown in the table.
    \(r\)12345
    \(\mathrm { P } ( \mathrm { X } = \mathrm { r } )\)\(\frac { 1 } { 35 }\)\(\frac { 2 } { 35 }\)\(\frac { 1 } { 7 }\)\(\frac { 2 } { 7 }\)\(\frac { 17 } { 35 }\)
  2. Comment briefly on the shape of the distribution.
  3. Find each of the following.
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\)
    The random variable \(Y\) is given by \(Y = 5 X - 10\).
  4. Find each of the following.
    • \(\mathrm { E } ( Y )\)
    • \(\operatorname { Var } ( Y )\)
OCR MEI Further Statistics Minor 2021 November Q2
2 A road transport researcher is investigating the link between the age of a person, a years, and the distance, \(d\) metres, at which the person can read a large road sign. The researcher selects 13 individuals of different ages between 20 and 80 and measures the value of \(d\) for each of them. The spreadsheet below shows the data which the researcher obtained, together with a scatter diagram which illustrates the data.
\includegraphics[max width=\textwidth, alt={}, center]{691e8b55-e9a1-4fff-b9ee-a71ff1f73ead-3_725_1566_495_251}
  1. Explain which of the two variables \(a\) and \(d\) is the independent variable.
  2. Find the equation of the regression line of \(d\) on \(a\).
  3. Use the regression line to predict the average distance at which a 60-year-old person can read the road sign.
  4. Explain why it might not be sensible to use the regression line to predict the average distance at which a 5 -year-old child can read the road sign.
  5. Determine the value of the residual for \(a = 40\).
  6. Explain why it would not be useful to find the equation of the regression line of \(a\) on \(d\).
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 Minor 2020 November Q3
3 In this question you must show detailed reasoning. In a survey into pet ownership, one of the questions was 'Do you own either a cat or a dog (or both)?’. A total of 121 people took part in the survey and you should assume that they form a random sample of people in a particular town. The results, classified by the age of the person being surveyed, are shown in Table 3. \begin{table}[h]
\multirow{2}{*}{}Ownership of cat or dog
Does ownDoes not own
\multirow{2}{*}{Age}Over 45 years3829
Under 45 years2331
\captionsetup{labelformat=empty} \caption{Table 3}
\end{table} Carry out a test at the 10\% significance level to investigate whether, for people in this town, there is any association between age and ownership of a cat or dog.