Suppose 40% of the population of pre-teens have a TV in their bedroom.If a random sample of 500 pre-teens is drawn from the population,then the probability that 44% or more of the pre-teens have a TV in their bedroom is _______.
A) 0.9644
B) 0.4644
C) 0.0356
D) 0.0400
E) 0.9600

Question

Quizplus · Accepted Answer

This problem requires the use of the binomial distribution. Let X be the number of pre-teens out of 500 who have a TV in their bedroom. Then X follows a binomial distribution with n=500 and p=0.4. We need to find P(X≥0.44×500)=P(X≥220). We can use the normal approximation to the binomial distribution since n is large and p is not too close to 0 or 1. We have np=200≥10 and n(1−p)=300≥10, so X is approximately normal with mean μ=np=200 and standard deviation σ=√(np(1−p))=10.9545 (rounded to four decimal places).

Then, we standardize X: Z=(X−μ)/σ=(X−200)/10.9545.

To find P(X≥220), we can find P(Z≥(220−200)/10.9545)=P(Z≥1.8197) using a standard normal table or calculator. This probability is approximately 0.0356, so the answer is (C).

Suppose 40% of the Population of Pre-Teens Have a TV