Questions — OCR MEI S2 (80 questions)

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OCR MEI S2 2016 June Q1
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
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
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
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
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?