Question

Use a normal approximation to find the probability of the indicated number of voters. In this​ case, assume that 123 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 31 voted

The probability that exactly 31 of 123 eligible voters voted is ?

.

Answer 1

Solution:

Given that,

P = 0.22

1 - P = 0.78

n = 123

Here, BIN ( n , P ) that is , BIN (123 , 0.22)

According to normal approximation binomial,

X Normal

Mean = = n*P = 27.06

Standard deviation = =n*p*(1-p) = 21.1068

We using continuity correction factor

P(X = a) = P( a - 0.5 < X < a + 0.5)

P(30.5 < x < 31.5) = P((30.5 - 37.06)/ 21.1068) < (x - ) /  < (31.5 - 27.06) / 21.1068) )

= P(-1.43 < z < 0.97)

= P(z < 0.97) - P(z < -1.43)

= 0.8340 - 0.0764

Probability = 0.7576

The probability that exactly 31 of 123 eligible voters voted is 0.7576.