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CAPE Applied Mathematics Unit 1 · 2014 · Paper 2

56 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)1 markState what the value 15 in the 'number of employees' row represents.
  2. 1(a)(ii)1 markState what the value 3 in the 'average weekly sales' row represents.
  3. 1(b)(i)1 markState the sampling method used when all employee names are put into a box and 20 are selected.
  4. 1(b)(ii)1 markState the sampling method used when names are listed alphabetically and, starting from the third name, every fifth name is drawn until 20 names are selected.
  5. 1(b)(iii)1 markState the sampling method used when a number of employees are randomly selected from each department proportional to its size to obtain 20 employees.
  6. 1(c)1 markIdentify which sampling method from 1(b)(i), (ii), and (iii) provides a sample that is most representative of the employees across all departments.
  7. 1(d)3 marksUsing the proportional allocation method from 1(b)(iii), calculate the number of employees in the sample of 20 drawn from the Jewellery and Perfume department.
  8. 1(e)(i)3 marksCalculate the average weekly sales for the entire store.
  9. 1(e)(ii)3 marksCalculate the standard deviation of the sales for the entire store.
  10. 1(f)(i)3 marksDetermine the number of shoppers who spent between 30 and 60 minutes in the Boutique.
  11. 1(f)(ii)2 marks60% of the shoppers spent t minutes or less in the Boutique. Find the value of t.
  12. 1(f)(iii)2 marksEstimate the median time spent in the Boutique.
  13. 1(f)(iv)3 marksUsing the provided graph sheet insert, construct a box-and-whisker diagram representing the cumulative frequency distribution.
  14. 2(a)(i)2 marksState the class boundaries of the third class (30 – 39).
  15. 2(a)(ii)2 marksCalculate the class width (size) of the third class.
  16. 2(a)(iii)1 markState one disadvantage of presenting data in a grouped frequency distribution.
  17. 2(b)(i)6 marksCalculate the estimated mean consultation time.
  18. 2(b)(ii)4 marksCalculate the estimated variance of the consultation time.
  19. 2(b)(iii)2 marksCalculate the estimated standard deviation of the consultation time.
  20. 2(c)4 marksDraw a histogram representing the consultation times using the provided graph paper.
  21. 2(d)(i)2 marksFrom the histogram or otherwise, determine an estimate for the mode of the distribution.
  22. 2(d)(ii)2 marksDetermine from the histogram or otherwise an estimate for the number of consultations that lasted 45 minutes or more.
  23. 3(a)(i)3 marksCopy and complete the tree diagram by inserting the missing probabilities.
  24. 3(a)(ii)2 marksCalculate the probability that a randomly chosen item was produced by Machine A and is defective.
  25. 3(a)(iii)3 marksShow that the probability that a randomly chosen item is defective is 0.016.
  26. 3(a)(iv)3 marksCalculate the conditional probability that a randomly selected item came from Machine A given that it is defective.
  27. 3(a)(v)4 marksTwo randomly selected items were tested. Calculate the probability that exactly one of them is defective.
  28. 3(b)(i)2 marksCalculate P(N).
  29. 3(b)(ii)2 marksCalculate P(N | M).
  30. 3(b)(iii)2 marksCalculate P(M ∩ N').
  31. 3(c)(i)2 marksState with a reason whether events M and N are mutually exclusive.
  32. 3(c)(ii)2 marksState with a reason whether events M and N are independent.
  33. 4(a)3 marksState three conditions necessary to model a variable using a binomial distribution.
  34. 4(b)(i)3 marksCalculate P(X = 3).
  35. 4(b)(ii)4 marksCalculate P(X >= 2).
  36. 4(c)5 marksThe lifespan of an insect species follows a normal distribution with mean 72 days and standard deviation 8 days. Calculate the probability that an insect lives for more than 84 days.
  37. 4(d)(i)2 marksFind the expected number of germinating seeds in the package.
  38. 4(d)(ii)2 marksFind the standard deviation of the number of seeds that will germinate.
  39. 4(d)(iii)6 marksUse an approximate distribution to calculate the probability that less than 175 seeds from the package will germinate.
  40. 5(a)(i)a)3 marksCalculate an unbiased estimator for the mean number of minutes students arrive late.
  41. 5(a)(i)b)4 marksCalculate an unbiased estimator for the variance of the minutes students arrive late.
  42. 5(a)(ii)a)2 marksState in statistical symbols the null and alternative hypotheses.
  43. 5(a)(ii)b)4 marksDetermine the critical region for the test.
  44. 5(a)(ii)c)3 marksCalculate the value of the test statistic.
  45. 5(a)(ii)d)2 marksClearly state the conclusion of this hypothesis test.
  46. 5(a)(ii)e)1 markState the assumption required to perform this test.
  47. 5(b)6 marksIn a sample of 52 students, 18 arrived late for class. Construct a 95% confidence interval for the population proportion of students who arrive late.
  48. 6(a)(i)2 marksState the null and alternative hypotheses for this test.
  49. 6(a)(ii)a)2 marksDetermine the number of degrees of freedom for the test.
  50. 6(a)(ii)b)2 marksDetermine the critical region for the test.
  51. 6(a)(iii)2 marksDetermine the expected frequency for the cell in the third row and second column (observed count 15).
  52. 6(a)(iv)2 marksThe calculated chi-square test statistic is 9.1625. State the conclusion of the test.
  53. 6(b)(i)6 marksObtain the regression equation of y on x in the form y = a + bx.
  54. 6(b)(ii)2 marksEstimate the performance score for a person whose aptitude score was 37.
  55. 6(b)(iii)2 marksInterpret the regression coefficient b in the context of the problem.
  56. 6(b)(iv)5 marksCalculate the product-moment correlation coefficient, r, and interpret its value.

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