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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 S3 2009 January Q1
4 marks Moderate -0.3
1 At a particular hospital, admissions of patients as a result of visits to the Accident and Emergency Department occur randomly at a uniform average rate of 0.75 per day. Independently, admissions that result from G.P. referrals occur randomly at a uniform average rate of 6.4 per week. The total number of admissions from these two causes over a randomly chosen period of four weeks is denoted by \(T\). State the distribution of \(T\) and obtain its expectation and variance.
OCR S3 2009 January Q2
5 marks Standard +0.3
2 The continuous random variable \(U\) has (cumulative) distribution function given by $$\mathrm { F } ( u ) = \begin{cases} \frac { 1 } { 5 } \mathrm { e } ^ { u } & u < 0 \\ 1 - \frac { 4 } { 5 } \mathrm { e } ^ { - \frac { 1 } { 4 } u } & u \geqslant 0 \end{cases}$$
  1. Find the upper quartile of \(U\).
  2. Find the probability density function of \(U\).
OCR S3 2009 January Q3
8 marks Moderate -0.3
3 In a random sample of credit card holders, it was found that \(28 \%\) of them used their card for internet purchases.
  1. Given that the sample size is 1200 , find a \(98 \%\) confidence interval for the percentage of all credit card holders who use their card for internet purchases.
  2. Estimate the smallest sample size for which a \(98 \%\) confidence interval would have a width of at most \(5 \%\), and state why the value found is only an estimate.
OCR S3 2009 January Q4
7 marks Standard +0.3
4 The weekly sales of petrol, \(X\) thousand litres, at a garage may be modelled by a continuous random variable with probability density function given by $$f ( x ) = \begin{cases} c & 25 \leqslant x \leqslant 45 \\ 0 & \text { otherwise } \end{cases}$$ where \(c\) is a constant. The weekly profit, in \(\pounds\), is given by \(( 400 \sqrt { X } - 240 )\).
  1. Obtain the value of \(c\).
  2. Find the expected weekly profit.
  3. Find the probability that the weekly profit exceeds \(\pounds 2000\).
OCR S3 2009 January Q5
10 marks Standard +0.8
5 The concentration level of mercury in a large lake is known to have a normal distribution with standard deviation 0.24 in suitable units. At the beginning of June 2008, the mercury level was measured at five randomly chosen places on the lake, and the sample mean is denoted by \(\bar { x } _ { 1 }\). Towards the end of June 2008 there was a spillage in the lake which may have caused the mercury level to rise. Because of this the level was then measured at six randomly chosen points of the lake, and the mean of this sample is denoted by \(\bar { x } _ { 2 }\).
  1. State hypotheses for a test based on the two samples for whether, on average, the level of mercury had increased. Define any parameters that you use.
  2. Find the set of values of \(\bar { x } _ { 2 } - \bar { x } _ { 1 }\) for which there would be evidence at the 5\% significance level that, on average, the level of mercury had increased.
  3. Given that the average level had actually increased by 0.3 units, find the probability of making a Type II error in your test, and comment on its value.
OCR S3 2009 January Q6
13 marks Standard +0.3
6 A mathematics examination is taken by 29 boys and 26 girls. Experience has shown that the probability that any boy forgets to bring a calculator to the examination is 0.3 , and that any girl forgets is 0.2 . Whether or not any student forgets to bring a calculator is independent of all other students. The numbers of boys and girls who forget to bring a calculator are denoted by \(B\) and \(G\) respectively, and \(F = B + G\).
  1. Find \(\mathrm { E } ( F )\) and \(\operatorname { Var } ( F )\).
  2. Using suitable approximations to the distributions of \(B\) and \(G\), which should be justified, find the smallest number of spare calculators that should be available in order to be at least \(99 \%\) certain that all 55 students will have a calculator.
OCR S3 2009 January Q7
11 marks Standard +0.3
7 A tutor gives a randomly selected group of 8 students an English Literature test, and after a term's further teaching, she gives the group a similar test. The marks for the two tests are given in the table.
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
First test3827554332245146
Second test3726574330265448
  1. Stating a necessary condition, show by carrying out a suitable \(t\)-test, at the \(1 \%\) significance level, that the marks do not give evidence of an improvement.
  2. The tutor later found that she had marked the second test too severely, and she decided to add a constant amount \(k\) to each mark. Find the least integer value of \(k\) for which the increased marks would give evidence of improvement at the \(1 \%\) significance level.
OCR S3 2009 January Q8
14 marks Standard +0.3
8 A soft drinks factory produces lemonade which is sold in packs of 6 bottles. As part of the factory's quality control, random samples of 75 packs are examined at regular intervals. The number of underfilled bottles in a pack of 6 bottles is denoted by the random variable \(X\). The results of one quality control check are shown in the following table.
Number of underfilled bottles0123
Number of packs442083
A researcher assumes that \(X \sim \mathrm {~B} ( 3 , p )\).
  1. By finding the sample mean, show that an estimate of \(p\) is 0.2 .
  2. Show that, at the \(5 \%\) significance level, there is evidence that this binomial distribution does not fit the data.
  3. Another researcher suggests that the goodness of fit test should be for \(\mathrm { B } ( 6 , p )\). She finds that the corresponding value of \(\chi ^ { 2 }\) is 2.74 , correct to 3 significant figures. Given that the number of degrees of freedom is the same as in part (ii), state the conclusion of the test at the same significance level.
OCR S3 2010 January Q1
8 marks Moderate -0.3
1 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 2 } { 5 } & - a \leqslant x < 0 \\ \frac { 2 } { 5 } \mathrm { e } ^ { - 2 x } & x \geqslant 0 \end{cases}$$ Find
  1. the value of the constant \(a\),
  2. \(\mathrm { E } ( X )\).
OCR S3 2010 January Q2
8 marks Moderate -0.3
2 The amount of tomato juice, \(X \mathrm { ml }\), dispensed into cartons of a particular brand has a normal distribution with mean 504 and standard deviation 3 . The juice is sold in packs of 4 cartons, filled independently. The total amount of juice in one pack is \(Y \mathrm { ml }\).
  1. Find \(\mathrm { P } ( Y < 2000 )\). The random variable \(V\) is defined as \(Y - 4 X\).
  2. Find \(\mathrm { E } ( V )\) and \(\operatorname { Var } ( V )\).
  3. What is the probability that the amount of juice in a randomly chosen pack is more than 4 times the amount of juice in a randomly chosen carton?
OCR S3 2010 January Q3
10 marks Standard +0.3
3 It is given that \(X _ { 1 }\) and \(X _ { 2 }\) are independent random variables with \(X _ { 1 } \sim \mathrm {~N} \left( \mu _ { 1 } , 2.47 \right)\) and \(X _ { 2 } \sim \mathrm {~N} \left( \mu _ { 2 } , 4.23 \right)\). Random samples of \(n _ { 1 }\) observations of \(X _ { 1 }\) and \(n _ { 2 }\) observations of \(X _ { 2 }\) are taken. The sample means are denoted by \(\bar { X } _ { 1 }\) and \(\bar { X } _ { 2 }\).
  1. State the distribution of \(\bar { X } _ { 1 } - \bar { X } _ { 2 }\), giving its parameters. For two particular samples, \(n _ { 1 } = 5 , \Sigma x _ { 1 } = 48.25 , n _ { 2 } = 10\) and \(\Sigma x _ { 2 } = 72.30\).
  2. Test at the \(2 \%\) significance level whether \(\mu _ { 1 }\) differs from \(\mu _ { 2 }\). A student stated that because of the Central Limit Theorem the sample means will have normal distributions so it is unnecessary for \(X _ { 1 }\) and \(X _ { 2 }\) to have normal distributions.
  3. Comment on the student's statement.
OCR S3 2010 January Q4
9 marks Standard +0.8
4 The continuous random variable \(V\) has (cumulative) distribution function given by $$\mathrm { F } ( v ) = \begin{cases} 0 & v < 1 \\ 1 - \frac { 8 } { ( 1 + v ) ^ { 3 } } & v \geqslant 1 \end{cases}$$ The random variable \(Y\) is given by \(Y = \frac { 1 } { 1 + V }\).
  1. Show that the (cumulative) distribution function of \(Y\) is \(8 y ^ { 3 }\), over an interval to be stated, and find the probability density function of \(Y\).
  2. Find \(\mathrm { E } \left( \frac { 1 } { Y ^ { 2 } } \right)\).
OCR S3 2010 January Q5
11 marks Standard +0.3
5 Each of a random sample of 200 steel bars taken from a production line was examined and 27 were found to be faulty.
  1. Find an approximate \(90 \%\) confidence interval for the proportion of faulty bars produced. A change in the production method was introduced which, it was claimed, would reduce the proportion of faulty bars. After the change, each of a further random sample of 100 bars was examined and 8 were found to be faulty.
  2. Test the claim, at the \(10 \%\) significance level.
OCR S3 2010 January Q6
12 marks Standard +0.3
6 The deterioration of a certain drug over time was investigated as follows. The drug strength was measured in each of a random sample of 8 bottles containing the drug. These were stored for two years and the strengths were then re-measured. The original and final strengths, in suitable units, are shown in the following table.
Bottle12345678
Original strength8.79.49.28.99.68.29.98.8
Final strength8.19.09.08.89.38.09.58.5
  1. Stating any required assumption, test at the \(5 \%\) significance level whether the mean strength has decreased by more than 0.2 over the two years.
  2. Calculate a 95\% confidence interval for the mean reduction in strength over the two years.
OCR S3 2010 January Q7
14 marks Standard +0.3
7 A chef wished to ascertain her customers' preference for certain vegetables. She asked a random sample of 120 customers for their preferred vegetable from asparagus, broad beans and cauliflower. The responses, classified according to the gender of the customer, are shown in the table.
  1. Test, at the \(5 \%\) significance level, whether vegetable preference and gender are independent.
  2. Determine whether, at the \(10 \%\) significance level, the vegetables are equally preferred.
OCR S3 2013 January Q1
6 marks Standard +0.3
1 The independent random variables \(X\) and \(Y\) have the distributions \(\mathrm { N } \left( 10 , \sigma ^ { 2 } \right)\) and \(\operatorname { Po } ( 2 )\) respectively. The random variable \(S\) is given by \(S = 5 X - 2 Y + c\), where \(c\) is a constant.
It is given that \(\mathrm { E } ( S ) = \operatorname { Var } ( S ) = 408\).
  1. Find the value of \(c\) and show that \(\sigma ^ { 2 } = 16\).
  2. Find \(\mathrm { P } ( X \geqslant \mathrm { E } ( Y ) )\).
OCR S3 2013 January Q2
8 marks Standard +0.3
2 A new running track has been developed and part of the testing procedure involves 7 randomly chosen athletes. They each run 100 m on both the old and new tracks.
The results are as follows.
Athlete1234567
Time on old track \(( s )\)12.210.311.513.011.811.711.9
Time on new track \(( s )\)11.110.511.012.611.010.912.0
The population mean times on the old and new tracks are denoted by \(\mu _ { \mathrm { O } }\) seconds and \(\mu _ { \mathrm { N } }\) seconds respectively. Stating any necessary assumption, carry out a suitable \(t\)-test of the null hypothesis \(\mu _ { \mathrm { O } } - \mu _ { \mathrm { N } } = 0\) against the alternative hypothesis \(\mu _ { \mathrm { O } } - \mu _ { \mathrm { N } } > 0\). Use a \(2 \frac { 1 } { 2 } \%\) significance level .
OCR S3 2013 January Q3
7 marks Standard +0.3
3 Two reading schemes, \(A\) and \(B\), are compared by using them with a random sample of 9 five-year-old children. The children are divided into two groups, 5 allotted to scheme \(A\) and 4 to scheme \(B\), and the schemes are taught under similar conditions.
After one year the children are given the same test and their scores, \(x _ { A }\) and \(x _ { B }\), are summarised below. With the usual notation, $$\begin{aligned} & n _ { A } = 5 , \bar { x } _ { A } = 52.0 , \sum \left( x _ { A } - \bar { x } _ { A } \right) ^ { 2 } = 248 , \\ & n _ { B } = 4 , \bar { x } _ { B } = 56.5 , \sum \left( x _ { B } - \bar { x } _ { B } \right) ^ { 2 } = 381 . \end{aligned}$$ It may be assumed that scores have normal distributions.
  1. Calculate an \(80 \%\) confidence interval for the difference in population mean scores for the two methods.
  2. State a further assumption required for the validity of the interval.
OCR S3 2013 January Q4
9 marks Challenging +1.2
4 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 2 } \sqrt { x } & 0 < x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is given by \(Y = \frac { 1 } { \sqrt { X } }\).
  1. Find the (cumulative) distribution function of \(Y\), and hence show that its probability density function is given by $$\mathrm { g } ( y ) = \frac { 3 } { y ^ { 4 } }$$ for a set of values of \(y\) to be stated.
  2. Find the value of \(\mathrm { E } \left( Y ^ { 2 } \right)\).
OCR S3 2013 January Q5
9 marks Standard +0.3
5 A constitutional change was proposed for a Golf Club with a large membership. This was to be voted on at the Annual General Meeting. A month before this meeting the secretary asked a random sample of 50 members for their opinions. Out of the 50 members \(70 \%\) said they approved.
  1. Calculate an approximate \(90 \%\) confidence interval for the proportion \(p\) of all members who would approve the proposal.
  2. Explain what is meant by a \(90 \%\) confidence interval in this context.
  3. Nearer the date of the meeting the secretary asked a random sample of \(n\) members, and, as before, \(70 \%\) said they approved. This time the secretary calculated an approximate \(99 \%\) confidence interval for \(p\). It is given that the confidence interval does not include 0.85 . Find the smallest possible value of \(n\).