Questions S2 (1597 questions)

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OCR MEI S2 2011 January Q2
20 marks Standard +0.3
2 A student is investigating the numbers of sultanas in a particular brand of biscuit. The data in the table show the numbers of sultanas in a random sample of 50 of these biscuits.
Number of sultanas012345\(> 5\)
Frequency815129420
  1. Show that the sample mean is 1.84 and calculate the sample variance.
  2. Explain why these results support a suggestion that a Poisson distribution may be a suitable model for the distribution of the numbers of sultanas in this brand of biscuit. For the remainder of the question you should assume that a Poisson distribution with mean 1.84 is a suitable model for the distribution of the numbers of sultanas in these biscuits.
  3. Find the probability of
    (A) no sultanas in a biscuit,
    (B) at least two sultanas in a biscuit.
  4. Show that the probability that there are at least 10 sultanas in total in a packet containing 5 biscuits is 0.4389 .
  5. Six packets each containing 5 biscuits are selected at random. Find the probability that exactly 2 of the six packets contain at least 10 sultanas.
  6. Sixty packets each containing 5 biscuits are selected at random. Use a suitable approximating distribution to find the probability that more than half of the sixty packets contain at least 10 sultanas.
OCR MEI S2 2011 January Q3
17 marks Standard +0.3
3 The random variable \(X\) represents the reaction times, in milliseconds, of men in a driving simulator. \(X\) is Normally distributed with mean 355 and standard deviation 52.
  1. Find
    (A) \(\mathrm { P } ( X < 325 )\),
    (B) \(\mathrm { P } ( 300 < X < 400 )\).
  2. Find the value of \(k\) for which \(\mathrm { P } ( X < k ) = 0.2\). It is thought that women may have a different mean reaction time from men. In order to test this, a random sample of 25 women is selected. The mean reaction time of these women in the driving simulator is 344 milliseconds. You may assume that women's reaction times are also Normally distributed with standard deviation 52 milliseconds. A hypothesis test is carried out to investigate whether women have a different mean reaction time from men.
  3. Carry out the test at the \(5 \%\) significance level.
OCR MEI S2 2011 January Q4
18 marks Standard +0.3
4 A researcher is investigating the sizes of pebbles at various locations in a river. Three sites in the river are chosen and each pebble sampled at each site is classified as large, medium or small. The results are as follows.
Site\multirow{2}{*}{
Row
totals
}
\cline { 3 - 6 } \multicolumn{2}{|c|}{}ABC
\multirow{3}{*}{
Pebble
size
}		Large15121037
\cline { 2 - 6 }Medium28174590
\cline { 2 - 6 }Small473336116
Column totals906291243
  1. Carry out a test at the \(5 \%\) significance level to examine whether there is any association between pebble size and site. Your working should include a table of the contributions of each cell to the test statistic.
  2. By referring to each site, comment briefly on how the size of the pebbles compares with what would be expected if there were no association. You should support your answers by referring to your table of contributions.
OCR MEI S2 2012 January Q1
17 marks Standard +0.3
1 Nine long-distance runners are starting an exercise programme to improve their strength. During the first session, each of them has to do a 100 metre run and to do as many push-ups as possible in one minute. The times taken for the run, together with the number of push-ups each runner achieves, are shown in the table.
RunnerABCDEFGHI
100 metre time (seconds)13.211.610.912.314.713.111.713.612.4
Push-ups achieved324222364127373833
  1. Draw a scatter diagram to illustrate the data.
  2. Calculate the value of Spearman's rank correlation coefficient.
  3. Carry out a hypothesis test at the \(5 \%\) significance level to examine whether there is any association between time taken for the run and number of push-ups achieved.
  4. Under what circumstances is it appropriate to carry out a hypothesis test based on the product moment correlation coefficient? State, with a reason, which test is more appropriate for these data.
OCR MEI S2 2012 January Q2
19 marks Moderate -0.3
2 The number of printing errors per page in a book is modelled by a Poisson distribution with a mean of 0.85 .
  1. State conditions for a Poisson distribution to be a suitable model for the number of printing errors per page.
  2. A page is chosen at random. Find the probability of
    (A) exactly 1 error on this page,
    (B) at least 2 errors on this page. 10 pages are chosen at random.
  3. Find the probability of exactly 10 errors in these 10 pages.
  4. Find the least integer \(k\) such that the probability of there being \(k\) or more errors in these 10 pages is less than \(1 \%\). 30 pages are chosen at random.
  5. Use a suitable approximating distribution to find the probability of no more than 30 errors in these 30 pages.
OCR MEI S2 2012 January Q3
19 marks Standard +0.3
3 The lifetime of a particular type of light bulb is \(X\) hours, where \(X\) is Normally distributed with mean 1100 and variance 2000.
  1. Find \(\mathrm { P } ( 1100 < X < 1200 )\).
  2. Use a suitable approximating distribution to find the probability that, in a random sample of 100 of these light bulbs, no more than 40 have a lifetime between 1100 and 1200 hours.
  3. A factory has a large number of these light bulbs installed. As soon as \(1 \%\) of the bulbs have come to the end of their lifetimes, it is company policy to replace all of the bulbs. After how many hours should the bulbs need to be replaced?
  4. The bulbs are to be replaced by low-energy bulbs. The lifetime of these bulbs is Normally distributed and the mean is claimed by the manufacturer to be 7000 hours. The standard deviation is known to be 100 hours. A random sample of 25 low-energy bulbs is selected. Their mean lifetime is found to be 6972 hours. Carry out a 2 -tail test at the \(10 \%\) level to investigate the claim.
    [0pt] [Question 4 is printed overleaf.]
OCR MEI S2 2012 January Q4
17 marks Moderate -0.3
4 Birds are observed at feeding stations in three different places - woodland, farm and garden. The numbers of finches, thrushes and tits observed at each site are summarised in the table. The birds observed are regarded as a random sample from the population of birds of these species that use these feeding stations.
\multirow{2}{*}{Observed Frequency}Place
FarmGardenWoodlandTotals
\multirow{4}{*}{Species}Thrushes1174792
Tits702688184
Finches1721029
Totals98102105305
The expected frequencies under the null hypothesis for the usual \(\chi ^ { 2 }\) test are shown in the table below.
\multirow{2}{*}{Expected Frequency}Place
FarmGardenWoodland
\multirow{3}{*}{Species}Thrushes29.560730.767231.6721
Tits59.121361.534463.3443
Finches9.31809.69849.9836
  1. Verify that the entry 9.3180 is correct. The corresponding contributions to the test statistic are shown in the table below.
    \multirow{2}{*}{Contribution}Place
    FarmGardenWoodland
    \multirow{3}{*}{Species}Thrushes11.653960.748919.2192
    Tits2.001720.52019.5969
    Finches6.33326.11080.0000
  2. Verify that the entry 6.3332 is correct.
  3. Carry out the test at the \(1 \%\) level of significance.
  4. For each place, use the table of contributions to comment briefly on the differences between the observed and expected distributions of species.
OCR MEI S2 2013 January Q1
19 marks Standard +0.3
1 A manufacturer of playground safety tiles is testing a new type of tile. Tiles of various thicknesses are tested to estimate the maximum height at which people would be unlikely to sustain injury if they fell onto a tile. The results of the test are as follows.
Thickness \(( t \mathrm {~mm} )\)20406080100
Maximum height \(( h \mathrm {~m} )\)0.721.091.621.972.34
  1. Draw a scatter diagram to illustrate these data.
  2. State which of the two variables is the independent variable, giving a reason for your answer.
  3. Calculate the equation of the regression line of maximum height on thickness.
  4. Use the equation of the regression line to calculate estimates of the maximum height for thicknesses of
    (A) 70 mm ,
    (B) 120 mm . Comment on the reliability of each of these estimates.
  5. Calculate the value of the residual for the data point at which \(t = 40\).
  6. In a further experiment, the manufacturer tests a tile with a thickness of 200 mm and finds that the corresponding maximum height is 2.96 m . What can be said about the relationship between tile thickness and maximum height?
OCR MEI S2 2013 January Q2
18 marks Standard +0.3
2 John is observing butterflies being blown across a fence in a strong wind. He uses the Poisson distribution with mean 2.1 to model the number of butterflies he observes in one minute.
  1. Find the probability that John observes
    (A) no butterflies in a minute,
    (B) at least 2 butterflies in a minute,
    (C) between 5 and 10 butterflies inclusive in a period of 5 minutes.
  2. Use a suitable approximating distribution to find the probability that John observes at least 130 butterflies in a period of 1 hour. In fact some of the butterflies John observes being blown across the fence are being blown in pairs.
  3. Explain why this invalidates one of the assumptions required for a Poisson distribution to be a suitable model. John decides to revise his model for the number of butterflies he observes in one minute. In this new model, the number of pairs of butterflies is modelled by the Poisson distribution with mean 0.2 , and the number of single butterflies is modelled by an independent Poisson distribution with mean 1.7.
  4. Find the probability that John observes no more than 3 butterflies altogether in a period of one minute.
OCR MEI S2 2013 January Q3
17 marks Moderate -0.3
3 The amount of data, \(X\) megabytes, arriving at an internet server per second during the afternoon is modelled by the Normal distribution with mean 435 and standard deviation 30.
  1. Find
    (A) \(\mathrm { P } ( X < 450 )\),
    (B) \(\mathrm { P } ( 400 < X < 450 )\).
  2. Find the probability that, during 5 randomly selected seconds, the amounts of data arriving are all between 400 and 450 megabytes. The amount of data, \(Y\) megabytes, arriving at the server during the evening is modelled by the Normal distribution with mean \(\mu\) and standard deviation \(\sigma\).
  3. Given that \(\mathrm { P } ( Y < 350 ) = 0.2\) and \(\mathrm { P } ( Y > 390 ) = 0.1\), find the values of \(\mu\) and \(\sigma\).
  4. Find values of \(a\) and \(b\) for which \(\mathrm { P } ( a < Y < b ) = 0.95\).
OCR MEI S2 2013 January Q4
18 marks Moderate -0.3
4
  1. A random sample of 60 students studying mathematics was selected. Their grades in the Core 1 module are summarised in the table below, classified according to whether they worked less than 5 hours per week or at least 5 hours per week. Test, at the \(5 \%\) significance level, whether there is any association between grade and hours worked.
    Hours worked
    \cline { 3 - 4 } \multicolumn{2}{|c|}{}Less than 5At least 5
    \multirow{2}{*}{Grade}A or B2011
    \cline { 2 - 4 }C or lower1316
  2. At a canning factory, cans are filled with tomato purée. The machine which fills the cans is set so that the volume of tomato purée in a can, measured in millilitres, is Normally distributed with mean 420 and standard deviation 3.5. After the machine is recalibrated, a quality control officer wishes to check whether the mean is still 420 millilitres. A random sample of 10 cans of tomato purée is selected and the volumes, measured in millilitres, are as follows. $$\begin{array} { l l l l l l l l l l } 417.2 & 422.6 & 414.3 & 419.6 & 420.4 & 410.0 & 418.3 & 416.9 & 418.9 & 419.7 \end{array}$$ Carry out a test at the \(1 \%\) significance level to investigate whether the mean is still 420 millilitres. You should assume that the volumes are Normally distributed with unchanged standard deviation.
OCR MEI S2 2009 June Q1
16 marks Standard +0.3
1 An investment analyst thinks that there may be correlation between the cost of oil, \(x\) dollars per barrel, and the price of a particular share, \(y\) pence. The analyst selects 50 days at random and records the values of \(x\) and \(y\). Summary statistics for these data are shown below, together with a scatter diagram. $$\Sigma x = 2331.3 \quad \Sigma y = 6724.3 \quad \Sigma x ^ { 2 } = 111984 \quad \Sigma y ^ { 2 } = 921361 \quad \Sigma x y = 316345 \quad n = 50$$ \includegraphics[max width=\textwidth, alt={}, center]{ae79cdd9-a57c-490e-a9f3-f47c7c8a1aa6-2_857_905_516_621}
  1. Calculate the sample product moment correlation coefficient.
  2. Carry out a hypothesis test at the \(5 \%\) significance level to investigate the analyst's belief. State your hypotheses clearly, defining any symbols which you use.
  3. An assumption that there is a bivariate Normal distribution is required for this test to be valid. State whether it is the sample or the population which is required to have such a distribution. State, with a reason, whether in this case the assumption appears to be justified.
  4. Explain why a 2-tail test is appropriate even though it is clear from the scatter diagram that the sample has a positive correlation coefficient.
OCR MEI S2 2009 June Q2
19 marks Moderate -0.3
2 Jess is watching a shower of meteors (shooting stars). During the shower, she sees meteors at an average rate of 1.3 per minute.
  1. State conditions required for a Poisson distribution to be a suitable model for the number of meteors which Jess sees during a randomly selected minute. You may assume that these conditions are satisfied.
  2. Find the probability that, during one minute, Jess sees
    (A) exactly one meteor,
    (B) at least 4 meteors.
  3. Find the probability that, in a period of 10 minutes, Jess sees exactly 10 meteors.
  4. Use a suitable approximating distribution to find the probability that Jess sees a total of at least 100 meteors during a period of one hour.
  5. Jess watches the shower for \(t\) minutes. She wishes to be at least \(99 \%\) certain that she will see one or more meteors. Find the smallest possible integer value of \(t\).
OCR MEI S2 2009 June Q3
20 marks Standard +0.3
3 Intensity of light is measured in lumens. The random variable \(X\) represents the intensity of the light from a standard 100 watt light bulb. \(X\) is Normally distributed with mean 1720 and standard deviation 90. You may assume that the intensities for different bulbs are independent.
  1. Show that \(\mathrm { P } ( X < 1700 ) = 0.4121\).
  2. These bulbs are sold in packs of 4 . Find the probability that the intensities of exactly 2 of the 4 bulbs in a randomly chosen pack are below 1700 lumens.
  3. Use a suitable approximating distribution to find the probability that the intensities of at least 20 out of 40 randomly selected bulbs are below 1700 lumens. A manufacturer claims that the average intensity of its 25 watt low energy light bulbs is 1720 lumens. A consumer organisation suspects that the true figure may be lower than this. The intensities of a random sample of 20 of these bulbs are measured. A hypothesis test is then carried out to check the claim.
  4. Write down a suitable null hypothesis and explain briefly why the alternative hypothesis should be \(\mathrm { H } _ { 1 } : \mu < 1720\). State the meaning of \(\mu\).
  5. Given that the standard deviation of the intensity of such bulbs is 90 lumens and that the mean intensity of the sample of 20 bulbs is 1703 lumens, carry out the test at the \(5 \%\) significance level.
OCR MEI S2 2009 June Q4
17 marks Standard +0.3
4 In a traffic survey a random sample of 400 cars passing a particular location during the rush hour is selected. The type of car and the sex of the driver are classified as follows.
\multirow{2}{*}{}Sex\multirow{2}{*}{Row totals}
MaleFemale
\multirow{5}{*}{Type of car}Hatchback9636132
Saloon7735112
People carrier384482
4WD19827
Sports car222547
Column totals252148400
  1. Carry out a test at the \(5 \%\) significance level to examine whether there is any association between type of car and sex of driver. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.
  2. For each type of car, comment briefly on how the number of drivers of each sex compares with what would be expected if there were no association. OCR is committed to seeking permission to reproduce all third-party content that it uses in its 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, is given to all schools that receive assessment material 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.
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OCR MEI S2 2010 June Q1
16 marks Standard +0.3
1 Two celebrities judge a talent contest. Each celebrity gives a score out of 20 to each of a random sample of 8 contestants. The scores, \(x\) and \(y\), given by the celebrities to each contestant are shown below.
ContestantABCDEFGH
\(x\)61792013151114
\(y\)6131011971215
  1. Calculate the value of Spearman's rank correlation coefficient.
  2. Carry out a hypothesis test at the \(5 \%\) significance level to determine whether there is positive association between the scores allocated by the two celebrities.
  3. State the distributional assumption required for a test based on the product moment correlation coefficient. Sketch a scatter diagram of the scores above, and discuss whether it appears that the assumption is likely to be valid.
OCR MEI S2 2010 June Q2
19 marks Moderate -0.8
2 A radioactive source is decaying at a mean rate of 3.4 counts per 5 seconds.
  1. State conditions for a Poisson distribution to be a suitable model for the rate of decay of the source. You may assume that a Poisson distribution with a mean rate of 3.4 counts per 5 seconds is a suitable model.
  2. State the variance of this Poisson distribution.
  3. Find the probability of
    (A) exactly 3 counts in a 5 -second period,
    (B) at least 3 counts in a 5 -second period.
  4. Find the probability of exactly 40 counts in a period of 60 seconds.
  5. Use a suitable approximating distribution to find the probability of at least 40 counts in a period of 60 seconds.
  6. The background radiation rate also, independently, follows a Poisson distribution and produces a mean count of 1.4 per 5 seconds. Find the probability that the radiation source together with the background radiation give a total count of at least 8 in a 5 -second period.
OCR MEI S2 2010 June Q3
19 marks Standard +0.3
3 In a men's cycling time trial, the times are modelled by the random variable \(X\) minutes which is Normally distributed with mean 63 and standard deviation 5.2.
  1. Find $$\begin{aligned} & \text { (A) } \mathrm { P } ( X < 65 ) \text {, } \\ & \text { (B) } \mathrm { P } ( 60 < X < 65 ) \text {. } \end{aligned}$$
  2. Find the probability that 5 riders selected at random all record times between 60 and 65 minutes.
  3. A competitor aims to be in the fastest \(5 \%\) of entrants (i.e. those with the lowest times). Find the maximum time that he can take. It is suggested that holding the time trial on a new course may result in lower times. To investigate this, a random sample of 15 competitors is selected. These 15 competitors do the time trial on the new course. The mean time taken by these riders is 61.7 minutes. You may assume that times are Normally distributed and the standard deviation is still 5.2 minutes. A hypothesis test is carried out to investigate whether times on the new course are lower.
  4. Write down suitable null and alternative hypotheses for the test. Carry out the test at the 5\% significance level.
OCR MEI S2 2010 June Q4
18 marks Standard +0.3
4 In a survey a random sample of 63 runners is selected. The category of runner and the type of running are classified as follows.
\multirow{2}{*}{}Category of runner\multirow{2}{*}{Row totals}
JuniorSeniorVeteran
\multirow{3}{*}{Type of running}Track98219
Road481224
Both410620
Column totals17262063
  1. Carry out a test at the \(5 \%\) significance level to examine whether there is any association between category of runner and the type of running. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.
  2. For each category of runner, comment briefly on how the type of running compares with what would be expected if there were no association.
OCR MEI S2 2011 June Q1
18 marks Easy -1.2
1 An experiment is performed to determine the response of maize to nitrogen fertilizer. Data for the amount of nitrogen fertilizer applied, \(x \mathrm {~kg} / \mathrm { hectare }\), and the average yield of maize, \(y\) tonnes/hectare, in 5 experimental plots are given in the table below.
\(x\)0306090120
\(y\)0.52.54.76.27.4
  1. Draw a scatter diagram to illustrate these data.
  2. Calculate the equation of the regression line of \(y\) on \(x\).
  3. Draw your regression line on your scatter diagram and comment briefly on its fit.
  4. Calculate the value of the residual for the data point where \(x = 30\) and \(y = 2.5\).
  5. Use the equation of the regression line to calculate estimates of average yield with nitrogen fertilizer applications of
    (A) \(45 \mathrm {~kg} / \mathrm { hectare }\),
    (B) \(150 \mathrm {~kg} /\) hectare.
  6. In a plot where \(150 \mathrm {~kg} /\) hectare of nitrogen fertilizer is applied, the average yield of maize is 8.7 tonnes/hectare. Comment on this result.
OCR MEI S2 2011 June Q2
16 marks Moderate -0.3
2 At a drive-through fast food takeaway, cars arrive independently, randomly and at a uniform average rate. The numbers of cars arriving per minute may be modelled by a Poisson distribution with mean 0.62.
  1. Briefly explain the meaning of each of the three terms 'independently', 'randomly' and 'at a uniform average rate', in the context of cars arriving at a fast food takeaway.
  2. Find the probability of at most 1 car arriving in a period of 1 minute.
  3. Find the probability of more than 5 cars arriving in a period of 10 minutes.
  4. State the exact distribution of the number of cars arriving in a period of 1 hour.
  5. Use a suitable approximating distribution to find the probability that at least 40 cars arrive in a period of 1 hour.
OCR MEI S2 2011 June Q3
20 marks Standard +0.3
3 The weights of Braeburn apples on display in a supermarket, measured in grams, are Normally distributed with mean 210.5 and standard deviation 15.2.
  1. Find the probability that a randomly selected apple weighs at least 220 grams.
  2. These apples are sold in packs of 3. You may assume that the weights of apples in each pack are independent. Find the probability that all 3 of the apples in a randomly selected pack weigh at least 220 grams.
  3. 100 packs are selected at random.
    (A) State the exact distribution of the number of these 100 packs in which all 3 apples weigh at least 220 grams.
    (B) Use a suitable approximating distribution to find the probability that in at most one of these packs all 3 apples weigh at least 220 grams.
    (C) Explain why this approximating distribution is suitable.
  4. The supermarket also sells Cox's Orange Pippin apples. The weights of these apples, measured in grams, are Normally distributed with mean 185 and standard deviation \(\sigma\).
    (A) Given that \(10 \%\) of randomly selected Cox's Orange Pippin apples weigh less than 170 grams, calculate the value of \(\sigma\).
    (B) Sketch the distributions of the weights of both types of apple on a single diagram.
OCR MEI S2 2011 June Q4
18 marks Standard +0.3
4
  1. In a survey on internet usage, a random sample of 200 people is selected. The people are asked how much they have spent on internet shopping during the last three months. The results, classified by amount spent and sex, are shown in the table.
    \multirow{2}{*}{}Sex\multirow{2}{*}{Row totals}
    MaleFemale
    \multirow{5}{*}{Amount spent}Nothing283462
    Less than £50172138
    £50 up to £200222648
    £200 up to £1000231639
    £1000 or more8513
    Column totals98102200
    1. Write down null and alternative hypotheses for a test to examine whether there is any association between amount spent and sex of person. The contributions to the test statistic for the usual \(\chi ^ { 2 }\) test are shown in the table below.
      \multirow{2}{*}{}Sex
      MaleFemale
      \multirow{5}{*}{Amount spent}Nothing0.18650.1791
      Less than £500.14090.1354
      £50 up to £2000.09820.0944
      £200 up to £10000.79180.7608
      £1000 or more0.41710.4007
      The sum of these contributions, correct to 3 decimal places, is 3.205.
    2. Calculate the expected frequency for females spending nothing. Verify the corresponding contribution, 0.1791 , to the test statistic.
    3. Carry out the test at the \(5 \%\) level of significance, stating your conclusion clearly.
  2. A bakery sells loaves specified as having a mean weight of 400 grams. It is known that the weights of these loaves are Normally distributed and that the standard deviation is 5.7 grams. An inspector suspects that the true mean weight may be less than 400 grams. In order to test this, the inspector takes a random sample of 6 loaves. Carry out a suitable test at the \(5 \%\) level, given that the weights, in grams, of the 6 loaves are as follows.
    \(\begin{array} { l l l l l l } 392.1 & 405.8 & 401.3 & 387.4 & 391.8 & 400.6 \end{array}\) RECOGNISING ACHIEVEMENT
OCR MEI S2 2012 June Q1
19 marks Standard +0.3
1 The times, in seconds, taken by ten randomly selected competitors for the first and last sections of an Olympic bobsleigh run are denoted by \(x\) and \(y\) respectively. Summary statistics for these data are as follows. $$\Sigma x = 113.69 \quad \Sigma y = 52.81 \quad \Sigma x ^ { 2 } = 1292.56 \quad \Sigma y ^ { 2 } = 278.91 \quad \Sigma x y = 600.41 \quad n = 10$$
  1. Calculate the sample product moment correlation coefficient.
  2. Carry out a hypothesis test at the \(10 \%\) significance level to investigate whether there is any correlation between times taken for the first and last sections of the bobsleigh run.
  3. State the distributional assumption which is necessary for this test to be valid. Explain briefly how a scatter diagram may be used to check whether this assumption is likely to be valid.
  4. A commentator says that in order to have a fast time on the last section, you must have a fast time on the first section. Comment briefly on this suggestion.
  5. (A) Would your conclusion in part (ii) have been different if you had carried out the hypothesis test at the \(1 \%\) level rather than the \(10 \%\) level? Explain your answer.
    (B) State one advantage and one disadvantage of using a \(1 \%\) significance level rather than a \(10 \%\) significance level in a hypothesis test.
OCR MEI S2 2012 June Q2
18 marks Moderate -0.8
2 A particular genetic mutation occurs in one in every 300 births on average. A random sample of 1200 births is selected.
  1. State the exact distribution of \(X\), the number of births in the sample which have the mutation.
  2. Explain why \(X\) has, approximately, a Poisson distribution.
  3. Use a Poisson approximating distribution to find
    (A) \(\mathrm { P } ( X = 1 )\),
    (B) \(\mathrm { P } ( X > 4 )\).
  4. Twenty independent samples, each of 1200 births, are selected. State the mean and variance of a Normal approximating distribution suitable for modelling the total number of births with the mutation in the twenty samples.
  5. Use this Normal approximating distribution to
    (A) find the probability that there are at least 90 births which have the mutation,
    ( \(B\) ) find the least value of \(k\) such that the probability that there are at most \(k\) births with this mutation is greater than 5\%.