Questions — OCR MEI S3 (71 questions)

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OCR MEI S3 2006 January Q1
1 A railway company is investigating operations at a junction where delays often occur. Delays (in minutes) are modelled by the random variable \(T\) with the following cumulative distribution function. $$\mathrm { F } ( t ) = \begin{cases} 0 & t \leqslant 0
1 - \mathrm { e } ^ { - \frac { 1 } { 3 } t } & t > 0 \end{cases}$$
  1. Find the median delay and the 90th percentile delay.
  2. Derive the probability density function of \(T\). Hence use calculus to find the mean delay.
  3. Find the probability that a delay lasts longer than the mean delay. You are given that the variance of \(T\) is 9 .
  4. Let \(\bar { T }\) denote the mean of a random sample of 30 delays. Write down an approximation to the distribution of \(\bar { T }\).
  5. A random sample of 30 delays is found to have mean 4.2 minutes. Does this cast any doubt on the modelling?
OCR MEI S3 2006 January Q3
3 A production line has two machines, A and B , for delivering liquid soap into bottles. Each machine is set to deliver a nominal amount of 250 ml , but it is not expected that they will work to a high level of accuracy. In particular, it is known that the ambient temperature affects the rate of flow of the liquid and leads to variation in the amounts delivered. The operators think that machine B tends to deliver a somewhat greater amount than machine A , no matter what the ambient temperature. This is being investigated by an experiment. A random sample of 10 results from the experiment is shown below. Each column of data is for a different ambient temperature.
Ambient temperature\(T _ { 1 }\)\(T _ { 2 }\)\(T _ { 3 }\)\(T _ { 4 }\)\(T _ { 5 }\)\(T _ { 6 }\)\(T _ { 7 }\)\(T _ { 8 }\)\(T _ { 9 }\)\(T _ { 10 }\)
Amount delivered by machine A246.2251.6252.0246.6258.4251.0247.5247.1248.1253.4
Amount delivered by machine B248.3252.6252.8247.2258.8250.0247.2247.9249.0254.5
  1. Use an appropriate \(t\) test to examine, at the \(5 \%\) level of significance, whether the mean amount delivered by machine B may be taken as being greater than that delivered by machine A , stating carefully your null and alternative hypotheses and the required distributional assumption.
  2. Using the data for machine A in the table above, provide a two-sided \(95 \%\) confidence interval for the mean amount delivered by this machine, stating the required distributional assumption. Explain whether you would conclude that the machine appears to be working correctly in terms of the nominal amount as set.
OCR MEI S3 2006 January Q4
4 Quality control inspectors in a factory are investigating the lengths of glass tubes that will be used to make laboratory equipment.
  1. Data on the observed lengths of a random sample of 200 glass tubes from one batch are available in the form of a frequency distribution as follows.
    Length
    \(x ( \mathrm {~mm} )\)
    Observed
    frequency
    \(x \leqslant 298\)1
    \(298 < x \leqslant 300\)30
    \(300 < x \leqslant 301\)62
    \(301 < x \leqslant 302\)70
    \(302 < x \leqslant 304\)34
    \(x > 304\)3
    The sample mean and standard deviation are 301.08 and 1.2655 respectively.
    The corresponding expected frequencies for the Normal distribution with parameters estimated by the sample statistics are
    Length
    \(x ( \mathrm {~mm} )\)
    Expected
    frequency
    \(x \leqslant 298\)1.49
    \(298 < x \leqslant 300\)37.85
    \(300 < x \leqslant 301\)55.62
    \(301 < x \leqslant 302\)58.32
    \(302 < x \leqslant 304\)44.62
    \(x > 304\)2.10
    Examine the goodness of fit of a Normal distribution, using a 5\% significance level.
  2. It is thought that the lengths of tubes in another batch have an underlying distribution similar to that for the batch in part (i) but possibly with different location and dispersion parameters. A random sample of 10 tubes from this batch gives the following lengths (in mm ). $$\begin{array} { l l l l l l l l l l } 301.3 & 301.4 & 299.6 & 302.2 & 300.3 & 303.2 & 302.6 & 301.8 & 300.9 & 300.8 \end{array}$$ (A) Discuss briefly whether it would be appropriate to use a \(t\) test to examine a hypothesis about the population mean length for this batch.
    (B) Use a Wilcoxon test to examine at the \(10 \%\) significance level whether the population median length for this batch is 301 mm .
OCR MEI S3 2007 January Q1
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
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
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
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
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
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
10 marks
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
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
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
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
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
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 2008 June Q1
1
  1. Sarah travels home from work each evening by bus; there is a bus every 20 minutes. The time at which Sarah arrives at the bus stop varies randomly in such a way that the probability density function of \(X\), the length of time in minutes she has to wait for the next bus, is given by $$\mathrm { f } ( x ) = k ( 20 - x ) \text { for } 0 \leqslant x \leqslant 20 \text {, where } k \text { is a constant. }$$
    1. Find \(k\). Sketch the graph of \(\mathrm { f } ( x )\) and use its shape to explain what can be deduced about how long Sarah has to wait.
    2. Find the cumulative distribution function of \(X\) and hence, or otherwise, find the probability that Sarah has to wait more than 10 minutes for the bus.
    3. Find the median length of time that Sarah has to wait.
    1. Define the term 'simple random sample'.
    2. Explain briefly how to carry out cluster sampling.
    3. A researcher wishes to investigate the attitudes of secondary school pupils to pollution. Explain why he might prefer to collect his data using a cluster sample rather than a simple random sample. An electronics company purchases two types of resistor from a manufacturer. The resistances of the resistors (in ohms) are known to be Normally distributed. Type A have a mean of 100 ohms and standard deviation of 1.9 ohms. Type B have a mean of 50 ohms and standard deviation of 1.3 ohms.
    4. Find the probability that the resistance of a randomly chosen resistor of type A is less than 103 ohms.
    5. Three resistors of type A are chosen at random. Find the probability that their total resistance is more than 306 ohms.
    6. One resistor of type A and one resistor of type B are chosen at random. Find the probability that their total resistance is more than 147 ohms.
    7. Find the probability that the total resistance of two randomly chosen type B resistors is within 3 ohms of one randomly chosen type A resistor.
    8. The manufacturer now offers type C resistors which are specified as having a mean resistance of 300 ohms. The resistances of a random sample of 100 resistors from the first batch supplied have sample mean 302.3 ohms and sample standard deviation 3.7 ohms. Find a \(95 \%\) confidence interval for the true mean resistance of the resistors in the batch. Hence explain whether the batch appears to be as specified.
OCR MEI S3 2008 June Q3
3
  1. A tea grower is testing two types of plant for the weight of tea they produce. A trial is set up in which each type of plant is grown at each of 8 sites. The total weight, in grams, of tea leaves harvested from each plant is measured and shown below.
    SiteABCDEFGH
    Type I225.2268.9303.6244.1230.6202.7242.1247.5
    Type II215.2242.1260.9241.7245.5204.7225.8236.0
    1. The grower intends to perform a \(t\) test to examine whether there is any difference in the mean yield of the two types of plant. State the hypotheses he should use and also any necessary assumption.
    2. Carry out the test using a \(5 \%\) significance level.
  2. The tea grower deals with many types of tea and employs tasters to rate them. The tasters do this by giving each tea a score out of 100. The tea grower wishes to compare the scores given by two of the tasters. Their scores for a random selection of 10 teas are as follows.
    TeaQRSTUVWXYZ
    Taster 169798563816585868977
    Taster 274759966756496949686
    Use a Wilcoxon test to examine, at the \(5 \%\) level of significance, whether it appears that, on the whole, the scores given to teas by these two tasters differ.
OCR MEI S3 2008 June Q4
4
  1. A researcher is investigating the feeding habits of bees. She sets up a feeding station some distance from a beehive and, over a long period of time, records the numbers of bees arriving each minute. For a random sample of 100 one-minute intervals she obtains the following results.
    Number of bees01234567\(\geqslant 8\)
    Number of intervals61619181714640
    1. Show that the sample mean is 3.1 and find the sample variance. Do these values support the possibility of a Poisson model for the number of bees arriving each minute? Explain your answer.
    2. Use the mean in part (i) to carry out a test of the goodness of fit of a Poisson model to the data.
  2. The researcher notes the length of time, in minutes, that each bee spends at the feeding station. The times spent are assumed to be Normally distributed. For a random sample of 10 bees, the mean is found to be 1.465 minutes and the standard deviation is 0.3288 minutes. Find a \(95 \%\) confidence interval for the overall mean time.
OCR MEI S3 2009 January Q1
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
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
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
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
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
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
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?