An opinion poll was taken before an election. The table shows the percentage of voters who said they would vote for parties A , B and C . Party A B C Votes % 36 41 23 Find an approximation for the probability that, in a random sample of 120 of these voters exactly 50 said they would vote for party B .

Given that table of opinion poll, which was taken before election, Party A B C Votes % 36 41 23 Taking a random sample of 120 of these voters. So, n = 120 Let random variable X is the number of voters, who said they would vote for party B . Let success is the voter said they would vote for party B . So, probability of success p = 41 % = 0 . 41 We know that for the binomial distribution X ~ B n , p , n is a number of trials and p is the probability of success. Here X ~ B 120 , 0 . 41 Probability of failure is q = 1 - p = 1 - 0 . 41 = 0 . 59 A binomial distribution can be well-approximated by a normal distribution if n p and n q are both greater than 5 . Here, n p = 120 × 0 . 41 = 49 . 2 > 5 , n q = 120 × 0 . 59 = 70 . 8 > 5 Hence, given binomial distribution can be well-approximated by a normal distribution. Mean μ = n p = 49 . 2 , Variance σ 2 = n p q = 120 × 0 . 41 × 0 . 59 = 29 . 028 Probability that exactly 50 voters said they would vote for party B is P X = 50 . P X = 50 = P 49 . 5 < X < 50 . 5 ⇒ P X = 50 = P X < 50 . 5 - P X < 49 . 5 Using the continuity correction, standardising the value of x = 50 . 5 and x = 49 . 5 . Standardising value z = x - μ σ Here, μ = 49 . 2 , σ 2 = 29 . 028 For x = 50 . 5 , ⇒ z = 50 . 5 - 49 . 2 29 . 028 ⇒ z = 1 . 3 5 . 388 = 0 . 241 For x = 49 . 5 , ⇒ z = 49 . 5 - 49 . 2 29 . 028 ⇒ z = 0 . 3 5 . 388 = 0 . 056 So, P X = 50 = P Z < 0 . 241 - P Z < 0 . 056 We know that, if z = a , P Z < a = ϕ a ⇒ P X = 50 = ϕ 0 . 241 - ϕ 0 . 056 ⇒ P X = 50 = 0 . 5952 - 0 . 5223 ⇒ P X = 50 = 0 . 0729

MEDIUM

AS and A Level

IMPORTANT

Earn 100

A company manufactures rubber and plastic washers in the ratio $4 : 1$ . The washers are randomly packed into boxes of $25$ .
Find the probability that a randomly selected box contains exactly $21$ rubber washers.

Important Questions on The Normal Distribution

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 8 - The Normal Distribution>EXERCISE 8E>Q 9

A company manufactures rubber and plastic washers in the ratio

4 : 1

. The washers are randomly packed into boxes of

25

.
Find the probability that a randomly selected box contains exactly

10

plastic washers.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 8 - The Normal Distribution>EXERCISE 8E>Q 9

A company manufactures rubber and plastic washers in the ratio

4 : 1

. The washers are randomly packed into boxes of

25

A retail pack contains $2000$ washers. Find the expectation and variance of the number of rubber washers in a retail pack.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 8 - The Normal Distribution>EXERCISE 8E>Q 9

A company manufactures rubber and plastic washers in the ratio $4 : 1$ . The washers are randomly packed into boxes of $25$ .

A retail pack contains $2000$ washers. Using a suitable approximation, find the probability that a retail pack contains at most $1620$ rubber washers.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 8 - The Normal Distribution>EXERCISE 8E>Q 10

In a certain town,

63 %

of homes have an Internet connection. In a random sample of

20

homes in this town, find the probability that exactly

15

have an Internet connection.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 8 - The Normal Distribution>EXERCISE 8E>Q 10

In a certain town,

63 %

of homes have an Internet connection. In a random sample of

20

homes in this town, find the probability that exactly nine do not have an Internet connection.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 8 - The Normal Distribution>EXERCISE 8E>Q 10

In a certain town, $63 %$ of homes have an Internet connection.

Use a suitable approximation to find the probability that more than $65 %$ of a random sample of $600$ homes in this town have an Internet connection.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 8 - The Normal Distribution>EXERCISE 8E>Q 11

17 %

of the people interviewed in a survey said they watch more than two hours of

TV

per day. A random sample of

300

of those who were interviewed is taken. Find an approximate value for the probability that at least one-fifth of those in the sample watch more than two hours of

TV

per day.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 8 - The Normal Distribution>EXERCISE 8E>Q 12

An opinion poll was taken before an election. The table shows the percentage of voters who said they would vote for parties $A, B$ and $C$ .

Party	$A$	$B$	$C$
Votes $(%)$	$36$	$41$	$23$

Find an approximation for the probability that, in a random sample of $120$ of these voters exactly $50$ said they would vote for party $B$ .

A company manufactures rubber and plastic washers in the ratio 4:1. The washers are randomly packed into boxes of 25. Find the probability that a randomly selected box contains exactly 21 rubber washers.

Important Questions on The Normal Distribution

Learn and Prepare for any exam you want !

A company manufactures rubber and plastic washers in the ratio $4 : 1$ . The washers are randomly packed into boxes of $25$ .
Find the probability that a randomly selected box contains exactly $21$ rubber washers.