Question

Suppose that we will randomly select a sample of n = 117 elements from a population and that we will compute the sample proportion

  of these elements that fall into a category of interest. If the true population proportion p equals .7:

(a) Describe the shape of the sampling distribution of

. Why can we validly describe the shape?

(b) Find the mean and the standard deviation of the sampling distribution of

. (Round the answers to 2 decimal places.)

Answer 1

n=117 and p=0.7 which is binomial distribution

We will do normal approximation as n is large

X ~ N(E(X),(Var(X))

X ~ N(np,np(1-p)) ( mean and variance of binomial distribution)

X/n ~ N(E(X/n) , Var(X/n)) ~ N(1/n E(X) , 1/n²  Var(X))

a. We know that sampling distribution of p is normally distributed if np>15

In our question np = 117*0.7 = 81.9 , So we can say that it is normally distributed

And we know that normal distribution have bell shaped curve , So sampling distribution will have bell shaped curve.

b. X/n~ N(1/n E(X),1/n²  Var(X))

X/n ~ N(p , p(1-p)/n)

Mean of Sample distribution is p and Standard deviation is (p(1-p)/n)^0.5

mean = 0.7

variance = 0.00179 or standard deviation = 0.00179^0.5 = 0.042