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CAPE Applied Mathematics Unit 1 · 2008 (second paper) · Paper 2

51 questions and parts from this paper. Open one to see it in full, then practise it on Quelpr and get it marked against the mark scheme.

Practise this paper
  1. 1(a)(i)3 marksGive THREE reasons for inspecting a sample of the printed newspapers rather than the entire population.
  2. 1(a)(ii)4 marksState TWO differences between a cluster sample and a stratified random sample in this context.
  3. 1(a)(iii)3 marksCalculate to the nearest integer the number of papers to select from Monday's production using stratified random sampling, given a total sample size of 5000.
  4. 1(b)6 marksConstruct a well-labelled pie chart representing the daily newspaper production, using a circle radius of 4 cm and angles rounded to the nearest whole degree.
  5. 1(c)(i)4 marksCalculate the mean daily newspaper production.
  6. 1(c)(ii)5 marksCalculate the standard deviation of the daily newspaper production.
  7. 2(a)4 marksConstruct the cumulative distribution table for the given data.
  8. 2(b)4 marksOn the provided answer graph sheet, plot the cumulative frequency curve representing the cumulative distribution table.
  9. 2(c)(i)2 marksUse the cumulative frequency curve to estimate the number of people whose commute time is under 35 minutes.
  10. 2(c)(ii)2 marksUse the cumulative frequency curve to estimate the median commuting time.
  11. 2(c)(iii)3 marksUse the cumulative frequency curve to estimate the interquartile range of the commute times.
  12. 2(d)3 marksConstruct a box-and-whisker plot displaying the commute times data.
  13. 2(e)5 marksCalculate an estimate of the mean daily commute time.
  14. 2(f)2 marksState the shape of the distribution of the commute times.
  15. 3(a)(i)2 marksDetermine the probability of selecting a green candle when one is chosen at random.
  16. 3(a)(ii)a)2 marksThree candles are randomly selected from the box. Using R for red and G for green, list the elements of the sample space.
  17. 3(a)(ii)b)3 marksDetermine the probability that exactly two of the three candles selected are red.
  18. 3(b)(i)2 marksState, with a statistical reason, whether events A and B are mutually exclusive.
  19. 3(b)(ii)2 marksState, with a statistical reason, whether events A and B are independent.
  20. 3(c)(i)3 marksFind the probability that a randomly chosen student studies neither French nor Spanish.
  21. 3(c)(ii)2 marksFind the probability that a randomly chosen student studies French only.
  22. 3(c)(iii)2 marksFind the probability that a student studies French given that they study Spanish.
  23. 3(d)(i)2 marksShow that X is a valid random variable.
  24. 3(d)(ii)5 marksCalculate the mean and standard deviation of X.
  25. 4(a)3 marksState THREE conditions required for a random variable to follow a binomial distribution.
  26. 4(b)(i)2 marksState with reason whether drawing three discs with replacement from a bag containing 4 yellow, 3 blue, and 2 white discs can be modelled by a binomial distribution.
  27. 4(b)(ii)2 marksState with reason whether drawing three discs without replacement from a bag containing 5 yellow and 4 blue discs can be modelled by a binomial distribution.
  28. 4(b)(iii)2 marksState with reason whether drawing three discs with replacement from a bag containing 5 yellow and 4 blue discs can be modelled by a binomial distribution.
  29. 4(c)(i)2 marksFor a group of 40 potential donors, calculate the expected number turned away due to tattoos.
  30. 4(c)(ii)a)3 marksFor a random sample of 7 prospective donors, calculate the probability that none of them have tattoos.
  31. 4(c)(ii)b)3 marksFor a random sample of 7 prospective donors, calculate the probability that exactly 3 of them have tattoos.
  32. 4(c)(ii)c)2 marksFor a random sample of 7 prospective donors, calculate the probability that at least one of them has tattoos.
  33. 4(d)6 marksThe mass of a loaf of bread follows a normal distribution with mean 480 g and standard deviation 20 g. Calculate the probability that a randomly chosen loaf has mass between 465 g and 500 g.
  34. 5(a)(i)a)2 marksCalculate the unbiased estimate of the population mean.
  35. 5(a)(i)b)3 marksCalculate the unbiased estimate of the population standard deviation.
  36. 5(a)(ii)a)3 marksFor testing H0: mu = 45 versus H1: mu < 45 at the 5% significance level, determine the critical region.
  37. 5(a)(ii)b)3 marksCalculate the value of the test statistic for this hypothesis test.
  38. 5(a)(ii)c)1 markClearly state the conclusion of the test.
  39. 5(b)(i)2 marksCalculate the theoretical probability of rolling a sum of 9 with a pair of fair dice.
  40. 5(b)(ii)2 marksState in symbols the null and alternative hypotheses for testing if the dice yield fewer 9s than expected.
  41. 5(b)(iii)3 marksDetermine the critical region for this hypothesis test at the 5% significance level.
  42. 5(b)(iv)5 marksCalculate the appropriate test statistic.
  43. 5(b)(v)1 markClearly state the conclusion drawn from the test.
  44. 6(a)5 marksPlot the scatter diagram for the distance and travel time data on the provided answer sheet.
  45. 6(b)5 marksCalculate and interpret the product-moment correlation coefficient between distance and travel time.
  46. 6(c)(i)4 marksFind the equation of the regression line of y on x in the form y = a + bx.
  47. 6(c)(ii)3 marksDraw the regression line on the scatter diagram.
  48. 6(d)(i)1.5 marksInterpret the value of the regression coefficient b in the context of the problem.
  49. 6(d)(ii)1.5 marksInterpret the value of the constant a in the context of the problem.
  50. 6(e)(i)3 marksEstimate the travel time for a delegate who travelled 145 km by car.
  51. 6(e)(ii)2 marksComment briefly on the reliability of the estimated travel time.

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