Exam-style Questions: The Normal Distribution
1. The time taken for a paperboy to deliver his papers is normally distributed with mean 52 minutes and standard deviation 6.5 minutes.
Find the probability on any given day the paperboy takes:
a) longer than 1 hour to deliver
(3 marks)
b) less than 45 minutes
(3 marks)
c) between 48 and 56 minutes
(3 marks)
(Marks available: 9)
Answer outline and marking scheme for question: 1
1. X ~ N(52, 6.52)
a) P(X > 60) = P(Z > 60 - 52) add bell on ans sheet 3 A1
6.5
= P(Z > 1.23) = 1 - F(1.23) A1
= 1 - 0.8907 = 0.1093 A1
(3 marks)
b) P(X
6.5
= P(Z = 1 - F(1.08)
= 1 - 0.8599 = 0.1401 A1
(3 marks)
c) P(48
6.5 6.5
= P(-0.62
= F(0.62) - [1-F(0.62)] A1
= 2F(0.62) - 1 = 2 ' 0.7324 - 1
= 0.4648 A1
(3 marks)
(Marks available: 9)
2. An Olympic high jumper jumps distances which are normally distributed with mean, μ = 2.45m and variance 0.49m.
a) find the probability the jumper manages a height over 2.35m
(2 marks)
b) What distance is exceeded by 5% of his jumps?
(4 marks)
(Marks available: 6)
Answer outline and marking scheme for question: 2
X ~ N(2.45,0.49) we are given the variance so s = v0.49= 0.7
a) P(X > 2.35) = P(Z > 2.35 - 2.45) add bell on sheet 3
0.7
= P(Z > -0.14) = F(0.14) = 0.5557
(2 marks)
b) we need the distance d where
P(X > d) = 0.05
P(Z > d - 2.45) = 0.05 sheet 3 for bell
0.7
1 - F(d - 2.45) = 0.05 hence F(d - 2.45) = 0.95
0.7 0.7
so d - 2.45 = 1.645 hence d = 3.6m ( a very high jump!)
0.7
(4 marks)
(Marks available: 6)
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