Question

A random sample of n=250 measurements is drawn from a binomial population with probability of success .85.

a). Find E(pˆ) and σ(pˆ).

b). Describe the shape of the sampling distribution of pˆ.

c). Find P(pˆ<.9).

Answer 1

Answer:

Given,

sample size n = 250

probability of success p = 0.85

q = 1 - p

= 1 - 0.85

= 0.15

a)

To determine the mean & standard deviation

Mean = np

= 250*0.85

= 212.5

Standard deviation = sqrt(npq)

= sqrt(250*0.85*0.15)

= 5.65

b)

Consider,

np = 250*0.85

= 212.5 >= 10

nq = 250*0.15

= 37.5 >= 10

So the shape of the sampling distribution of p is approximately normal due to the large sample size.

c)

P(p^ < 0.9) = P((p - p^)/sqrt(p*q/n) < (0.85 - 0.9)/sqrt(0.85*0.15/250))

= P(z < -2.214)

= 0.0134144 [since from z table]

= 0.0134