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 46 voted

The probability that exactly 46 of 187 eligible voters voted is ______
​(Round to four decimal places as​ needed.)

Answer 1

Solution:

Given that,

P = 22% = 0.22

1 - P = 1 - 0.22 = 0.78

n = 187

Here, BIN ( n , P ) that is , BIN (187 , 0.22)

then,

n*p = 187*0.22 = 41.14 > 5

n(1- P) = 187*0.78 = 145.86 > 5

According to normal approximation binomial,

X Normal

Mean = = n*P = 41.14

Standard deviation = =n*p*(1-p) = 187*0.22*0.78 = 32.089

We using countinuity correction factor

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

P(45.5 < x < 46.5) = P((45.5 - 41.14)/ 32.089) < (x - ) /  < (46.5 - 41.14) / 32.089) )

= P(0.770 < z < 0.946)

= P(z < 0.946) - P(z < 0.770)

= 0.8279 - 0.7793

= 0.0486

Probability = 0.0486