Question

Binomial Distribution

According to a survey, one out of four investors in the United States has exchange-traded funds in their portfolios (USA Today, 11 January 2007). Consider a random sample of 20 investors drawn from the US population.

a. Compute the probability that at least 2 investors have exchange-traded funds in their portfolios.

b. If you found that exactly 12 investors have exchange-traded funds in their portfolios, would you doubt the accuracy of the survey results?

c. Compute the expected number of investors who have exchange-traded funds in their portfolios.

Answer 1

Solution:-

p = 1/4 = 0.25

a) The probability that at least 2 investors have exchange-traded funds in their portfolios is 0.9757.

x = 2

By applying binomial distribution:-

P(x,n) = ⁿC_x*p^x*(1-p)^(n-x)

P(x > 2) = 0.9757

b) If we found that exactly 12 investors have exchange-traded funds in their portfolios, we should not doubt the accuracy of the survey results, becuase the probbaility of this occurance is very small, less than 0.05.

x = 12

By applying binomial distribution:-

P(x,n) = ⁿC_x*p^x*(1-p)^(n-x)

P(x = 12) = 0.00075

c) The expected number of investors who have exchange-traded funds in their portfolios is 5.0.

E(x) = n*p

E(x) = 20*0.25

E(x) = 5.0