A public opinion poll in Northern California was conducted t

A public opinion poll in Northern California was conducted to see if Northern Californians are prepared for the “big earthquake” that experts predict will devastate the region sometime in the next 50 years. It was learned that 62% “have not secured objects in their homes that might fall and cause injury and damage during a temblor.” A random sample of 864 Northern Californians was taken from the entire amount surveyed. Assume that each person’s answer to the survey is independent.

Use the approximation for the following—given that number of Northern Californians who have not secured objects is at least 520, what is the probability that it is more than 570?

Solution

Normal Approximation to Binomial Distribution
Mean ( np ) =864 * 0.62 = 535.68
Standard Deviation ( npq )= 864*0.62*0.38 = 14.2674
Normal Distribution = Z= X- u / sd                   
a)
P(X < 520) = (520-535.68)/14.2674
= -15.68/14.2674= -1.099
= P ( Z <-1.099) From Standard NOrmal Table
= 0.1359                  
P(X >=520) = 1 - P(X < 520) = 1 - 0.1359 = 0.8641

b)
P(X > 570) = (570-535.68)/14.2674
= 34.32/14.2674 = 2.4055
= P ( Z >2.405) From Standard Normal Table
= 0.0081