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CAPE Applied Mathematics Unit 1 · 2012 · Paper 2

62 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)A2 marksIdentify the sampling technique used where 3 supermarkets are randomly selected and 20 workers sampled from each, and state whether it is random or non-random.
  2. 1(a)(i)B2 marksIdentify the sampling technique where workers are selected from each supermarket in proportion to the number employed there, and state whether it is random or non-random.
  3. 1(a)(i)C2 marksIdentify the sampling technique where one person is selected alphabetically from payroll and then every 6th person after that until 60 are chosen, and state whether it is random or non-random.
  4. 1(a)(ii)1 markState ONE disadvantage of using Method A.
  5. 1(a)(iii)1 markState ONE disadvantage of using Method C.
  6. 1(a)(iv)3 marksOf the 350 workers, 90 are employed at a particular location. Calculate the number of these persons that will be in the sample if Method B is used.
  7. 1(b)(i)2 marksDefine the term 'sampling frame'.
  8. 1(b)(ii)1 markExplain why the telephone directory may not be a representative sampling frame for a survey about household detergent usage.
  9. 1(b)(iii)1 markExplain why having radio listeners call in to state which station they were listening to produces a biased sample of a programme's popularity.
  10. 1(c)(i)5 marksUsing groups of 10, starting at 40, construct a stem and leaf diagram to display these data.
  11. 1(c)(ii)1 markState ONE advantage of using a stem and leaf diagram to display data.
  12. 1(c)(iii)a)1 markFrom your diagram, state the median mark obtained in the test.
  13. 1(c)(iii)b)3 marksFrom your diagram, state the inter-quartile range of the marks obtained.
  14. 2(a)(i)1 markThe 300 members of the sports club constitute a ________________________.
  15. 2(a)(ii)1 markThe conducting of a poll on all the club members is known as a _____________________.
  16. 2(a)(iii)1 markThe 25 members from which data were collected are known as a _____________________.
  17. 2(a)(iv)1 markThe average age, 24, of the 25 members is known as a _____________________.
  18. 2(a)(v)1 markThe average age, 31, of the entire club is known as a ________________________.
  19. 2(b)(i)a)1 markDetermine the mode of the distribution.
  20. 2(b)(i)b)1 markDetermine the median of the distribution.
  21. 2(b)(ii)a)3 marksCalculate the mean number of times that persons used their cell phones.
  22. 2(b)(ii)b)3 marksCalculate the standard deviation of the number of times that persons used their cell phones.
  23. 2(b)(iii)1 markDescribe the shape of the distribution.
  24. 2(c)(i)1 markUse the graph to determine the number of walkers that took part in the race.
  25. 2(c)(ii)2 marksUse the graph to determine the number of walkers that took less than 25 minutes to complete the race.
  26. 2(c)(iii)2 marksUse the graph to determine the percentage of walkers that took longer than 55 minutes to complete the race.
  27. 2(c)(iv)3 marksUse the graph to determine an approximate value of the inter-quartile range of the times that walkers took to complete the race.
  28. 2(c)(v)3 marksUse the graph to determine the value of x for which 60 per cent of the walkers took more than x minutes to complete the race.
  29. 3(a)(i)a)2 marksCalculate P(R ∪ Q).
  30. 3(a)(i)b)2 marksCalculate P(R|Q).
  31. 3(a)(ii)a)2 marksState, with reason, whether R and Q are independent.
  32. 3(a)(ii)b)2 marksState, with reason, whether R and Q are mutually exclusive.
  33. 3(b)(i)a)3 marksCalculate the probability that a randomly chosen student does neither Finance nor Economics.
  34. 3(b)(i)b)3 marksCalculate the probability that a randomly chosen student does Economics only or Finance only.
  35. 3(b)(i)c)3 marksCalculate the probability that a randomly chosen student does Economics, given that he or she does Finance.
  36. 3(b)(ii)3 marksTwo students are chosen at random from the class. Calculate the probability that one does Economics only and the other does Finance only.
  37. 3(c)(i)2 marksDetermine the probability that the bakery will receive more than 6 orders on a given day.
  38. 3(c)(ii)3 marksCalculate the mean number of orders for special cakes that the bakery will expect on any day.
  39. 4(a)(i)2 marksState the distribution and its parameters for which X can be modelled.
  40. 4(a)(ii)a)3 marksCalculate the probability that exactly THREE customers pay with cash for their petrol.
  41. 4(a)(ii)b)2 marksCalculate the probability that at least ONE customer pays with cash.
  42. 4(a)(iii)2 marksIf 40 persons buy petrol, determine the number of them that are expected to pay with cash.
  43. 4(b)(i)4 marksCalculate the probability that a cabbage chosen at random will have a weight greater than 650 grams.
  44. 4(b)(ii)5 marksTwenty per cent of cabbages were classified as large. Calculate the weight that these cabbages must exceed to be classified as large.
  45. 4(b)(iii)5 marksCalculate the probability that a cabbage weighs between 610 grams and 650 grams.
  46. 4(b)(iv)2 marksA supermarket buys cabbages between 610 and 650 grams. If a farmer has 65 cabbages for sale, determine the number expected to be sold to the supermarket.
  47. 5(a)(i)3 marksState the mean, the variance and the distribution of the mean, X̄, of the sample.
  48. 5(a)(ii)4 marksCalculate the probability that the sample mean is less than 417 grams.
  49. 5(b)4 marksCalculate a 98% confidence interval for the mean length of the pencils produced by the machine.
  50. 5(c)(i)4 marksCalculate an unbiased estimate for the standard deviation of the lifetime of the battery of the computer.
  51. 5(c)(ii)2 marksState TWO conditions that are necessary for the valid use of a t-test to test a hypothesis about the mean of X.
  52. 5(c)(iii)8 marksAssuming a t-test is valid, test at the 5% significance level whether the advertisement was overstating the battery life. Clearly state: a) null and alternative hypotheses, b) test value, c) critical region, d)…
  53. 6(a)(i)2 marksState appropriate null and alternative hypotheses for this test.
  54. 6(a)(ii)2 marksDetermine the number of degrees of freedom for the test.
  55. 6(a)(iii)2 marksDetermine the critical region for the test.
  56. 6(a)(iv)3 marksCopy and complete the table showing the expected frequencies for EACH cell.
  57. 6(a)(v)3 marksThe calculated χ² test value is 9.534. Clearly state the conclusion that can be drawn from this test.
  58. 6(b)(i)3 marksOn the graph sheet provided, plot these values as a scatter diagram.
  59. 6(b)(ii)a)2 marksInterpret the value 0.8 in the regression equation as it relates to the rainfall at the two stations.
  60. 6(b)(ii)b)3 marksCalculate the mean rainfall for Station A and for Station B.
  61. 6(b)(ii)c)3 marksDraw the regression line y = 0.8x + 0.38 on the same graph as the scatter diagram.
  62. 6(b)(ii)d)2 marksUse the regression line to estimate the rainfall at Station B when the rainfall at Station A is 4.5 cm.

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