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 800 is surveyed. Calculate the following using the normal approximation to the binomial distribution. (Round your answers to four decimal places.)

(a) Find the probability that less than 340 favor a charter school for grades K through 5.

(b) Find the probability that 415 or more favor a charter school for grades K through 5.

(c) Find the probability that no more than 390 favor a charter school for grades K through 5.


(d) Find the probability that there are fewer than 375 that favor a charter school for grades K through 5.


(e) Find the probability that exactly 400 favor a charter school for grades K through 5.

Answer 1

n = 800

p = 0.53

= n * p = 800 * 0.53 = 424

= sqrt(np(1 - p))

    = sqrt(800 * 0.53 * (1 - 0.53))

    = 14.1167

a) P(X < 340)

= P(X < 339.5)

= P((X - )/< (339.5 - )/)

= P(Z < (339.5 - 424)/14.1167)

= P(Z < -5.99)

= 0

b) P(X > 415)

= P(X > 414.5)

= P((X - )/> (414.5 - )/)

= P(Z > (414.5 - 424)/14.1167)

= 1 - P(Z < (414.5 - 424)/14.1167)

= 1 - P(Z < -0.67)

= 1 - 0.2514

= 0.7486

c) P(X < 390)

= P(X < 390.5)

= P((X - )/< (390.5 - )/)

= P(Z < (390.5 - 424)/14.1167)

= P(Z < -2.37)

= 0.0089

d) P(X < 375)

= P(X < 374.5)

= P((X - )/< (374.5 - )/)

= P(Z < (374.5 - 424)/14.1167)

= P(Z < -3.51)

= 0

e) P(X = 400)

= P(399.5 < X < 400.5)

= P((399.5 - )/< (X - )/< (400.5 - )/)

= P((399.5 - 424)/14.1167 < Z < (400.5 - 424)/14.1167)

= P(-1.74 < Z < -1.66)

= P(Z < -1.66) - P(Z < -1.74)

= 0.0485 - 0.0409

= 0.0076