Question

5. It has been reported that 20% of the voters were not in support of immigration reform in the country. A sample of 400 voters were randomly selected. (a) (3 points) Find the mean number of voters that are not in support of immi- gration reform in the country. (a) (b) (4 points) Find the standard deviation of the number of voters that are not in support of immigration reform in the country. (b) (c) (6 points) Find the probability of getting exactly 75 of them that are not in support of immigration reform in the country. (c) (d) (5 points) Find the probability of getting more than 100 of them that are not in support of immigration reform in the country. (d) (e) (5 points) Find the probability of getting between 70 and 90, inclusive, of them that are not in support of immigration reform in the country,

Answer 1

The number of voters who are not in support of immigration reform out of the 400 voters selected is modelled here as:

$X \sim Bin(n = 400, p= 0.2)$

a) The mean number is computed here as:
E(X) = np = 400*0.2 = 80
Therefore 80 is the required mean here.

b) The standard deviation here is computed as:

$SD(X) = \sqrt{np(1-p)} = \sqrt{400*0.2*0.8} = 8$

Therefore 8 is the standard deviation here.

c) The probability that exactly 75 of them that are not in support of immigration reform in the country is computed here as:

$P(X = 75) = \binom{400}{75}0.2^{75}0.8^{325} = 0.0419$

Therefore 0.0419 is the required probability here.

d) The probability of getting more than 100 of them that are not in support of immigration reform in the country is computed here as:

P(X > 100) = 1 - P(X <= 100)

This is computed in EXCEL as:
=1-BINOM.DIST(100,400,0.2,TRUE)

0.0062 is the output here.

Therefore 0.0062 is the required probability here.

e) The probability here is computed as:
P(70 <= X <= 90)

= P(X <= 90) - P(X <= 69)

This is computed in EXCEL as:
=binom.dist(90,400,0.2,TRUE)-binom.dist(69,400,0.2,TRUE)

0.8109 is the output here.

Therefore 0.8109 is the required probability here.