CAPE Applied Mathematics Unit 1 · 2009 (second paper) · Paper 2

41 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(d)(i)a)1 markDetermine the mode of the distribution.
2. 1(d)(i)b)1 markDetermine the median of the distribution.
3. 1(d)(ii)a)2 marksCalculate the mean number of hours that students slept.
4. 1(d)(ii)b)2 marksCalculate the inter-quartile range of the hours slept.
5. 1(d)(iii)3 marksUsing the graph sheet provided, draw a box and whisker diagram to show this information.
6. 1(d)(iv)1 markState the shape of the distribution.
7. 3(a)(i)2 marksCalculate P(A ∪ B).
8. 3(a)(ii)2 marksDetermine P(A | B).
9. 3(b)(i)2 marksCalculate P(G ∩ H).
10. 3(b)(ii)2 marksCalculate P(G ∪ H).
11. 3(c)(i)4 marksShow this information on a well-labelled tree diagram.
12. 3(c)(ii)3 marksCalculate the probability that a randomly selected article is defective.
13. 3(c)(iii)3 marksGiven that a randomly selected item is defective, determine the probability that it was produced by machine A.
14. 3(d)(i)2 marksCalculate the value of k.
15. 3(d)(ii)3 marksCalculate the number of overtime hours that attendants will be expected to work each week.
16. 3(d)(iii)2 marksCalculate the probability that an attendant will work less than 8 hours overtime in any week.
17. 4(a)(i)1 markA card is taken from an ordinary pack of cards, its suite (Ace, Spade, Diamond, Heart) noted, and it is replaced in the pack. The process is repeated 12 times. X is the number of spades that were examined.
18. 4(a)(ii)1 markA fair die is tossed. This process is repeated a number of times. X is the number of throws of the fair die until a 6 is obtained.
19. 4(a)(iii)1 mark10 seeds are planted. The probability of a seed germinating is 0.8. X is the number of seeds that germinate.
20. 4(a)(iv)1 markA bag contains 8 white and 12 red marbles, all identical except for colour. A marble is taken from the bag and placed in another bag. This process is done 6 times. X is the number of red marbles taken.
21. 4(b)(i)2 marksDetermine the expected number of patients in the sample who would not keep their follow-up appointment.
22. 4(b)(ii)a)3 marksCalculate the probability that exactly 4 patients did not keep the follow-up appointment.
23. 4(b)(ii)b)3 marksCalculate the probability that at most 2 patients did not keep their follow-up appointment.
24. 4(c)3 marksThe random variable X has a normal distribution with mean 12 and variance 4. Find the constant c such that P(X < c) = 0.9332.
25. 4(d)(i)5 marksWhat is the probability that a rod selected at random will measure more than 65 centimetres?
26. 4(d)(ii)5 marksRods that are less than 54 centimetres are cut and sold as a smaller length. What percentage of rods will have to be cut?
27. 5(a)2 marksExplain what is meant by the term "a 95% confidence interval" in the context of a population mean.
28. 5(b)(i)2 marksDetermine an unbiased estimate of the standard deviation.
29. 5(b)(ii)4 marksConstruct, without using a continuity correction, a 95% confidence interval for the mean amount of money saved.
30. 5(b)(iii)a)2 marksIf the consultant considers that a 95% confidence interval is too wide, state TWO methods that can be taken to reduce this width.
31. 5(b)(iii)b)2 marksState which method is better and give a reason for your answer.
32. 5(b)(iv)4 marksDetermine the least number of farmers that must be sampled so that the width of a 99% confidence interval is less than 40.
33. 5(c)2 marksA 90% confidence interval for population mean, μ, is found for each sample when 60 random samples of size 45 are taken. Determine the expected number of intervals that do NOT contain μ.
34. 5(d)(i)2 marksState the distribution modelled by X̄ giving its parameters.
35. 5(d)(ii)5 marksCalculate P(X̄ > 4.5).
36. 6(a)(i)2 marksState appropriate null and alternative hypotheses.
37. 6(a)(ii)11 marksOn the answer sheet provided, complete the table where O denotes the observed frequency, E denotes the expected frequency and N denotes the total frequency, 400, when the null hypothesis is true.
38. 6(a)(iii)a)2 marksDetermine the number of degrees of freedom for this hypothesis test.
39. 6(a)(iii)b)2 marksDetermine the critical region for this hypothesis test.
40. 6(a)(iv)3 marksClearly state the conclusion that may be drawn from this test.
41. 6(b)5 marksIt is found that the regression line of y on x passes through the point (0, 2). If the means of the x and y values are 6 and 8 respectively, find the equation of the regression line of y on x and express it in the form…