Question

Check My Work (1 remaining) A market research firm conducts telephone surveys with a 38% historical response rate. What is the probability that in a new sample of 400 telephone numbers, at least 150 individuals will cooperate and respond to the questions? In other words, what is the probability that the sample proportion will be at least 150/400 = .375 (to 4 decimals)? Use z-table.

Answer 1

Solution:

We are given

Sample size = n = 400,

Population proportion = p = 0.38, q = 1 – p = 1 – 0.38 = 0.62

np = 400*.38 = 152 > 5

nq = 400*.62 = 248 > 5

So, we can use normal approximation.

Mean = np = 400*.38 = 152

SD = sqrt(npq) = sqrt(400*.38*.62) = 9.707729

We have to find P(X≥150)

P(X≥150) = P(X>149.5) (by using continuity correction)

P(X>149.5) = 1 – P(X<149.5)

Z = (X – mean) / SD

Z = (149.5 - 152) / 9.707729

Z = -0.25753

P(Z<-0.25753) = P(X<149.5) = 0.398386

(by using z-table)

P(X>149.5) = 1 – P(X<149.5)

P(X>149.5) = 1 – 0.398386

P(X>149.5) = 0.601614

Required probability = 0.6016