HARD

AS and A Level

IMPORTANT

Earn 100

A continuous random variable, $X$ , has a normal distribution with mean $8$ and standard deviation $σ$ . Given that $P (X > 5) = 0.9772$ , find $P (X < 9.5)$ .

Important Questions on The Normal Distribution

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 8 - The Normal Distribution>END-OF-CHAPTER REVIEW EXERCISE 8>Q 2

The variable

Y

is normally distributed. Given that

10 σ = 3 μ

and

P (Y < 10) = 0.75

, find

P (Y ⩾ 6)

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 8 - The Normal Distribution>END-OF-CHAPTER REVIEW EXERCISE 8>Q 3

In Scotland, in November, on average

\Rightarrow - 2 = \frac{5 - 8}{σ} \Rightarrow - 2 σ = - 3 \Rightarrow σ = 1.5

of days are cloudy. Assume that the weather on any one day is independent of the weather on other days.
Use a normal approximation to find the probability of there being fewer than

25

cloudy days in Scotland in November (

30

days).

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 8 - The Normal Distribution>END-OF-CHAPTER REVIEW EXERCISE 8>Q 3

In Scotland, in November, on average

80 %

of days are cloudy. Assume that the weather on any one day is independent of the weather on other days.
Give a reason why the use of a normal approximation is justified.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 8 - The Normal Distribution>END-OF-CHAPTER REVIEW EXERCISE 8>Q 4

At a store, it is known that

1

out of every

9

customers uses a gift voucher in part-payment for purchases. A randomly selected sample of

72

customers is taken. Use a suitable approximation and continuity correction to find the probability that at most

6

of these customers use a gift voucher in part-payment for their purchases.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 8 - The Normal Distribution>END-OF-CHAPTER REVIEW EXERCISE 8>Q 5

A survey shows that

54 %

of parents believe mathematics to be the most important subject that their children study. Use a suitable approximation to find the probability that at least

30

out of a sample of

50

parents believe mathematics to be the most important subject studied.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 8 - The Normal Distribution>END-OF-CHAPTER REVIEW EXERCISE 8>Q 6

Two normally distributed continuous random variables are

X

and

Y

. It is given that

X ~ N (1.5, 0 . 2^{2})

and that

Y ~ N (2.0, 0 . 5^{2})

. On the same diagram, sketch graphs showing the probability density functions of

X

and of

Y

. Indicate the line of symmetry of each clearly labelled graph.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 8 - The Normal Distribution>END-OF-CHAPTER REVIEW EXERCISE 8>Q 7

The random variable $X$ is such that $X ~ N (82, 126)$ . A value of $X$ is chosen at random and rounded to the nearest whole number. Find the probability that this whole number is $84$ .

(Write answer up to $3$ decimal places)

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 8 - The Normal Distribution>END-OF-CHAPTER REVIEW EXERCISE 8>Q 7

The random variable $X$ is such that $X ~ N (82, 126)$ . Five independent observations of $X$ are taken. Find the probability that at most one of them is greater than $87$ .

(Write answer up to $3$ decimal places)

A continuous random variable, X, has a normal distribution with mean 8 and standard deviation σ. Given that PX>5=0.9772, find PX<9.5.

Important Questions on The Normal Distribution

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A continuous random variable, $X$ , has a normal distribution with mean $8$ and standard deviation $σ$ . Given that $P (X > 5) = 0.9772$ , find $P (X < 9.5)$ .