Questions — OCR MEI S3 (73 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
OCR MEI S3 2014 June Q3
19 marks Standard +0.3
3
  1. A personal trainer believes that drinking a glass of beetroot juice an hour before exercising enables endurance tests to be completed more quickly. To test his belief he takes a random sample of 12 of his trainees and, on two occasions, asks them to carry out 100 repetitions of a particular exercise as quickly as possible. Each trainee drinks a glass of water on one occasion and a glass of beetroot juice on the other occasion. The times in seconds taken by the trainees are given in the table.
    TraineeWaterBeetroot juice
    A75.172.9
    B86.279.9
    C77.371.6
    D89.190.2
    E67.968.2
    F101.595.2
    G82.576.5
    H83.380.2
    I102.599.1
    J91.382.2
    K92.590.1
    L77.277.9
    The trainer wishes to test his belief using a paired \(t\) test at the \(1 \%\) level of significance. Assuming any necessary assumptions are valid, carry out a test of the hypotheses \(\mathrm { H } _ { 0 } : \mu _ { D } = 0 , \mathrm { H } _ { 1 } : \mu _ { D } < 0\), where \(\mu _ { D }\) is the population mean difference in times (time with beetroot juice minus time with water).
  2. An ornithologist believes that the number of birds landing on the bird feeding station in her garden in a given interval of time during the morning should follow a Poisson distribution. In order to test her belief, she makes the following observations in 60 randomly chosen minutes one morning.
    Number of birds0123456\(\geqslant 7\)
    Frequency25101714741
    Given that the data in the table have a mean value of 3.3, use a goodness of fit test, with a significance level of \(5 \%\), to investigate whether the ornithologist is justified in her belief.
OCR MEI S3 2014 June Q4
17 marks Challenging +1.2
4 The probability density function of a random variable \(X\) is given by $$\mathrm { f } ( x ) = \begin{cases} k x & 0 \leqslant x \leqslant a \\ k ( 2 a - x ) & a < x \leqslant 2 a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(k\) are positive constants.
  1. Sketch \(\mathrm { f } ( x )\). Hence explain why \(\mathrm { E } ( X ) = a\).
  2. Show that \(k = \frac { 1 } { a ^ { 2 } }\).
  3. Find \(\operatorname { Var } ( X )\) in terms of \(a\). In order to estimate the value of \(a\), a random sample of size 50 is taken from the distribution. It is found that the sample mean and standard deviation are \(\bar { x } = 1.92\) and \(s = 0.8352\).
  4. Construct a symmetrical \(95 \%\) confidence interval for \(a\). Give one reason why the answer is only approximate.
  5. A non-statistician states that the probability that \(a\) lies in the interval found in part (iv) is 0.95 . Comment on this statement. \section*{END OF QUESTION PAPER} \section*{OCR \(^ { \text {® } }\)}
OCR MEI S3 2016 June Q1
18 marks Standard +0.8
1 A game consists of 20 rounds. Each round is denoted as either a starter, middle or final round. The times taken for each round are independently and Normally distributed with the following parameters (given in seconds).
Type of roundMeanStandard deviation
Starter20015
Middle22025
Final25020
The game consists of 4 starter, 12 middle and 4 final rounds. Find the probability that
  1. the mean time per round for the 4 final rounds will exceed 260 seconds,
  2. all 20 rounds will be completed in a total time of 75 minutes or less,
  3. the 12 middle rounds will take at least 3.5 times as long in total as the 4 starter rounds,
  4. the mean time per round for the 12 middle rounds will be at least 25 seconds less than the mean time per round for the 4 final rounds.
OCR MEI S3 2016 June Q2
18 marks Standard +0.3
2
  1. A genetic model involving body colour and eye colour of fruit flies predicts that offspring will consist of four phenotypes in the ratio \(9 : 3 : 3 : 1\). A random sample of 200 such offspring is taken. Their phenotypes are found to be as follows.
    PhenotypeBrown body Red eyeBrown body Brown eyeBlack body Red eyeBlack body Brown eye
    Frequency12537326
    Relative proportion from model9331
    Carry out a test, using a \(2.5 \%\) level of significance, of the goodness of fit of the genetic model to these data.
  2. The median length of European fruit flies is 2.5 mm . South American fruit flies are believed to be larger than European fruit flies. A random sample of 12 South American fruit flies is taken. The flies are found to have the following lengths (in mm). \(1.7 \quad 1.4\) \(3.1 \quad 3.5\) 3.8
    4.2
    2.2
    2.9
    4.4
    2.6 \(3.9 \quad 3.2\) Carry out a Wilcoxon signed rank test, using a \(5 \%\) level of significance, to test this belief.
OCR MEI S3 2016 June Q3
18 marks Standard +0.3
3 The random variable \(X\) has the following probability density function: $$\mathrm { f } ( x ) = \begin{cases} k \left( 1 - x ^ { 2 } \right) & - 1 \leqslant x \leqslant 1 \\ 0 & \text { elsewhere } \end{cases}$$ where \(k\) is a positive constant.
  1. Calculate the value of \(k\).
  2. Sketch the probability density function.
  3. Calculate \(\operatorname { Var } ( X )\).
  4. Find a cubic equation satisfied by the upper quartile \(q\), and hence verify that \(q = 0.35\) to 2 decimal places.
  5. A random sample of 40 values of \(X\) is taken. Using a suitable approximating distribution, calculate the probability that the mean of these values is greater than 0.125 . Justify your choice of distribution.
OCR MEI S3 2016 June Q4
18 marks Standard +0.3
4 An insurance company is investigating a new system designed to reduce the average time taken to process claim forms. The company has decided to use 10 experienced employees to process claims using the old system and the new system. Two procedures for comparing the systems are proposed.
Procedure \(A\) There are two sets of claim forms, set 1 and set 2. Each contains the same number of forms. Each employee processes set 1 on the old system and set 2 on the new system. The times taken are compared. Procedure \(B\) There is just one set of claim forms which each employee processes firstly on the old system and then on the new system. The times taken are compared.
  1. State one weakness of each of these procedures. In fact a third procedure which avoids these two weaknesses is adopted. In this procedure each employee is given a randomly selected set of claim forms. Each set contains the same number of forms. The employees each process their set of claim forms on both systems. The times taken, in minutes, are shown in the table.
    Employee12345678910
    Old system40.542.952.851.777.266.765.249.255.658.3
    New system39.240.750.650.771.470.571.147.752.155.5
  2. Carry out a paired \(t\) test at the \(5 \%\) level of significance to investigate whether the mean length of time taken to process a set of forms has reduced using the new system.
  3. State fully the usual conditions for a paired \(t\) test.
  4. Construct a \(99 \%\) confidence interval for the mean reduction in time taken to process a set of forms using the new system.
OCR MEI S3 2008 January Q1
18 marks Moderate -0.3
1
  1. The time (in milliseconds) taken by my computer to perform a particular task is modelled by the random variable \(T\). The probability that it takes more than \(t\) milliseconds to perform this task is given by the expression \(\mathrm { P } ( T > t ) = \frac { k } { t ^ { 2 } }\) for \(t \geqslant 1\), where \(k\) is a constant.
    1. Write down the cumulative distribution function of \(T\) and hence show that \(k = 1\).
    2. Find the probability density function of \(T\).
    3. Find the mean time for the task.
  2. For a different task, the times (in milliseconds) taken by my computer on 10 randomly chosen occasions were as follows. $$\begin{array} { c c c c c c c c c c } 6.4 & 5.9 & 5.0 & 6.2 & 6.8 & 6.0 & 5.2 & 6.5 & 5.7 & 5.3 \end{array}$$ From past experience it is thought that the median time for this task is 5.4 milliseconds. Carry out a test at the \(5 \%\) level of significance to investigate this, stating your hypotheses carefully.
OCR MEI S3 2008 January Q2
18 marks Standard +0.3
2 In the vegetable section of a local supermarket, leeks are on sale either loose (and unprepared) or prepared in packs of 4 . The weights of unprepared leeks are modelled by the random variable \(X\) which has the Normal distribution with mean 260 grams and standard deviation 24 grams. The prepared leeks have had \(40 \%\) of their weight removed, so that their weights, \(Y\), are modelled by \(Y = 0.6 X\).
  1. Find the probability that a randomly chosen unprepared leek weighs less than 300 grams.
  2. Find the probability that a randomly chosen prepared leek weighs more than 175 grams.
  3. Find the probability that the total weight of 4 randomly chosen prepared leeks in a pack is less than 600 grams.
  4. What total weight of prepared leeks in a randomly chosen pack of 4 is exceeded with probability 0.975 ?
  5. Sandie is making soup. She uses 3 unprepared leeks and 2 onions. The weights of onions are modelled by the Normal distribution with mean 150 grams and standard deviation 18 grams. Find the probability that the total weight of her ingredients is more than 1000 grams.
  6. A large consignment of unprepared leeks is delivered to the supermarket. A random sample of 100 of them is taken. Their weights have sample mean 252.4 grams and sample standard deviation 24.6 grams. Find a \(99 \%\) confidence interval for the true mean weight of the leeks in this consignment.
OCR MEI S3 2008 January Q3
18 marks Standard +0.3
3 Engineers in charge of a chemical plant need to monitor the temperature inside a reaction chamber. Past experience has shown that when functioning correctly the temperature inside the chamber can be modelled by a Normal distribution with mean \(380 ^ { \circ } \mathrm { C }\). The engineers are concerned that the mean operating temperature may have fallen. They decide to test the mean using the following random sample of 12 recent temperature readings.
374.0378.1363.0357.0377.9388.4
379.6372.4362.4377.3385.2370.6
  1. Give three reasons why a \(t\) test would be appropriate.
  2. Carry out the test using a \(5 \%\) significance level. State your hypotheses and conclusion carefully.
  3. Find a 95\% confidence interval for the true mean temperature in the reaction chamber.
  4. Describe briefly one advantage and one disadvantage of having a 99\% confidence interval instead of a 95\% confidence interval.
OCR MEI S3 2008 January Q4
18 marks Standard +0.3
4
  1. In Germany, towards the end of the nineteenth century, a study was undertaken into the distribution of the sexes in families of various sizes. The table shows some data about the numbers of girls in 500 families, each with 5 children. It is thought that the binomial distribution \(\mathrm { B } ( 5 , p )\) should model these data.
    Number of girlsNumber of families
    032
    1110
    2154
    3125
    463
    516
    1. Use this information to calculate an estimate for the mean number of girls per family of 5 children. Hence show that 0.45 can be taken as an estimate of \(p\).
    2. Investigate at a \(5 \%\) significance level whether the binomial model with \(p\) estimated as 0.45 fits the data. Comment on your findings and also on the extent to which the conditions for a binomial model are likely to be met.
  2. A researcher wishes to select 50 families from the 500 in part (a) for further study. Suggest what sort of sample she might choose and describe how she should go about choosing it.
OCR MEI S3 Q2
20 marks Standard +0.3
2 Geoffrey is a university lecturer. He has to prepare five questions for an examination. He knows by experience that it takes about 3 hours to prepare a question, and he models the time (in minutes) taken to prepare one by the Normally distributed random variable \(X\) with mean 180 and standard deviation 12, independently for all questions.
  1. One morning, Geoffrey has a gap of 2 hours 50 minutes ( 170 minutes) between other activities. Find the probability that he can prepare a question in this time.
  2. One weekend, Geoffrey can devote 14 hours to preparing the complete examination paper. Find the probability that he can prepare all five questions in this time. A colleague, Helen, has to check the questions.
  3. She models the time (in minutes) to check a question by the Normally distributed random variable \(Y\) with mean 50 and standard deviation 6, independently for all questions and independently of \(X\). Find the probability that the total time for Geoffrey to prepare a question and Helen to check it exceeds 4 hours.
  4. When working under pressure of deadlines, Helen models the time to check a question in a different way. She uses the Normally distributed random variable \(\frac { 1 } { 4 } X\), where \(X\) is as above. Find the length of time, as given by this model, which Helen needs to ensure that, with probability 0.9 , she has time to check a question. Ian, an educational researcher, suggests that a better model for the time taken to prepare a question would be a constant \(k\) representing "thinking time" plus a random variable \(T\) representing the time required to write the question itself, independently for all questions.
  5. Taking \(k\) as 45 and \(T\) as Normally distributed with mean 120 and standard deviation 10 (all units are minutes), find the probability according to Ian's model that a question can be prepared in less than 2 hours 30 minutes. Juliet, an administrator, proposes that the examination should be reduced in time and shorter questions should be used.
  6. Juliet suggests that Ian's model should be used for the time taken to prepare such shorter questions but with \(k = 30\) and \(T\) replaced by \(\frac { 3 } { 5 } T\). Find the probability as given by this model that a question can be prepared in less than \(1 \frac { 3 } { 4 }\) hours.
OCR MEI S3 2006 January Q1
18 marks Standard +0.3
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. $$F(t) = \begin{cases} 0 & t \leq 0 \\ 1 - e^{-\frac{1}{t}} & t > 0 \end{cases}$$
  1. Find the median delay and the 90th percentile delay. [5]
  2. Derive the probability density function of \(T\). Hence use calculus to find the mean delay. [5]
  3. Find the probability that a delay lasts longer than the mean delay. [2]
You are given that the variance of \(T\) is 9.
  1. Let \(\overline{T}\) denote the mean of a random sample of 30 delays. Write down an approximation to the distribution of \(\overline{T}\). [3]
  2. A random sample of 30 delays is found to have mean 4.2 minutes. Does this cast any doubt on the modelling? [3]
OCR MEI S3 2006 January Q2
18 marks Standard +0.3
Geoffrey is a university lecturer. He has to prepare five questions for an examination. He knows by experience that it takes about 3 hours to prepare a question, and he models the time (in minutes) taken to prepare one by the Normally distributed random variable \(X\) with mean 180 and standard deviation 12, independently for all questions.
  1. One morning, Geoffrey has a gap of 2 hours 50 minutes (170 minutes) between other activities. Find the probability that he can prepare a question in this time. [3]
  2. One weekend, Geoffrey can devote 14 hours to preparing the complete examination paper. Find the probability that he can prepare all five questions in this time. [3]
A colleague, Helen, has to check the questions.
  1. She models the time (in minutes) to check a question by the Normally distributed random variable \(Y\) with mean 50 and standard deviation 6, independently for all questions and independently of \(X\). Find the probability that the total time for Geoffrey to prepare a question and Helen to check it exceeds 4 hours. [3]
  2. When working under pressure of deadlines, Helen models the time to check a question in a different way. She uses the Normally distributed random variable \(\frac{1}{2}X\), where \(X\) is as above. Find the length of time, as given by this model, which Helen needs to ensure that, with probability 0.9, she has time to check a question. [4]
Ian, an educational researcher, suggests that a better model for the time taken to prepare a question would be a constant \(k\) representing "thinking time" plus a random variable \(T\) representing the time required to write the question itself, independently for all questions.
  1. Taking \(k\) as 45 and \(T\) as Normally distributed with mean 120 and standard deviation 10 (all units are minutes), find the probability according to Ian's model that a question can be prepared in less than 2 hours 30 minutes. [2]
Juliet, an administrator, proposes that the examination should be reduced in time and shorter questions should be used.
  1. Juliet suggests that Ian's model should be used for the time taken to prepare such shorter questions but with \(k = 30\) and \(T\) replaced by \(\frac{2}{3}T\). Find the probability as given by this model that a question can be prepared in less than \(1\frac{1}{4}\) hours. [3]
OCR MEI S3 2006 January Q3
18 marks Standard +0.3
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. [11]
  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. [7]
OCR MEI S3 2006 January Q4
18 marks Standard +0.3
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 \(x\) (mm)Observed frequency
    \(x \leq 298\)1
    \(298 < x \leq 300\)30
    \(300 < x \leq 301\)62
    \(301 < x \leq 302\)70
    \(302 < x \leq 304\)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 \(x\) (mm)Expected frequency
    \(x \leq 298\)1.49
    \(298 < x \leq 300\)37.85
    \(300 < x \leq 301\)55.62
    \(301 < x \leq 302\)58.32
    \(302 < x \leq 304\)44.62
    \(x > 304\)2.10
    Examine the goodness of fit of a Normal distribution, using a 5\% significance level. [7]
  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). 301.3 \quad 301.4 \quad 299.6 \quad 302.2 \quad 300.3 \quad 303.2 \quad 302.6 \quad 301.8 \quad 300.9 \quad 300.8
    1. Discuss briefly whether it would be appropriate to use a \(t\) test to examine a hypothesis about the population mean length for this batch. [2]
    2. Use a Wilcoxon test to examine at the 10\% significance level whether the population median length for this batch is 301 mm. [9]
OCR MEI S3 2008 June Q1
19 marks Moderate -0.8
  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 $$f(x) = k(20-x) \text{ for } 0 \leq x \leq 20, \text{ where } k \text{ is a constant.}$$
    1. Find \(k\). Sketch the graph of \(f(x)\) and use its shape to explain what can be deduced about how long Sarah has to wait. [5]
    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. [4]
    3. Find the median length of time that Sarah has to wait. [3]
    1. Define the term 'simple random sample'. [2]
    2. Explain briefly how to carry out cluster sampling. [3]
    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. [2]
OCR MEI S3 2008 June Q2
18 marks Standard +0.3
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.
  1. Find the probability that the resistance of a randomly chosen resistor of type A is less than 103 ohms. [3]
  2. Three resistors of type A are chosen at random. Find the probability that their total resistance is more than 306 ohms. [3]
  3. 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. [3]
  4. 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. [5]
  5. 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. [4]
OCR MEI S3 2008 June Q3
18 marks Standard +0.3
  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. [3]
    2. Carry out the test using a 5\% significance level. [7]
  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. [8]
OCR MEI S3 2008 June Q4
17 marks Standard +0.3
  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\(\geq 8\)
    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. [3]
    2. Use the mean in part (i) to carry out a test of the goodness of fit of a Poisson model to the data. [10]
  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. [4]
OCR MEI S3 2010 June Q1
18 marks Moderate -0.8
  1. The manager of a company that employs 250 travelling sales representatives wishes to carry out a detailed analysis of the expenses claimed by the representatives. He has an alphabetical (by surname) list of the representatives. He chooses a sample of representatives by selecting the 10th, 20th, 30th and so on. Name the type of sampling the manager is attempting to use. Describe a weakness in his method of using it, and explain how he might overcome this weakness. [3]
The representatives each use their own cars to drive to meetings with customers. The total distance, in miles, travelled by a representative in a month is Normally distributed with mean 2018 and standard deviation 96.
  1. Find the probability that, in a randomly chosen month, a randomly chosen representative travels more than 2100 miles. [3]
  2. Find the probability that, in a randomly chosen 3-month period, a randomly chosen representative travels less than 6000 miles. What assumption is needed here? Give a reason why it may not be realistic. [5]
  3. Each month every representative submits a claim for travelling expenses plus commission. Travelling expenses are paid at the rate of 45 pence per mile. The commission is 10\% of the value of sales in that month. The value, in £, of the monthly sales has the distribution N(21200, 1100²). Find the probability that a randomly chosen claim lies between £3000 and £3300. [7]
OCR MEI S3 2010 June Q2
18 marks Standard +0.3
William Sealy, a biochemistry student, is doing work experience at a brewery. One of his tasks is to monitor the specific gravity of the brewing mixture during the brewing process. For one particular recipe, an initial specific gravity of 1.040 is required. A random sample of 9 measurements of the specific gravity at the start of the process gave the following results. 1.046 \quad 1.048 \quad 1.039 \quad 1.055 \quad 1.038 \quad 1.054 \quad 1.038 \quad 1.051 \quad 1.038
  1. William has to test whether the specific gravity of the mixture meets the requirement. Why might a \(t\) test be used for these data and what assumption must be made? [3]
  2. Carry out the test using a significance level of 10\%. [9]
  3. Find a 95\% confidence interval for the true mean specific gravity of the mixture and explain what is meant by a 95\% confidence interval. [6]
OCR MEI S3 2010 June Q3
18 marks Standard +0.3
  1. In order to prevent and/or control the spread of infectious diseases, the Government has various vaccination programmes. One such programme requires people to receive a booster injection at the age of 18. It is felt that the proportion of people receiving this booster could be increased and a publicity campaign is undertaken for this purpose. In order to assess the effectiveness of this campaign, health authorities across the country are asked to report the percentage of 18-year-olds receiving the booster before and after the campaign. The results for a randomly chosen sample of 9 authorities are as follows.
    AuthorityABCDEFGHI
    Before769888818684839380
    After829793778395919589
    This sample is to be tested to see whether the campaign appears to have been successful in raising the percentage receiving the booster.
    1. Explain why the use of paired data is appropriate in this context. [1]
    2. Carry out an appropriate Wilcoxon signed rank test using these data, at the 5\% significance level. [10]
  2. Benford's Law predicts the following probability distribution for the first significant digit in some large data sets.
    Digit123456789
    Probability0.3010.1760.1250.0970.0790.0670.0580.0510.046
    On one particular day, the first significant digits of the stock market prices of the shares of a random sample of 200 companies gave the following results.
    Digit123456789
    Frequency55342716151712159
    Test at the 10\% level of significance whether Benford's Law provides a reasonable model in the context of share prices. [7]
OCR MEI S3 2010 June Q4
18 marks Moderate -0.3
A random variable \(X\) has an exponential distribution with probability density function \(f(x) = \lambda e^{-\lambda x}\) for \(x \geq 0\), where \(\lambda\) is a positive constant.
  1. Verify that \(\int_0^{\infty} f(x) \, dx = 1\) and sketch \(f(x)\). [5]
  2. In this part of the question you may use the following result. $$\int_0^{\infty} x^r e^{-\lambda x} \, dx = \frac{r!}{\lambda^{r+1}} \text{ for } r = 0, 1, 2, \ldots$$ Derive the mean and variance of \(X\) in terms of \(\lambda\). [6]
The random variable \(X\) is used to model the lifetime, in years, of a particular type of domestic appliance. The manufacturer of the appliance states that, based on past experience, the mean lifetime is 6 years.
  1. Let \(\overline{X}\) denote the mean lifetime, in years, of a random sample of 50 appliances. Write down an approximate distribution for \(\overline{X}\). [4]
  2. A random sample of 50 appliances is found to have a mean lifetime of 7.8 years. Does this cast any doubt on the model? [3]