Question

Use a normal approximation to find the probability of the indicated number of voters. In this​ case, assume that 187 eligible voters aged​ 18-24 are randomly selected. Suppose a previous study showed that among eligible voters aged​ 18-24, 22% of them voted. Probability that exactly 47 voted The probability that exactly 47 of 187 eligible voters voted is nothing.

Answer 1

X~ Binomial (n,p)

Where n = 187 , p = 0.22

mean = np = 187 * 0.22 = 41.14

Standard deviation = sqrt( np(1-p) )

= sqrt( 187 * 0.22 * 0.78)

= 5.66473

Using normal approximtion

P( X < x) = P ( Z < x - mean / SD)

For P( X = x) = P(x-0.5 < X < x +0.5)

So

P( X = 47) = P( 46.5 < X < 47.5)

= P( X < 47.5) - P( X < 46.5)

= P( Z < 47.5 - 41.14 / 5.66473) - P( Z < 46.5 - 41.14 / 5.66473)

= P (Z < 1.1227) - P( Z < 0.9462)

= 0.8692 - 0.8280

= 0.04125