Edexcel S1 — Question 5

Exam BoardEdexcel
ModuleS1 (Statistics 1)
TopicConditional Probability
TypeIndependence test requiring preliminary calculations
  1. The events \(A\) and \(B\) are such that \(\mathrm { P } ( A \cap B ) = 0.24 , \mathrm { P } ( A \cup B ) = 0.88\) and \(\mathrm { P } ( B ) = 0.52\).
    1. Find \(\mathrm { P } ( A )\).
    2. Determine, with reasons, whether \(A\) and \(B\) are
      1. mutually exclusive,
      2. independent.
    3. Find \(\mathrm { P } ( B \mid A )\).
    4. Find \(\mathrm { P } \left( A ^ { \prime } \mid B ^ { \prime } \right)\).
    5. The times taken by a group of people to complete a task are modelled by a normal distribution with mean 8 hours and standard deviation 2 hours.
      Use this model to calculate
    6. the probability that a person chosen at random took between 5 and 9 hours to complete the task,
    7. the range, symmetrical about the mean, within which \(80 \%\) of the people's times lie.
      (5 marks)
      It is found that, in fact, \(80 \%\) of the people take more than 5 hours. The model is modified so that the mean is still 8 hours but the standard deviation is no longer 2 hours.
    8. Find the standard deviation of the times in the modified model.
    9. The following data was collected for seven cars, showing their engine size, \(x\) litres, and their fuel consumption, \(y \mathrm {~km}\) per litre, on a long journey.
    Car\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)
    \(x\)0.951.201.371.762.252.502.875
    \(y\)21.317.215.519.114.711.49.0
    \(\sum x = 12 \cdot 905 , \sum x ^ { 2 } = 26 \cdot 8951 , \sum y = 108 \cdot 2 , \sum y ^ { 2 } = 1781 \cdot 64 , \sum x y = 183 \cdot 176\).
  2. Calculate the equation of the regression line of \(x\) on \(y\), expressing your answer in the form \(x = a y + b\).
  3. Calculate the product moment correlation coefficient between \(y\) and \(x\) and give a brief interpretation of its value.
  4. Use the equation of the regression line to estimate the value of \(x\) when \(y = 12\). State, with a reason, how accurate you would expect this estimate to be.
  5. Comment on the use of the line to find values of \(x\) as \(y\) gets very small.
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