Questions S2 (1690 questions)

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OCR MEI S2 2015 June Q4
20 marks Standard +0.3
4
  1. As part of an investigation into smoking, a random sample of 120 students was selected. The students were asked whether they were smokers, and also whether either of their parents were smokers. The results are summarised in the table below. Test, at the \(5 \%\) significance level, whether there is any association between the smoking habits of the students and their parents.
    At least one
    parent smokes
    Neither parent
    smokes
    Student smokes2127
    Student does not smoke1755
  2. The manufacturer of a particular brand of cigarette claims that the nicotine content of these cigarettes is Normally distributed with mean 0.87 mg . A researcher suspects that the mean nicotine content of this brand is higher than the value claimed by the manufacturer. The nicotine content, \(x \mathrm { mg }\), is measured for a random sample of 100 cigarettes. The data are summarised as follows. $$\sum x = 88.20 \quad \sum x ^ { 2 } = 78.68$$ Carry out a test at the \(1 \%\) significance level to investigate the researcher's belief. \section*{END OF QUESTION PAPER}
OCR MEI S2 2016 June Q1
18 marks Standard +0.3
1 A researcher believes that there may be negative association between the quantity of fertiliser used and the percentage of the population who live in rural areas in different countries. The data below show the percentage of the population who live in rural areas and the fertiliser use measured in kg per hectare, for a random sample of 11 countries.
Percentage of population33658358169617747117
Fertiliser use764466831071765137157
  1. Draw a scatter diagram to illustrate the data.
  2. Explain why it might not be valid to carry out a test based on the product moment correlation coefficient in this case.
  3. Calculate the value of Spearman's rank correlation coefficient.
  4. Carry out a hypothesis test at the \(1 \%\) significance level to investigate the researcher's belief.
  5. Explain the meaning of ' \(1 \%\) significance level'.
  6. In order to carry out a test based on Spearman's rank correlation coefficient, what modelling assumptions, if any, are required about the underlying distribution?
OCR MEI S2 2016 June Q2
16 marks Standard +0.3
2 When a genetic sequence of plant DNA is given a dose of radiation, some of the genes may mutate. The probability that a gene mutates is 0.012 . Mutations occur randomly and independently.
  1. Explain the meanings of the terms 'randomly' and 'independently' in this context. A short stretch of DNA containing 20 genes is given a dose of radiation.
  2. Find the probability that exactly 1 out of the 20 genes mutates. A longer stretch of DNA containing 500 genes is given a dose of radiation.
  3. Explain why a Poisson distribution is an appropriate approximating distribution for the number of genes that mutate.
  4. Use this Poisson distribution to find the probability that there are
    (A) exactly two genes that mutate,
    (B) at least two genes that mutate. A third stretch of DNA containing 50000 genes is given a dose of radiation.
  5. Use a suitable approximating distribution to find the probability that there are at least 650 genes that mutate.
OCR MEI S2 2016 June Q3
18 marks Moderate -0.3
3 Many types of computer have cooling fans. The random variable \(X\) represents the lifetime in hours of a particular model of cooling fan. \(X\) is Normally distributed with mean 50600 and standard deviation 3400.
  1. Find \(\mathrm { P } ( 50000 < X < 55000 )\).
  2. The manufacturers claim that at least \(95 \%\) of these fans last longer than 45000 hours. Is this claim valid?
  3. Find the value of \(h\) for which \(99.9 \%\) of these fans last \(h\) hours or more.
  4. The random variable \(Y\) represents the lifetime in hours of a different model of cooling fan. \(Y\) is Normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is known that \(\mathrm { P } ( Y < 60000 ) = 0.6\) and \(\mathrm { P } ( Y > 50000 ) = 0.9\). Find the values of \(\mu\) and \(\sigma\).
  5. Sketch the distributions of lifetimes for both types of cooling fan on a single diagram.
OCR MEI S2 2016 June Q4
20 marks Moderate -0.3
4
  1. A random sample of 80 GCSE students was selected to take part in an investigation into whether attitudes to mathematics differ between girls and boys. The students were asked if they agreed with the statement 'Mathematics is one of my favourite subjects'. They were given three options 'Agree', 'Disagree', 'Neither agree nor disagree'. The results, classified according to sex, are summarised in the table below.
    AgreeDisagreeNeither
    Male17138
    Female121119
    The contributions to the test statistic for the usual \(\chi ^ { 2 }\) test are shown in the table below.
    AgreeDisagreeNeither
    Male0.75500.22461.8153
    Female0.68310.20321.6424
    1. Calculate the expected frequency for females who agree. Verify the corresponding contribution, 0.6831 , to the test statistic.
    2. Carry out the test at the \(5 \%\) level of significance.
  2. The level of radioactivity in limpets (a type of shellfish) in the sea near to a nuclear power station is regularly monitored. Over a period of years it has been found that the level (measured in suitable units) is Normally distributed with mean 5.64. Following an incident at the power station, a researcher suspects that the mean level of radioactivity in limpets may have increased. The researcher selects a random sample of 60 limpets. Their levels of radioactivity, \(x\) (measured in the same units), are summarised as follows. $$\sum x = 373 \quad \sum x ^ { 2 } = 2498$$ Carry out a test at the \(5 \%\) significance level to investigate the researcher's belief.
OCR MEI S2 Q3
18 marks Standard +0.3
3 In a triathlon, competitors have to swim 600 metres, cycle 40 kilometres and run 10 kilometres. To improve her strength, a triathlete undertakes a training programme in which she carries weights in a rucksack whilst running. She runs a specific course and notes the total time taken for each run. Her coach is investigating the relationship between time taken and weight carried. The times taken with eight different weights are illustrated on the scatter diagram below, together with the summary statistics for these data. The variables \(x\) and \(y\) represent weight carried in kilograms and time taken in minutes respectively. \includegraphics[max width=\textwidth, alt={}, center]{d138173d-c70c-46db-b9b9-d5f19334c5f1-04_627_1536_630_281} Summary statistics: \(n = 8 , \Sigma x = 36 , \Sigma y = 214.8 , \Sigma x ^ { 2 } = 204 , \Sigma y ^ { 2 } = 5775.28 , \Sigma x y = 983.6\).
  1. Calculate the equation of the regression line of \(y\) on \(x\). On one of the eight runs, the triathlete was carrying 4 kilograms and took 27.5 minutes. On this run she was delayed when she tripped and fell over.
  2. Calculate the value of the residual for this weight.
  3. The coach decides to recalculate the equation of the regression line without the data for this run. Would it be preferable to use this recalculated equation or the equation found in part (i) to estimate the delay when the triathlete tripped and fell over? Explain your answer. The triathlete's coach claims that there is positive correlation between cycling and swimming times in triathlons. The product moment correlation coefficient of the times of twenty randomly selected competitors in these two sections is 0.209 .
  4. Carry out a hypothesis test at the \(5 \%\) level to examine the coach's claim, explaining your conclusions clearly.
  5. What distributional assumption is necessary for this test to be valid? How can you use a scatter diagram to decide whether this assumption is likely to be true?
Edexcel S2 2014 January Q1
8 marks Moderate -0.3
  1. The probability of a leaf cutting successfully taking root is 0.05
Find the probability that, in a batch of 10 randomly selected leaf cuttings, the number taking root will be
    1. exactly 1
    2. more than 2 A second random sample of 160 leaf cuttings is selected.
  2. Using a suitable approximation, estimate the probability of at least 10 leaf cuttings taking root.
Edexcel S2 2014 January Q2
10 marks Moderate -0.3
2. Bill owns a restaurant. Over the next four weeks Bill decides to carry out a sample survey to obtain the customers' opinions.
  1. Suggest a suitable sampling frame for the sample survey.
  2. Identify the sampling units.
  3. Give one advantage and one disadvantage of taking a census rather than a sample survey. Bill believes that only \(30 \%\) of customers would like a greater choice on the menu. He takes a random sample of 50 customers and finds that 20 of them would like a greater choice on the menu.
  4. Test, at the \(5 \%\) significance level, whether or not the percentage of customers who would like a greater choice on the menu is more than Bill believes. State your hypotheses clearly.
Edexcel S2 2014 January Q3
11 marks Standard +0.3
  1. The continuous random variable \(X\) has cumulative distribution function given by
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0 \\ \frac { 1 } { 6 } x ( x + 1 ) & 0 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{array} \right.$$
  1. Find the value of \(a\) such that \(\mathrm { P } ( X > a ) = 0.4\) Give your answer to 3 significant figures.
  2. Use calculus to find (i) \(\mathrm { E } ( X )\) (ii) \(\operatorname { Var } ( X )\).
Edexcel S2 2014 January Q4
7 marks Standard +0.3
  1. The number of telephone calls per hour received by a business is a random variable with distribution \(\operatorname { Po } ( \lambda )\).
Charlotte records the number of calls, \(C\), received in 4 hours. A test of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 1.5\) is carried out. \(\mathrm { H } _ { 0 }\) is rejected if \(C > 10\)
  1. Write down the alternative hypothesis.
  2. Find the significance level of the test. Given that \(\mathrm { P } ( C > 10 ) < 0.1\)
  3. find the largest possible value of \(\lambda\) that can be found by using the tables.
Edexcel S2 2014 January Q5
12 marks Standard +0.8
5. A school photocopier breaks down randomly at a rate of 15 times per year.
  1. Find the probability that there will be exactly 3 breakdowns in the next month.
  2. Show that the probability that there will be at least 2 breakdowns in the next month is 0.355 to 3 decimal places.
  3. Find the probability of at least 2 breakdowns in each of the next 4 months. The teachers would like a new photocopier. The head teacher agrees to monitor the situation for the next 12 months. The head teacher decides he will buy a new photocopier if there is more than 1 month when the photocopier has at least 2 breakdowns.
  4. Find the probability that the head teacher will buy a new photocopier.
Edexcel S2 2014 January Q6
15 marks Standard +0.3
  1. The continuous random variable \(X\) has probability density function given by
$$f ( x ) = \left\{ \begin{array} { c c } k ( x + 1 ) ^ { 2 } & - 1 \leqslant x \leqslant 1 \\ k ( 6 - 2 x ) & 1 < x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a positive constant.
  1. Sketch the graph of \(\mathrm { f } ( x )\).
  2. Show that the value of \(k\) is \(\frac { 3 } { 20 }\)
  3. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
  4. Find the median of \(X\), giving your answer to 3 significant figures.
Edexcel S2 2014 January Q7
12 marks Challenging +1.2
  1. The random variable \(Y \sim \mathrm {~B} ( n , p )\).
Using a normal approximation the probability that \(Y\) is at least 65 is 0.2266 and the probability that \(Y\) is more than 52 is 0.8944 Find the value of \(n\) and the value of \(p\).
Edexcel S2 2015 January Q1
16 marks Standard +0.8
The number of cars caught speeding per day, by a particular camera, has a Poisson distribution with mean 0.8
  1. Find the probability that in a given 4 day period exactly 3 cars will be caught speeding by this camera. A car has been caught speeding by this camera.
  2. Find the probability that the period of time that elapses before the next car is caught speeding by this camera is less than 48 hours. Given that 4 cars were caught speeding by this camera in a two day period,
  3. find the probability that 1 was caught on the first day and 3 were caught on the second day. Each car that is caught speeding by this camera is fined \(\pounds 60\)
  4. Using a suitable approximation, find the probability that, in 90 days, the total amount of fines issued will be more than \(\pounds 5000\)
Edexcel S2 2015 January Q2
11 marks Moderate -0.8
2. A continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1 \\ \frac { 1 } { 5 } ( x - 1 ) & 1 \leqslant x \leqslant 6 \\ 1 & x > 6 \end{array} \right.$$
  1. Find \(\mathrm { P } ( X > 4 )\)
  2. Write down the value of \(\mathrm { P } ( X \neq 4 )\)
  3. Find the probability density function of \(X\), specifying it for all values of \(X\)
  4. Write down the value of \(\mathrm { E } ( X )\)
  5. Find \(\operatorname { Var } ( X )\)
  6. Hence or otherwise find \(\mathrm { E } \left( 3 X ^ { 2 } + 1 \right)\)
Edexcel S2 2015 January Q3
11 marks Moderate -0.8
3. Explain what you understand by
  1. a statistic,
  2. a sampling distribution. A factory stores screws in packets. A small packet contains 100 screws and a large packet contains 200 screws. The factory keeps small and large packets in the ratio 4:3 respectively.
  3. Find the mean and the variance of the number of screws in the packets stored at the factory. A random sample of 3 packets is taken from the factory and \(Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\) denote the number of screws in each of these packets.
  4. List all the possible samples.
  5. Find the sampling distribution of \(\bar { Y }\)
Edexcel S2 2015 January Q4
7 marks Standard +0.3
4. Accidents occur randomly at a crossroads at a rate of 0.5 per month. A researcher records the number of accidents, \(X\), which occur at the crossroads in a year.
  1. Find \(\mathrm { P } ( 5 \leqslant X < 7 )\) A new system is introduced at the crossroads. In the first 18 months, 4 accidents occur at the crossroads.
  2. Test, at the \(5 \%\) level of significance, whether or not there is reason to believe that the new system has led to a reduction in the mean number of accidents per month. State your hypotheses clearly.
Edexcel S2 2015 January Q5
9 marks Standard +0.8
5. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } k \left( x ^ { 2 } + a \right) & - 1 < x \leqslant 2 \\ 3 k & 2 < x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are constants.
Given that \(\mathrm { E } ( X ) = \frac { 17 } { 12 }\)
  1. find the value of \(k\) and the value of \(a\)
  2. Write down the mode of \(X\)
Edexcel S2 2015 January Q6
13 marks Standard +0.8
6. The Headteacher of a school claims that \(30 \%\) of parents do not support a new curriculum. In a survey of 20 randomly selected parents, the number, \(X\), who do not support the new curriculum is recorded. Assuming that the Headteacher's claim is correct, find
  1. the probability that \(X = 5\)
  2. the mean and the standard deviation of \(X\) The Director of Studies believes that the proportion of parents who do not support the new curriculum is greater than \(30 \%\). Given that in the survey of 20 parents 8 do not support the new curriculum,
  3. test, at the \(5 \%\) level of significance, the Director of Studies' belief. State your hypotheses clearly. The teachers believe that the sample in the original survey was biased and claim that only \(25 \%\) of the parents are in support of the new curriculum. A second random sample, of size \(2 n\), is taken and exactly half of this sample supports the new curriculum. A test is carried out at a 10\% level of significance of the teachers' belief using this sample of size \(2 n\) Using the hypotheses \(\mathrm { H } _ { 0 } : p = 0.25\) and \(\mathrm { H } _ { 1 } : p > 0.25\)
  4. find the minimum value of \(n\) for which the outcome of the test is that the teachers' belief is rejected.
Edexcel S2 2015 January Q7
8 marks Standard +0.8
7. A multiple choice examination paper has \(n\) questions where \(n > 30\) Each question has 5 answers of which only 1 is correct. A pass on the paper is obtained by answering 30 or more questions correctly. The probability of obtaining a pass by randomly guessing the answer to each question should not exceed 0.0228 Use a normal approximation to work out the greatest number of questions that could be used.
Edexcel S2 2017 January Q1
7 marks Easy -1.2
The continuous random variable \(W\) has the normal distribution \(\mathrm { N } \left( 32,4 { } ^ { 2 } \right)\)
  1. Write down the value of \(\mathrm { P } ( W = 36 )\) The discrete random variable \(X\) has the binomial distribution \(\mathrm { B } ( 20,0.45 )\)
  2. Find \(\mathrm { P } ( X = 8 )\)
  3. Find the probability that \(X\) lies within one standard deviation of its mean.
Edexcel S2 2017 January Q2
7 marks Moderate -0.8
2. The continuous random variable \(X\) is uniformly distributed over the interval \([ \alpha , \beta ]\) where \(\beta > \alpha\) Given that \(\mathrm { E } ( X ) = 8\)
  1. write down an equation involving \(\alpha\) and \(\beta\) Given also that \(\mathrm { P } ( X \leqslant 13 ) = 0.7\)
  2. find the value of \(\alpha\) and the value of \(\beta\)
  3. find \(\operatorname { Var } ( X )\)
  4. find \(\mathrm { P } ( 5 \leqslant X \leqslant 35 )\)
Edexcel S2 2017 January Q3
16 marks Standard +0.3
3.
  1. State the condition under which the normal distribution may be used as an approximation to the Poisson distribution. The number of reported first aid incidents per week at an airport terminal has a Poisson distribution with mean 3.5
  2. Find the modal number of reported first aid incidents in a randomly selected week. Justify your answer. The random variable \(X\) represents the number of reported first aid incidents at this airport terminal in the next 2 weeks.
  3. Find \(\mathrm { P } ( X > 5 )\)
  4. Given that there were exactly 6 reported first aid incidents in a 2 week period, find the probability that exactly 4 were reported in the first week.
  5. Using a suitable approximation, find the probability that in the next 40 weeks there will be at least 120 reported first aid incidents.
Edexcel S2 2017 January Q4
10 marks Standard +0.3
  1. The time, in thousands of hours, that a certain electrical component will last is modelled by the random variable \(X\), with probability density function
$$f ( x ) = \begin{cases} \frac { 3 } { 64 } x ^ { 2 } ( 4 - x ) & 0 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ Using this model, find, by algebraic integration,
  1. the mean number of hours that a component will last,
  2. the standard deviation of \(X\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ce1f9aa7-cf16-4293-98b1-157eed35b761-06_478_974_1069_479} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of the probability density function of the random variable \(X\).
  3. Give a reason why the random variable \(X\) might be unsuitable as a model for the time, in thousands of hours, that these electrical components will last.
  4. Sketch a probability density function of a more realistic model.
Edexcel S2 2017 January Q5
14 marks Standard +0.8
  1. In the manufacture of cloth in a factory, defects occur randomly in the production process at a rate of 2 per \(5 \mathrm {~m} ^ { 2 }\)
The quality control manager randomly selects 12 pieces of cloth each of area \(15 \mathrm {~m} ^ { 2 }\).
  1. Find the probability that exactly half of these 12 pieces of cloth will contain at most 7 defects. The factory introduces a new procedure to manufacture the cloth. After the introduction of this new procedure, the manager takes a random sample of \(25 \mathrm {~m} ^ { 2 }\) of cloth from the next batch produced to test if there has been any change in the rate of defects.
    1. Write down suitable hypotheses for this test.
    2. Describe a suitable test statistic that the manager should use.
    3. Explain what is meant by the critical region for this test.
  3. Using a 5\% level of significance, find the critical region for this test. You should choose the largest critical region for which the probability in each tail is less than 2.5\%
  4. Find the actual significance level for this test.