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 500 is surveyed. Calculate the following using the normal approximation to the binomial distribution.

(Round your answers to four decimal places.)

(b) Find the probability that 260 or more favor a charter school for grades K through 5.

(c) Find the probability that no more than 235 favor a charter school for grades K through 5.
(e) Find the probability that exactly 250 favor a charter school for grades K through 5.

Answer 1

X ~ bin ( n , p )

Where n = 500 , p = 0.53

Mean = np = 500 * 0.53 = 265

Standard deviation = sqrt [ n p ( 1 - p) ] = sqrt [ 500 * 0.53 ( 1 - 0.53) ] = 11.1602

a)

Using normal approximation with continuity correction,

P(X >= 260 ) = P(X > 259.5)
P ( X > 259.5 ) = P(Z > (259.5 - 265 ) / 11.1602 )
= P ( Z > -0.49 )
= 1 - P ( Z < -0.49 )
= 1 - 0.3121 (From Z table)
= 0.6879

b)

Using normal approximation with continuity correction,  

P(X <= 235) = P(X < 235.5)

P ( ( X < 235.5 ) = P ( Z < 235.5 - 265 ) / 11.1602 )
= P ( Z < -2.64 )
P ( X < 235.5 ) = 0.0041 (From Z table)

c)

Using normal approximation with continuity correction,

P(X = 250) = P(249.5 < X < 250.5)
P ( 249.5 < X < 250.5 ) = P ( Z < ( 250.5 - 265 ) / 11.1602 ) - P ( Z < ( 249.5 - 265 ) / 11.1602 )
= P ( Z < -1.3) - P ( Z < -1.39 )
= 0.0968 - 0.0823 (From Z table)
= 0.0145