CAPE Applied Mathematics Unit 1 · 2021 · Paper 2

63 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.

1. 1(a)2 marksState ONE similarity and ONE difference between a stratified random sample and a quota sample.
2. 1(b)(i)1 markIdentify the sampling method T: selecting equal numbers of male and female contestants, even though there are twice as many female applicants as male applicants.
3. 1(b)(ii)1 markIdentify the sampling method U: grouping applicants by age and selecting contestants at random from within each group in proportion to the number of applicants in that group.
4. 1(b)(iii)1 markIdentify the sampling method V: assigning a unique number from 001 to 500 to each applicant and using random numbers to select the contestants.
5. 1(b)(iv)1 markIdentify the sampling method W: selecting a contestant randomly from among the first k applicants on a numbered list, then selecting every kth applicant on the list.
6. 1(b)(v)1 markIdentify by letter the sampling method among T, U, V, and W that would result in a sample that is MOST representative of the 500 applicants.
7. 1(b)(vi)2 marksIf sampling method U is used, 4 of the 20 contestants selected from the 500 applicants are over 50 years old. Determine the total number of applicants who are older than 50 years of age.
8. 1(b)(vii)3 marksUsing a random sampling numbers table, starting at row 1, column 3 with 36 and moving from left to right along the row, list the first FIVE numbers selected to choose 5 of the 20 contestants under method V.
9. 1(b)(viii)2 marksCalculate the value of k in sampling method W when selecting 20 contestants from 500 applicants.
10. 1(c)(i)1 markComplete the cumulative frequency row in the provided table.
11. 1(c)(ii)4 marksOn the grid provided, draw a cumulative frequency curve to represent the data in the table, using a scale of 2 cm to represent 10 minutes on the x-axis and 2 cm to represent 10 children on the y-axis.
12. 1(c)(iii)1 markUsing your cumulative frequency curve, estimate the median time taken to complete the reading assignment.
13. 1(c)(iv)3 marksUsing your cumulative frequency curve, estimate the interquartile range.
14. 1(c)(v)2 marksUsing your cumulative frequency curve, estimate the proportion of children who took less than 35 minutes to complete the reading assignment.
15. 2(a)(i)1 markIdentify the measure of variability that is influenced most by extreme values.
16. 2(a)(ii)1 markIdentify the measure of variability obtained by squaring the standard deviation.
17. 2(a)(iii)1 markIdentify the measure of variability that represents the middle 50 per cent of the distribution.
18. 2(b)(i)4 marksCalculate the 10% trimmed mean of the scores.
19. 2(b)(ii)1 markDetermine the modal score(s) of the examination data.
20. 2(b)(iii)3 marksCalculate the interquartile range of the 20 scores.
21. 2(b)(iv)4 marksCalculate the standard deviation of the scores.
22. 2(b)(v)2 marksInterpret the value of the standard deviation calculated in (b)(iv).
23. 2(c)(i)2 marksDetermine the median number of eggs per tray.
24. 2(c)(ii)1 markDetermine the mode of the distribution of eggs per tray.
25. 2(c)(iii)4 marksCalculate the mean number of eggs per tray.
26. 2(c)(iv)1 markDescribe the shape of the distribution.
27. 3(a)(i)3 marksCalculate P(A ∩ B).
28. 3(a)(ii)3 marksCalculate P(B).
29. 3(a)(iii)2 marksCalculate P(A ∪ B).
30. 3(a)(iv)2 marksState whether Event A and Event B are independent, giving ONE reason for your answer.
31. 3(b)(i)3 marksCalculate the value of the constant b.
32. 3(b)(ii)3 marksDetermine the expected value of X, E(X).
33. 3(b)(iii)3 marksDetermine the variance of X, Var(X).
34. 3(c)(i)4 marksDraw a probability tree diagram to represent the scenario.
35. 3(c)(ii)2 marksUse the tree diagram to determine the probability that the counters drawn are of different colours.
36. 4(a)(i)1 markIdentify the distribution that may be used to model this situation.
37. 4(a)(ii)1 markState the parameters of the distribution.
38. 4(a)(iii)2 marksDetermine the expected number of customers in the sample who do not make their monthly payments on time.
39. 4(a)(iv)2 marksCalculate the probability that exactly 5 customers do not make their monthly payments on time.
40. 4(a)(v)4 marksCalculate the probability that at most 3 customers do not make their monthly payments on time.
41. 4(a)(vi)7 marksFor a sample of 100 customers, use a suitable approximation to determine the probability that no more than 35 customers make their monthly payments on time, stating all your assumptions.
42. 4(b)(i)4 marksDetermine the probability that a javelin selected at random will measure more than 255 cm.
43. 4(b)(ii)4 marksJavelins less than 220 cm are cut and sold as fence posts. Calculate the percentage of javelins that will have to be cut and sold as fence posts.
44. 5(a)(i)5 marksCalculate unbiased estimates for the mean and standard deviation of X.
45. 5(a)(ii)2 marksState an appropriate distribution for the sample mean, X-bar.
46. 5(a)(iii)3 marksDetermine an approximate 95% confidence interval for the unknown population mean, mu.
47. 5(a)(iv)2 marksFormulate the null and alternative hypotheses.
48. 5(a)(v)1 markDetermine the critical value for the test.
49. 5(a)(vi)1 markDetermine the rejection region.
50. 5(a)(vii)2 marksGiven that the test statistic is -1.59, state clearly a valid conclusion for the test.
51. 5(b)(i)2 marksIdentify the distribution that can be used to model this situation, giving ONE reason.
52. 5(b)(ii)2 marksDetermine the critical value.
53. 5(b)(iii)1 markDetermine the rejection region.
54. 5(b)(iv)2 marksGiven that the mean is 30.5 and the standard deviation is 10.6, calculate the test statistic.
55. 5(b)(v)2 marksState clearly a valid conclusion for the test.
56. 6(a)(i)2 marksFormulate the null and alternative hypotheses for the chi-square test.
57. 6(a)(ii)4 marksComplete the table to show the missing expected frequencies and marginal totals.
58. 6(a)(iii)3 marksCalculate the chi-square test statistic value.
59. 6(a)(iv)3 marksDetermine the critical region of the test at the 5% level of significance.
60. 6(a)(v)2 marksState clearly the conclusion that may be drawn from this test.
61. 6(b)4 marksA random sample of 384 provided a 95% confidence interval for a population proportion, p, with a margin of error of 0.05. Approximate the value of p based on this information.
62. 6(c)(i)3 marksCalculate the sample mean and sample variance for these observations.
63. 6(c)(ii)4 marksDetermine the critical region for a test of H0: mu = 14.0 against H1: mu != 14.0 at the 5% significance level.