CAPE Applied Mathematics Unit 1 · 2013 · Paper 2

57 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)(i)1.5 marksIndicate whether the brands of coffee served at a restaurant is quantitative or qualitative, and if quantitative, whether it is discrete or continuous.
2. 1(a)(ii)1.5 marksIndicate whether the number of television sets owned by households in a district is quantitative or qualitative, and if quantitative, whether it is discrete or continuous.
3. 1(a)(iii)1.5 marksIndicate whether the volume of water used weekly by a laundry is quantitative or qualitative, and if quantitative, whether it is discrete or continuous.
4. 1(a)(iv)1.5 marksIndicate whether the types of fish sold in the market is quantitative or qualitative, and if quantitative, whether it is discrete or continuous.
5. 1(b)(i)1 markState whether the 5 buses owned by "Village Tours" is a population or a sample.
6. 1(b)(ii)1 markState whether 5 of the buses that were sent on the island tour last week is a population or a sample.
7. 1(c)(i)1 markState whether asking the 500 persons who attended the jazz show on Thursday night to name their favourite performer is a census or a sample survey.
8. 1(c)(ii)1 markState whether asking 500 of the persons who attended the jazz concert last week to name their favourite performer is a census or a sample survey.
9. 1(d)2 marksState TWO reasons why it may be necessary to conduct a sample survey rather than use a census.
10. 1(e)(i)2 marksExplain why a stratified random sample will NOT be an appropriate method to use.
11. 1(e)(ii)6 marksExplain how a random number table could be used to select the sample of 15 students from the programme.
12. 1(e)(iii)5 marksUse Table 4 (Random Sampling Numbers) provided in the Statistical Tables to obtain a random sample of size 15, without replacement, from numbers 00 to 90 inclusive, starting at the third row from the top with 72 and…
13. 2(a)(i)2 marksState the boundaries of the modal class.
14. 2(a)(ii)3 marksApproximately what percentage of students in the class is taller than 67 inches?
15. 2(a)(iii)4 marksCalculate the mean height of the students in the class.
16. 2(a)(iv)4 marksCalculate the standard deviation of the heights of the students in the class.
17. 2(b)(i)1 markWhat advantage does preparing a stem and leaf diagram have over grouping a data set using a grouped frequency distribution?
18. 2(b)(ii)a)1 markHow many patients were in the sample?
19. 2(b)(ii)b)2 marksDetermine the median waiting time for the sample.
20. 2(b)(ii)c)3 marksCalculate the inter-quartile range for the data.
21. 2(b)(ii)d)3 marksOn the graph paper provided, draw the box and whiskers diagram to show this data.
22. 2(b)(ii)e)2 marksDescribe the shape of the distribution of the data.
23. 3(a)(i)a)3 marksFind P(R ∩ T).
24. 3(a)(i)b)3 marksFind P(R ∪ T).
25. 3(a)(ii)3 marksDraw a clearly labelled Venn diagram showing the sets (R ∩ T'), (R ∩ T), (R' ∩ T), (R ∪ T)' and their respective probabilities.
26. 3(a)(iii)a)2 marksFind P(Q).
27. 3(a)(iii)b)2 marksShow that R and Q are mutually exclusive.
28. 3(b)(i)2 marksHow many people were in the survey?
29. 3(b)(ii)a)3 marksDetermine the probability that a randomly selected patron thought that there was an improvement in service.
30. 3(b)(ii)b)3 marksDetermine the probability that the patron was a female and she thought that there was no improvement in the service.
31. 3(b)(ii)c)4 marksGiven that the patron thought there was an improvement, what is the probability that the patron was a male?
32. 4(a)(i)2 marksDetermine the value of a.
33. 4(a)(ii)a)3 marksCalculate E[X].
34. 4(a)(ii)b)3 marksCalculate Var[X].
35. 4(a)(ii)c)2 marksCalculate P(X ≥ 2).
36. 4(b)(i)2 marksCalculate the value of k.
37. 4(b)(ii)2 marksShow that P(X > 2) = 4/5.
38. 4(c)(i)3 marksWhat is the probability that he sells exactly 3 defective tyres in the next 10 tyres of that brand?
39. 4(c)(ii)2 marksThe supplier expects a shipment of 500 tyres of that brand. How many of these tyres will be expected to be defective?
40. 4(c)(iii)6 marksUsing an approximate distribution, calculate an estimate for the probability that a shipment of 500 tyres will have more than 90 defective tyres.
41. 5(a)(i)3 marksState the distribution of the sample mean, X̄, giving the value of its parameters.
42. 5(a)(ii)5 marksCalculate P(X̄ ≥ 38.7).
43. 5(b)(i)4 marksCalculate a 94 per cent confidence interval for the true mean mass, µ, of the packages.
44. 5(b)(ii)2 marks40 random samples of the packages of biscuits were taken, and a 94 per cent confidence interval for µ was found for each. Determine the expected number of intervals that will contain µ.
45. 5(c)(i)2 marksState TWO conditions why a t-test will be appropriate to test the hypothesis that the machine is not dispensing on average 10 ounces of coffee.
46. 5(c)(ii)2 marksFormulate suitable null and alternative hypotheses (using statistical symbols) to test whether the mean amount of coffee dispensed is 10 ounces or whether it is less than 10 ounces.
47. 5(c)(iii)7 marksTest at the 5% level of significance whether the machine is dispensing on average 10 ounces of coffee.
48. 6(a)(i)2 marksInterpret the value of 0.09 in this equation.
49. 6(a)(ii)2 marksInterpret the value of 6.6 in this equation.
50. 6(a)(iii)2 marksUse the regression equation to estimate the value of y when x = 15.
51. 6(a)(iv)2 marksThe product moment correlation coefficient for the data is 0.32. Interpret this value relative to the variables x and y.
52. 6(b)(i)2 marksState appropriate null and alternative hypotheses for this test.
53. 6(b)(ii)4 marksCopy and complete the table of observed (O) and expected (E) frequencies of the grades obtained in the examination.
54. 6(b)(iii)a)2 marksDetermine the number of degrees of freedom for this hypothesis test.
55. 6(b)(iii)b)2 marksDetermine the critical region of the test.
56. 6(b)(iii)c)5 marksCalculate the χ² test statistic.
57. 6(b)(iv)2 marksClearly state the conclusion which may be drawn from this test.