Suppose a simple random sample of size n 50 is obtained fro

Suppose a simple random sample of size n = 50 is obtained from a population whose size is N = 30,000 and whose population proportion with a specified characteristic is p = 0.4. Complete parts (a) through (c) below. Determine the mean of the sampling distribution of p. (Round to one decimal place as needed.) Determine the standard deviation of the sampling distribution of p. (Round to six decimal places as needed.) What is the probability of obtaining x = 22 or more individuals with the characteristic? That is, what is P(p > 0.44)? (Round to four decimal places as needed.) What is the probability of obtaining x = 17 or fewer individuals with the characteristic? That is, what is P(p

Solution

a.

n*p>5, 50*0.4> 5 => 20>5
n*(1-p)>5, 50*0.6> 5 => 20>5
Can Use Normal Approximation                  
                  
Approximately Normal because n<0.05 N , np(1-q)>=10

b.
Mean = Proportion ( P ) =0.4
Standard Deviation ( sd )= Sqrt (P*Q /n) = Sqrt(0.4*0.6/50) = 0.06928

c.
P(X >= 0.44) = (0.44-0.4)/0.0693
= 0.04/0.0693 = 0.5772
= P ( Z >0.577) From Standard Normal Table
= 0.2819                  

d.
P(X <= 0.34) = 1 - P(X>0.34)
P(X > 0.34) = (0.34-0.4)/0.0693
= -0.06/0.0693 = -0.8658
= P ( Z >-0.866) From Standard Normal Table
= 0.8067  

P(X <= 0.34) = 1 - 0.8067
= 0.1933