Sample Assessment

1. Sample Assessment Probability

2. Question 1a(i) • Highlight important information (words) as you read

3. Question 1a(i) • At the time of the 2006 Census, 65% of people aged 15 years and over were employed. • Of the people employed at the time of the 2006 Census, 77% were in full-time work and the rest were in part-time work.

4. 1a(i) • Of the people employed full-time: • 18 % had no qualification • 33% had a school qualification as their highest qualification • 49% had a post-school qualification as their highest qualification.

5. Of the people employed part-time: • 20% had no qualification • 43% had a school qualification as their highest qualification • 37% had a post-school qualification as their highest qualification. • Calculate the proportion of people from the 2006 Census who were employed part-time with no qualification.

6. 65% were employed • 77% were in full-time work so 23% were in part time work • Of the people employed part-time: • 20% had no qualification

7. If two people were randomly selected from the 2006 Census, calculate the probability that both were employed full-time. Justify any assumptions that you have made in your calculation of this probability.

8. 65% of people (aged 15 years and over) were employed • 77% were in full-time work

9. Achieved answer • Assumption is that of independence

10. Merit answer • Assumption is that of independence. This is reasonable as the people were selected randomly and a census involves a large number of people and hence the proportion would not change significantly when one person is taken out.

11. Question 1(b) (i) • It was found that of a group of 120 people from Wellington: • 32 are at least 65 years old • 83 earn at least $50 000 • 17 are under 65 years old and earn under $50 000. • Calculate the probability that a randomly selected person from this group earns at least $50 000 and is under 65 years old. Hence determine with reasoning if the events ‘a person from this group earns at least $50 000’ and ‘a person from this group is under 65 years old’ are mutually exclusive.

12. 2-way table • group of 120

13. 32 are at least 65 years old

14. 83 earn at least $50 000

15. 17 are under 65 years old and earn under • $50 000.

16. Finish table

17. 71/120 • person from this group earns at least $50 000 and is under 65 years old

18. Achieved answer • For events to be mutually exclusive • As this is not zero, the two events are NOT mutually exclusive

19. It was also found that of this group: • 98 have a post-school qualification • everyone is either at least 65 years old, or earns at least $50 000, or has a post-school qualification • 19 are at least 65 years old but do not have a post-school qualification • 4 are at least 65 years old, earn at least $50 000, and have a post-school qualification.

20. Calculate the probability that a randomly selected person from this group has a post-school qualification and earns at least $50 000. You should explain your reasoning in calculating your answer.

21. Venn Diagram c a b d

22. 32 are at least 65 years old ≥ 65 years old 32 – (a + b+ c)

23. 83 earn at least $50 000 ≥ 65 years old ≥ $50 000 83-(a+c+d)

24. everyone is either at least 65 years old, or earns at least $50 000, or has a post-school qualification 0 ≥ 65 years old ≥ $50 000 Post-school qualification

25. 17 are under 65 years old and earn under $50 000 0 ≥ 65 years old ≥ $50 000 17

26. 98 have a post-school qualification98 – 17 = a + b + d = 81 0 ≥ 65 years old ≥ $50 000 17 17

27. 19 are at least 65 years old but do not have a post-school qualification32 – (a + b) = 19  (a + b) = 13 0 ≥ 65 years old ≥ $50 000 68 17 17

28. 4 are at least 65 years old, earn at least $50 000, and have a post-school qualification 0 ≥ 65 years old ≥ $50 000 4 68 17 17

29. 0 ≥ 65 years old ≥ $50 000 4 68 9 17 17

30. Adding everything = 120, so b = 8 0 ≥ 65 years old b 11 - b ≥ $50 000 19 - b 4 68 9 17 17

31. Adding everything = 120, so b = 8 (not needed) 0 ≥ 65 years old 8 3 ≥ $50 000 11 4 68 9 17 17

32. Excellence

33. Merit: Candidate determines the count/probability of another event not given in the question (partial completion of Venn diagram) for part (ii)

34. Question 2(a) The data below shows internet access by household type for the West Coast of the South Island. Consider the events ‘a household of a privately owned house’ and ‘a household has access to the internet’. Explain whether these events are independent.

35. Consider the events ‘a household of a privately owned house’ and ‘a household has access to the internet’. Explain whether these events are independent.

36. Add the totals needed

37. P(household of a privately owned house) = 0.398 • P(access to internet) = 0.665 • P(household of a privately owned house ∩ access to internet) = 0.261 • P(household of a privately owned house) x P(access to internet) = 0.398 x 0.645 = 0.265 • (3 s.f.) • Therefore the events are not independent as • P(A) x P(B) ≠ P(A ∩ B)

38. Is the household of a privately owned house more likely to have access to the internet than the household of a rented house? Support your answer with appropriate statistical statements.

39. P(internet/household of a privately owned house) = 0.261/0.398 = 0.656 (3 s.f.)P(internet/household of a rented house) = 0.404/0.602 = 0.671 (3 s.f.) No, the household of a privately owned house is not more likely to have access to the internet than the household of a rented house.

40. Question 2(b) • The Newborn Metabolic Screening Programme screens newborn babies for different metabolic disorders. • Screening attempts to identify babies who have a metabolic disorder. These babies are then referred for diagnostic testing to confirm or rule out having a metabolic disorder.

41. For a particular metabolic disorder: • screening identifies about 18 in 20 000 babies as being more likely than others to have this disorder • subsequent diagnostic testing (including retesting) finds around 10% of babies identified through screening actually have this disorder • approximately 6 in 66 400 babies actually have this disorder.

42. Start here • For a particular metabolic disorder: • screening identifies about 18 in 20 000 babies as being more likely than others to have this disorder • subsequent diagnostic testing (including retesting) finds around 10% of babies identified through screening actually have this disorder • approximately 6 in 66 400 babies actually have this disorder.

43. A table may be useful approximately 6 in 66 400 babies actually have this disorder

44. screening identifies about 18 in 20 000 babies as being more likely than others to have this disorder

45. subsequent diagnostic testing (including retesting) finds around 10% of babies identified through screening actually have this disorder

46. If a baby actually has this disorder, what is the approximate risk of not being diagnosed with this disorder through the screening process?

47. If a baby actually has this disorder, what is the approximate risk of not being diagnosed with this disorder through the screening process?

48. If a baby actually has this disorder, what is the approximate risk of not being diagnosed with this disorder through the screening process?

49. Question 3 • Emma was given a set of six different keys to her new house. One of the keys opened the dead lock on the front door, and a different key opened the door lock on the front door. • Emma did not know which keys were the correct keys for each lock, and was able to open both locks after four key attempts.

50. Emma’s friend Sene thought this was a low number of key attempts, and wondered what process Emma used to find the correct keys. • Sene designed and carried out a simulation to estimate how many key attempts it would take before both locks were open, using a ‘trial and error’ process, to see if this might have been the process Emma used.