Question

Suppose in a local Kindergarten through 12^th grade (K - 12) school district, 53 percent of the population favor a charter school for grades K through five. A simple random sample of 600 is surveyed. Calculate the following using the normal approximation to the binomial distribution. (Round your answers to four decimal places.

Find the probability that 315 or more favor a charter school for grades K through 5.

Find the probability that exactly 300 favor a charter school for grades K through 5.

Answer 1

Solution:

Given that,

P = 0.53

1 - P = 0.47

n = 600

Here, BIN ( n , P ) that is , BIN (600 , 0.53)

According to normal approximation binomial,

X Normal

Mean = = n*P = 318

Standard deviation = =n*p*(1-p) = 149.46

We using continuity correction factor

a)

P(X a ) = P(X > a - 0.5)

P(x > 314.5) = 1 - P(x < 314.5)

= 1 - P((x - ) / < (314.5 - 318) / 149.46)

= 1 - P(z < -0.29)

= 1 - 0.3859   

= 0.6141

Probability = 0.6141

b)

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

P(299.5 < x < 300.5) = P((299.5 - 318)/ 149.46) < (x - ) /  < (300.5 - 318) / 149.46) )

= P(-1.51 < z < -1.43)

= P(z < -1.43) - P(z < -1.51)

= 0.0764 - 0.0655

Probability = 0.0109