Questions — OCR MEI S3 (73 questions)

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OCR MEI S3 2007 January Q1
18 marks Standard +0.3
1 The continuous random variable \(X\) has probability density function $$f ( x ) = k ( 1 - x ) \quad \text { for } 0 \leqslant x \leqslant 1$$ where \(k\) is a constant.
  1. Show that \(k = 2\). Sketch the graph of the probability density function.
  2. Find \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = \frac { 1 } { 18 }\).
  3. Derive the cumulative distribution function of \(X\). Hence find the probability that \(X\) is greater than the mean.
  4. Verify that the median of \(X\) is \(1 - \frac { 1 } { \sqrt { 2 } }\).
  5. \(\bar { X }\) is the mean of a random sample of 100 observations of \(X\). Write down the approximate distribution of \(\bar { X }\).
OCR MEI S3 2007 January Q2
18 marks Standard +0.3
2 The manager of a large country estate is preparing to plant an area of woodland. He orders a large number of saplings (young trees) from a nursery. He selects a random sample of 12 of the saplings and measures their heights, which are as follows (in metres). $$\begin{array} { l l l l l l l l l l l l } 0.63 & 0.62 & 0.58 & 0.56 & 0.59 & 0.62 & 0.64 & 0.58 & 0.55 & 0.61 & 0.56 & 0.52 \end{array}$$
  1. The manager requires that the mean height of saplings at planting is at least 0.6 metres. Carry out the usual \(t\) test to examine this, using a \(5 \%\) significance level. State your hypotheses and conclusion carefully. What assumption is needed for the test to be valid?
  2. Find a \(95 \%\) confidence interval for the true mean height of saplings. Explain carefully what is meant by a \(95 \%\) confidence interval.
  3. Suppose the assumption needed in part (i) cannot be justified. Identify an alternative test that the manager could carry out in order to check that the saplings meet his requirements, and state the null hypothesis for this test.
OCR MEI S3 2007 January Q3
18 marks Standard +0.3
3 Bill and Ben run their own gardening company. At regular intervals throughout the summer they come to work on my garden, mowing the lawns, hoeing the flower beds and pruning the bushes. From past experience it is known that the times, in minutes, spent on these tasks can be modelled by independent Normally distributed random variables as follows.
MeanStandard deviation
Mowing444.8
Hoeing322.6
Pruning213.7
  1. Find the probability that, on a randomly chosen visit, it takes less than 50 minutes to mow the lawns.
  2. Find the probability that, on a randomly chosen visit, the total time for hoeing and pruning is less than 50 minutes.
  3. If Bill mows the lawns while Ben does the hoeing and pruning, find the probability that, on a randomly chosen visit, Ben finishes first. Bill and Ben do my gardening twice a month and send me an invoice at the end of the month.
  4. Write down the mean and variance of the total time (in minutes) they spend on mowing, hoeing and pruning per month.
  5. The company charges for the total time spent at 15 pence per minute. There is also a fixed charge of \(\pounds 10\) per month. Find the probability that the total charge for a month does not exceed \(\pounds 40\).
OCR MEI S3 2007 January Q4
18 marks Standard +0.3
4
  1. An amateur weather forecaster has been keeping records of air pressure, measured in atmospheres. She takes the measurement at the same time every day using a barometer situated in her garden. A random sample of 100 of her observations is summarised in the table below. The corresponding expected frequencies for a Normal distribution, with its two parameters estimated by sample statistics, are also shown in the table.
    Pressure ( \(a\) atmospheres)Observed frequencyFrequency as given by Normal model
    \(a \leqslant 0.98\)41.45
    \(0.98 < a \leqslant 0.99\)65.23
    \(0.99 < a \leqslant 1.00\)913.98
    \(1.00 < a \leqslant 1.01\)1523.91
    \(1.01 < a \leqslant 1.02\)3726.15
    \(1.02 < a \leqslant 1.03\)2118.29
    \(1.03 < a\)810.99
    Carry out a test at the \(5 \%\) level of significance of the goodness of fit of the Normal model. State your conclusion carefully and comment on your findings.
  2. The forecaster buys a new digital barometer that can be linked to her computer for easier recording of observations. She decides that she wishes to compare the readings of the new barometer with those of the old one. For a random sample of 10 days, the readings (in atmospheres) of the two barometers are shown below.
    DayABCDEFGHIJ
    Old0.9921.0051.0011.0111.0260.9801.0201.0251.0421.009
    New0.9851.0031.0021.0141.0220.9881.0301.0161.0471.025
    Use an appropriate Wilcoxon test to examine at the \(10 \%\) level of significance whether there is any reason to suppose that, on the whole, readings on the old and new barometers do not agree.
OCR MEI S3 2006 June Q1
18 marks Standard +0.3
1 Design engineers are simulating the load on a particular part of a complex structure. They intend that the simulated load, measured in a convenient unit, should be given by the random variable \(X\) having probability density function $$f ( x ) = 12 x ^ { 3 } - 24 x ^ { 2 } + 12 x , \quad 0 \leqslant x \leqslant 1 .$$
  1. Find the mean and the mode of \(X\).
  2. Find the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\). $$\text { Verify that } \mathrm { F } \left( \frac { 1 } { 4 } \right) = \frac { 67 } { 256 } , \mathrm {~F} \left( \frac { 1 } { 2 } \right) = \frac { 11 } { 16 } \text { and } \mathrm { F } \left( \frac { 3 } { 4 } \right) = \frac { 243 } { 256 } .$$ The engineers suspect that the process for generating simulated loads might not be working as intended. To investigate this, they generate a random sample of 512 loads. These are recorded in a frequency distribution as follows.
    Load \(x\)\(0 \leqslant x \leqslant \frac { 1 } { 4 }\)\(\frac { 1 } { 4 } < x \leqslant \frac { 1 } { 2 }\)\(\frac { 1 } { 2 } < x \leqslant \frac { 3 } { 4 }\)\(\frac { 3 } { 4 } < x \leqslant 1\)
    Frequency12620913146
  3. Use a suitable statistical procedure to assess the goodness of fit of \(X\) to these data. Discuss your conclusions briefly.
OCR MEI S3 2006 June Q2
18 marks Standard +0.3
2 A bus route runs from the centre of town A through the town's urban area to a point B on its boundary and then through the country to a small town C . Because of traffic congestion and general road conditions, delays occur on both the urban and the country sections. All delays may be considered independent. The scheduled time for the journey from A to B is 24 minutes. In fact, journey times over this section are given by the Normally distributed random variable \(X\) with mean 26 minutes and standard deviation 3 minutes. The scheduled time for the journey from B to C is 18 minutes. In fact, journey times over this section are given by the Normally distributed random variable \(Y\) with mean 15 minutes and standard deviation 2 minutes. Journey times on the two sections of route may be considered independent. The timetable published to the public does not show details of times at intermediate points; thus, if a bus is running early, it merely continues on its journey and is not required to wait.
  1. Find the probability that a journey from A to B is completed in less than the scheduled time of 24 minutes.
  2. Find the probability that a journey from A to C is completed in less than the scheduled time of 42 minutes.
  3. It is proposed to introduce a system of bus lanes in the urban area. It is believed that this would mean that the journey time from A to B would be given by the random variable \(0.85 X\). Assuming this to be the case, find the probability that a journey from A to B would be completed in less than the currently scheduled time of 24 minutes.
  4. An alternative proposal is to introduce an express service. This would leave out some bus stops on both sections of the route and its overall journey time from A to C would be given by the random variable \(0.9 X + 0.8 Y\). The scheduled time from A to C is to be given as a whole number of minutes. Find the least possible scheduled time such that, with probability 0.75 , buses would complete the journey on time or early.
  5. A programme of minor road improvements is undertaken on the country section. After their completion, it is thought that the random variable giving the journey time from B to C is still Normally distributed with standard deviation 2 minutes. A random sample of 15 journeys is found to have a sample mean journey time from B to C of 13.4 minutes. Provide a two-sided \(95 \%\) confidence interval for the population mean journey time from B to C .
OCR MEI S3 2006 June Q3
18 marks Moderate -0.3
3 An employer has commissioned an opinion polling organisation to undertake a survey of the attitudes of staff to proposed changes in the pension scheme. The staff are categorised as management, professional and administrative, and it is thought that there might be considerable differences of opinion between the categories. There are 60,140 and 300 staff respectively in the categories. The budget for the survey allows for a sample of 40 members of staff to be selected for in-depth interviews.
  1. Explain why it would be unwise to select a simple random sample from all the staff.
  2. Discuss whether it would be sensible to consider systematic sampling.
  3. What are the advantages of stratified sampling in this situation?
  4. State the sample sizes in each category if stratified sampling with as nearly as possible proportional allocation is used. The opinion polling organisation needs to estimate the average wealth of staff in the categories, in terms of property, savings, investments and so on. In a random sample of 11 professional staff, the sample mean is \(\pounds 345818\) and the sample standard deviation is \(\pounds 69241\).
  5. Assuming the underlying population is Normally distributed, test at the \(5 \%\) level of significance the null hypothesis that the population mean is \(\pounds 300000\) against the alternative hypothesis that it is greater than \(\pounds 300000\). Provide also a two-sided \(95 \%\) confidence interval for the population mean.
    [0pt] [10]
OCR MEI S3 2006 June Q4
18 marks Moderate -0.3
4 A company has many factories. It is concerned about incidents of trespassing and, in the hope of reducing if not eliminating these, has embarked on a programme of installing new fencing.
  1. Records for a random sample of 9 factories of the numbers of trespass incidents in typical weeks before and after installation of the new fencing are as follows.
    FactoryABCDEFGHI
    Number before installation81264142241314
    Number after installation6110118101154
    Use a Wilcoxon test to examine at the \(5 \%\) level of significance whether it appears that, on the whole, the number of trespass incidents per week is lower after the installation of the new fencing than before.
  2. Records are also available of the costs of damage from typical trespass incidents before and after the introduction of the new fencing for a random sample of 7 factories, as follows (in £).
    FactoryTUVWXYZ
    Cost before installation1215955464672356236550
    Cost after installation12681105784802417318620
    Stating carefully the required distributional assumption, provide a two-sided \(99 \%\) confidence interval based on a \(t\) distribution for the population mean difference between costs of damage before and after installation of the new fencing. Explain why this confidence interval should not be based on the Normal distribution.
OCR MEI S3 2007 June Q1
18 marks Standard +0.3
1 A manufacturer of fireworks is investigating the lengths of time for which the fireworks burn. For a particular type of firework this length of time, in minutes, is modelled by the random variable \(T\) with probability density function $$\mathrm { f } ( t ) = k t ^ { 3 } ( 2 - t ) \quad \text { for } 0 < t \leqslant 2$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 5 } { 8 }\).
  2. Find the modal time.
  3. Find \(\mathrm { E } ( T )\) and show that \(\operatorname { Var } ( T ) = \frac { 8 } { 63 }\).
  4. A large random sample of \(n\) fireworks of this type is tested. Write down in terms of \(n\) the approximate distribution of \(\bar { T }\), the sample mean time.
  5. For a random sample of 100 such fireworks the times are summarised as follows. $$\Sigma t = 145.2 \quad \Sigma t ^ { 2 } = 223.41$$ Find a 95\% confidence interval for the mean time for this type of firework and hence comment on the appropriateness of the model.
OCR MEI S3 2007 June Q2
18 marks Standard +0.3
2 The operator of a section of motorway toll road records its weekly takings according to the types of vehicles using the motorway. For purposes of charging, there are three types of vehicle: cars, coaches, lorries. The weekly takings (in thousands of pounds) for each type are assumed to be Normally distributed. These distributions are independent of each other and are summarised in the table.
Vehicle typeMeanStandard deviation
Cars60.25.2
Coaches33.96.3
Lorries52.44.9
  1. Find the probability that the weekly takings for coaches are less than \(\pounds 40000\).
  2. Find the probability that the weekly takings for lorries exceed the weekly takings for cars.
  3. Find the probability that over a 4 -week period the total takings for cars exceed \(\pounds 225000\). What assumption must be made about the four weeks?
  4. Each week the operator allocates part of the takings for repairs. This is determined for each type of vehicle according to estimates of the long-term damage caused. It is calculated as follows: \(5 \%\) of takings for cars, \(10 \%\) for coaches and \(20 \%\) for lorries. Find the probability that in any given week the total amount allocated for repairs will exceed \(\pounds 20000\).
OCR MEI S3 2007 June Q3
18 marks Standard +0.3
3 The management of a large chain of shops aims to reduce the level of absenteeism among its workforce by means of an incentive bonus scheme. In order to evaluate the effectiveness of the scheme, the management measures the percentage of working days lost before and after its introduction for each of a random sample of 11 shops. The results are shown below.
ShopABCDEFGHIJK
\% days lost before3.55.03.53.24.54.94.16.06.88.16.0
\% days lost after1.84.32.94.54.45.83.56.76.45.45.1
  1. The management decides to carry out a \(t\) test to investigate whether there has been a reduction in absenteeism.
    1. State clearly the hypotheses that should be used together with any necessary assumptions.
    2. Carry out the test using a \(5 \%\) significance level.
  2. Find a 95\% confidence interval for the true mean percentage of days lost after the introduction of the incentive scheme and state any assumption needed. The management has set a target that the mean percentage should be 3.5. Do you think this has been achieved? Explain your answer.
OCR MEI S3 2007 June Q4
18 marks Standard +0.3
4 A machine produces plastic strip in a continuous process. Occasionally there is a flaw at some point along the strip. The length of strip (in hundreds of metres) between successive flaws is modelled by a continuous random variable \(X\) with probability density function \(\mathrm { f } ( x ) = \frac { 18 } { ( 3 + x ) ^ { 3 } }\) for \(x > 0\). The table below gives the frequencies for 100 randomly chosen observations of \(X\). It also gives the probabilities for the class intervals using the model.
Length \(x\) (hundreds of metres)Observed frequencyProbability
\(0 < x \leqslant 0.5\)210.2653
\(0.5 < x \leqslant 1\)240.1722
\(1 < x \leqslant 2\)120.2025
\(2 < x \leqslant 3\)150.1100
\(3 < x \leqslant 5\)130.1094
\(5 < x \leqslant 10\)90.0874
\(x > 10\)60.0532
  1. Examine the fit of this model to the data at the \(5 \%\) level of significance. You are given that the median length between successive flaws is 124 metres. At a later date the following random sample of ten lengths (in metres) between flaws is obtained. $$\begin{array} { l l l l l l l l l l } 239 & 77 & 179 & 221 & 100 & 312 & 52 & 129 & 236 & 42 \end{array}$$
  2. Test at the \(10 \%\) level of significance whether the median length may still be assumed to be 124 metres.
OCR MEI S3 2009 January Q1
18 marks Standard +0.3
1
  1. A continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \lambda x ^ { c } , \quad 0 \leqslant x \leqslant 1 ,$$ where \(c\) is a constant and the parameter \(\lambda\) is greater than 1 .
    1. Find \(c\) in terms of \(\lambda\).
    2. Find \(\mathrm { E } ( X )\) in terms of \(\lambda\).
    3. Show that \(\operatorname { Var } ( X ) = \frac { \lambda } { ( \lambda + 2 ) ( \lambda + 1 ) ^ { 2 } }\).
  2. Every day, Godfrey does a puzzle from the newspaper and records the time taken in minutes. Last year, his median time was 32 minutes. His times for a random sample of 12 puzzles this year are as follows. $$\begin{array} { l l l l l l l l l l l l } 40 & 20 & 18 & 11 & 47 & 36 & 38 & 35 & 22 & 14 & 12 & 21 \end{array}$$ Use an appropriate test, with a 5\% significance level, to examine whether Godfrey's times this year have decreased on the whole.
OCR MEI S3 2009 January Q2
18 marks Standard +0.3
2 A factory manufactures paperweights consisting of glass mounted on a wooden base. The volume of glass, in \(\mathrm { cm } ^ { 3 }\), in a paperweight has a Normal distribution with mean 56.5 and standard deviation 2.9. The volume of wood, in \(\mathrm { cm } ^ { 3 }\), also has a Normal distribution with mean 38.4 and standard deviation 1.1. These volumes are independent of each other. For the purpose of quality control, paperweights for testing are chosen at random from the factory's output.
  1. Find the probability that the volume of glass in a randomly chosen paperweight is less than \(60 \mathrm {~cm} ^ { 3 }\).
  2. Find the probability that the total volume of a randomly chosen paperweight is more than \(100 \mathrm {~cm} ^ { 3 }\). The glass has a mass of 3.1 grams per \(\mathrm { cm } ^ { 3 }\) and the wood has a mass of 0.8 grams per \(\mathrm { cm } ^ { 3 }\).
  3. Find the probability that the total mass of a randomly chosen paperweight is between 200 and 220 grams.
  4. The factory manager introduces some modifications intended to reduce the mean mass of the paperweights to 200 grams or less. The variance is also affected but not the Normality. Subsequently, for a random sample of 10 paperweights, the sample mean mass is 205.6 grams and the sample standard deviation is 8.51 grams. Is there evidence, at the \(5 \%\) level of significance, that the intended reduction of the mean mass has not been achieved?
OCR MEI S3 2009 January Q3
18 marks Standard +0.3
3 Pathology departments in hospitals routinely analyse blood specimens. Ideally the analysis should be done while the specimens are fresh to avoid any deterioration, but this is not always possible. A researcher decides to study the effect of freezing specimens for later analysis by measuring the concentrations of a particular hormone before and after freezing. He collects and divides a sample of 15 specimens. One half of each specimen is analysed immediately, the other half is frozen and analysed a month later. The concentrations of the particular hormone (in suitable units) are as follows.
Immediately15.2113.3615.9721.0712.8210.8011.5012.05
After freezing15.9610.6513.3815.0012.1112.6512.488.49
Immediately10.9018.4813.4313.1616.6214.9117.08
After freezing9.1315.5311.848.9916.2414.0316.13
A \(t\) test is to be used in order to see if, on average, there is a reduction in hormone concentration as a result of being frozen.
  1. Explain why a paired test is appropriate in this situation.
  2. State the hypotheses that should be used, together with any necessary assumptions.
  3. Carry out the test using a \(1 \%\) significance level.
  4. A \(p \%\) confidence interval for the true mean reduction in hormone concentration is found to be ( \(0.4869,2.8131\) ). Determine the value of \(p\).
OCR MEI S3 2009 January Q4
18 marks Standard +0.3
4
  1. Explain the meaning of 'opportunity sampling'. Give one reason why it might be used and state one disadvantage of using it. A market researcher is conducting an 'on-street' survey in a busy city centre, for which he needs to stop and interview 100 people. For each interview the researcher counts the number of people he has to ask until one agrees to be interviewed. The data collected are as follows.
    No. of people asked1234567 or more
    Frequency261917131186
    A model for these data is proposed as follows, where \(p\) (assumed constant throughout) is the probability that a person asked agrees to be interviewed, and \(q = 1 - p\).
    No. of people asked1234567 or more
    Probability\(p\)\(p q\)\(p q ^ { 2 }\)\(p q ^ { 3 }\)\(p q ^ { 4 }\)\(p q ^ { 5 }\)\(q ^ { 6 }\)
  2. Verify that these probabilities add to 1 whatever the value of \(p\).
  3. Initially it is thought that on average 1 in 4 people asked agree to be interviewed. Test at the \(10 \%\) level of significance whether it is reasonable to suppose that the model applies with \(p = 0.25\).
  4. Later an estimate of \(p\) obtained from the data is used in the analysis. The value of the test statistic (with no combining of cells) is found to be 9.124 . What is the outcome of this new test? Comment on your answer in relation to the outcome of the test in part (iii).
OCR MEI S3 2010 January Q1
17 marks Standard +0.3
1 Coastal wildlife wardens are monitoring populations of herring gulls. Herring gulls usually lay 3 eggs per nest and the wardens wish to model the number of eggs per nest that hatch. They assume that the situation can be modelled by the binomial distribution \(\mathrm { B } ( 3 , p )\) where \(p\) is the probability that an egg hatches. A random sample of 80 nests each containing 3 eggs has been observed with the following results.
Number of eggs hatched0123
Number of nests7232921
  1. Initially it is assumed that the value of \(p\) is \(\frac { 1 } { 2 }\). Test at the \(5 \%\) level of significance whether it is reasonable to suppose that the model applies with \(p = \frac { 1 } { 2 }\).
  2. The model is refined by estimating \(p\) from the data. Find the mean of the observed data and hence an estimate of \(p\).
  3. Using the estimated value of \(p\), the value of the test statistic \(X ^ { 2 }\) turns out to be 2.3857 . Is it reasonable to suppose, at the \(5 \%\) level of significance, that this refined model applies?
  4. Discuss the reasons for the different outcomes of the tests in parts (i) and (iii).
OCR MEI S3 2010 January Q2
19 marks Standard +0.3
2
  1. A continuous random variable, \(X\), has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 72 } \left( 8 x - x ^ { 2 } \right) & 2 \leqslant x \leqslant 8 \\ 0 & \text { otherwise } \end{cases}$$
    1. Find \(\mathrm { F } ( x )\), the cumulative distribution function of \(X\).
    2. Sketch \(\mathrm { F } ( x )\).
    3. The median of \(X\) is \(m\). Show that \(m\) satisfies the equation \(m ^ { 3 } - 12 m ^ { 2 } + 148 = 0\). Verify that \(m \approx 4.42\).
  2. The random variable in part (a) is thought to model the weights, in kilograms, of lambs at birth. The birth weights, in kilograms, of a random sample of 12 lambs, given in ascending order, are as follows. $$\begin{array} { l l l l l l l l l l l l } 3.16 & 3.62 & 3.80 & 3.90 & 4.02 & 4.72 & 5.14 & 6.36 & 6.50 & 6.58 & 6.68 & 6.78 \end{array}$$ Test at the 5\% level of significance whether a median of 4.42 is consistent with these data.
OCR MEI S3 2010 January Q3
18 marks Standard +0.3
3 Cholesterol is a lipid (fat) which is manufactured by the liver from the fatty foods that we eat. It plays a vital part in allowing the body to function normally. However, when high levels of cholesterol are present in the blood there is a risk of arterial disease. Among the factors believed to assist with achieving and maintaining low cholesterol levels are weight loss and exercise. A doctor wishes to test the effectiveness of exercise in lowering cholesterol levels. For a random sample of 12 of her patients, she measures their cholesterol levels before and after they have followed a programme of exercise. The measurements obtained are as follows.
PatientABCDEFGHIJKL
Before5.75.74.06.87.45.56.76.47.27.27.14.4
After5.84.05.25.76.05.05.84.27.35.26.44.1
  1. A \(t\) test is to be used in order to see if, on average, the exercise programme seems to be effective in lowering cholesterol levels. State the distributional assumption necessary for the test, and carry out the test using a \(1 \%\) significance level.
  2. A second random sample of 12 patients gives a \(95 \%\) confidence interval of \(( - 0.5380,1.4046 )\) for the true mean reduction (before - after) in cholesterol level. Find the mean and standard deviation for this sample. How might the doctor interpret this interval in relation to the exercise programme?
OCR MEI S3 2010 January Q4
18 marks Standard +0.3
4 The weights of a particular variety (A) of tomato are known to be Normally distributed with mean 80 grams and standard deviation 11 grams.
  1. Find the probability that a randomly chosen tomato of variety A weighs less than 90 grams. The weights of another variety (B) of tomato are known to be Normally distributed with mean 70 grams. These tomatoes are packed in sixes using packaging that weighs 15 grams.
  2. The probability that a randomly chosen pack of 6 tomatoes of variety B , including packaging, weighs less than 450 grams is 0.8463 . Show that the standard deviation of the weight of single tomatoes of variety B is 6 grams, to the nearest gram.
  3. Tomatoes of variety A are packed in fives using packaging that weighs 25 grams. Find the probability that the total weight of a randomly chosen pack of variety A is greater than the total weight of a randomly chosen pack of variety B .
  4. A new variety (C) of tomato is introduced. The weights, \(c\) grams, of a random sample of 60 of these tomatoes are measured giving the following results. $$\Sigma c = 3126.0 \quad \Sigma c ^ { 2 } = 164223.96$$ Find a \(95 \%\) confidence interval for the true mean weight of these tomatoes.
OCR MEI S3 2011 January Q1
19 marks Standard +0.3
1 Each month the amount of electricity, measured in kilowatt-hours ( kWh ), used by a particular household is Normally distributed with mean 406 and standard deviation 12.
  1. Find the probability that, in a randomly chosen month, less than 420 kWh is used. The charge for electricity used is 14.6 pence per kWh .
  2. Write down the distribution of the total charge for the amount of electricity used in any one month. Hence find the probability that, in a randomly chosen month, the total charge is more than \(\pounds 60\).
  3. The household receives a bill every three months. Assume that successive months may be regarded as independent of each other. Find the value of \(b\) such that the probability that a randomly chosen bill is less than \(\pounds b\) is 0.99 . In a different household, the amount of electricity used per month was Normally distributed with mean 432 kWh . This household buys a new washing machine that is claimed to be cheaper to run than the old one. Over the next six months the amounts of electricity used, in kWh , are as follows. $$\begin{array} { l l l l l l } 404 & 433 & 420 & 423 & 413 & 440 \end{array}$$
  4. Treating this as a random sample, carry out an appropriate test, with a \(5 \%\) significance level, to see if there is any evidence to suggest that the amount of electricity used per month by this household has decreased on average.
OCR MEI S3 2011 January Q2
18 marks Standard +0.3
2
    1. What is stratified sampling? Why would it be used?
    2. A local authority official wishes to conduct a survey of households in the borough. He decides to select a stratified sample of 2000 households using Council Tax property bands as the strata. At the time of the survey there are 79368 households in the borough. The table shows the numbers of households in the different tax bands.
      Tax bandA - BC - DE - FG - H
      Number of households322983321197394120
      Calculate the number of households that the official should choose from each stratum in order to obtain his sample of 2000 households so that each stratum is represented proportionally.
    1. What assumption needs to be made when using a Wilcoxon single sample test?
    2. As part of an investigation into trends in local authority spending, one of the categories of expenditure considered was 'Highways and the Environment'. For a random sample of 10 local authorities, the percentages of their total expenditure spent on Highways and the Environment in 1999 and then in 2009 are shown in the table.
      Local authorityABCDEFGHIJ
      19999.608.408.679.329.899.357.918.089.618.55
      20098.948.427.878.4110.1710.118.319.769.549.67
      Use a Wilcoxon test, with a significance level of \(10 \%\), to determine whether there appears to be any change to the average percentage of total expenditure spent on Highways and the Environment between 1999 and 2009.
OCR MEI S3 2011 January Q3
18 marks Standard +0.3
3 The masses, in kilograms, of a random sample of 100 chickens on sale in a large supermarket were recorded as follows.
Mass \(( m \mathrm {~kg} )\)\(m < 1.6\)\(1.6 \leqslant m < 1.8\)\(1.8 \leqslant m < 2.0\)\(2.0 \leqslant m < 2.2\)\(2.2 \leqslant m < 2.4\)\(2.4 \leqslant m < 2.6\)\(2.6 \leqslant m\)
Frequency2830421152
  1. Assuming that the first and last classes are the same width as the other classes, calculate an estimate of the sample mean and show that the corresponding estimate of the sample standard deviation is 0.2227 kg . A Normal distribution using the mean and standard deviation found in part (i) is to be fitted to these data. The expected frequencies for the classes are as follows.
    Mass \(( m \mathrm {~kg} )\)\(m < 1.6\)\(1.6 \leqslant m < 1.8\)\(1.8 \leqslant m < 2.0\)\(2.0 \leqslant m < 2.2\)\(2.2 \leqslant m < 2.4\)\(2.4 \leqslant m < 2.6\)\(2.6 \leqslant m\)
    Expected
    frequency
    		2.1710.92\(f\)33.8519.225.130.68
  2. Use the Normal distribution to find \(f\).
  3. Carry out a goodness of fit test of this Normal model using a significance level of 5\%.
  4. Discuss the outcome of the test with reference to the contributions to the test statistic and to the possibility of other significance levels.
OCR MEI S3 2011 January Q4
17 marks Standard +0.3
4 A timber supplier cuts wooden fence posts from felled trees. The posts are of length \(( k + X ) \mathrm { cm }\) where \(k\) is a constant and \(X\) is a random variable which has probability density function $$f ( x ) = \begin{cases} 1 + x & - 1 \leqslant x < 0 \\ 1 - x & 0 \leqslant x \leqslant 1 \\ 0 & \text { elsewhere } \end{cases}$$
  1. Sketch \(\mathrm { f } ( x )\).
  2. Write down the value of \(\mathrm { E } ( X )\) and find \(\operatorname { Var } ( X )\).
  3. Write down, in terms of \(k\), the approximate distribution of \(\bar { L }\), the mean length of a random sample of 50 fence posts. Justify your choice of distribution.
  4. In a particular sample of 50 posts, the mean length is 90.06 cm . Find a \(95 \%\) confidence interval for the true mean length of the fence posts.
  5. Explain whether it is reasonable to suppose that \(k = 90\).
OCR MEI S3 2012 January Q1
18 marks Standard +0.3
1
  1. Define simple random sampling. Describe briefly one difficulty associated with simple random sampling.
  2. Freeze-drying is an economically important process used in the production of coffee. It improves the retention of the volatile aroma compounds. In order to maintain the quality of the coffee, technologists need to monitor the drying rate, measured in suitable units, at regular intervals. It is known that, for best results, the mean drying rate should be 70.3 units and anything substantially less than this would be detrimental to the coffee. Recently, a random sample of 12 observations of the drying rate was as follows. $$\begin{array} { l l l l l l l l l l l l } 66.0 & 66.1 & 59.8 & 64.0 & 70.9 & 71.4 & 66.9 & 76.2 & 65.2 & 67.9 & 69.2 & 68.5 \end{array}$$
    1. Carry out a test to investigate at the \(5 \%\) level of significance whether the mean drying rate appears to be less than 70.3. State the distributional assumption that is required for this test.
    2. Find a 95\% confidence interval for the true mean drying rate.