Question

(4) According to a survey conducted by Harrah's Home Testing Institute, adults from 50% of U.S. households have gambled in a casino at some time in their life (American Demographics, October 1993). Assume that this percentage is true for the current population of adults from all U.S. households. A random sample of 400 U.S. households is selected. (a) Using normal approximation to find the probability that at least 320 of the 400 selected houscholds have gambled in a casino at some time in their life. (b) Suppose a sample of 400 households shows that 320 have gambled in a casino at some time in their life. Is 320 an unusual number? Make your comments on this result.

Answer 1

Solution:

We are given n=400, p = 50% = 0.5, q = 1 – p = 1 – 0.5 = 0.5

We have to use normal approximation for finding the required probability.

We have to find P(X≥320)

P(X≥320) = P(X>319.5) (by using continuity correction)

P(X>319.5) = 1 – P(X<319.5)

Mean = np = 400*0.5 = 200

SD = sqrt(npq) = sqrt(400*0.5*0.5) = 10

Z = (X – mean) / SD

Z = (319.5 - 200)/10

Z = 11.95

P(Z<11.95) = 1.00

(by using z-table)

P(X>319.5) = 1 – P(X<319.5)

P(X>319.5) = 1 – 1

P(X>319.5) = 0.00

Required probability = 0.00

Part b

Yes, 320 is an unusual number because the required probability is less than 0.05.