At UGA 26 of students are members of a fraternity of sororit

At UGA, 26% of students are members of a fraternity of sorority. Suppose random samplesof 78 students are selected, and for each sample, the proportion of students in a fraternityor sorority is calculated.
(a) What is the mean of the sampling distribution of the sample proportion?(b) What is the standard error of the sampling distribution of the sample proportion?(c) If, for a random sample of 78 students, the sample proportion ends up being exactly
one standard error above the population proportion, what is the sample proportion forthat sample of 78 students?
(d) What is the probability that the among the sample of 78 students less than 13 of themare members of a fraternity of sorority?

Solution

Given p=0.26; n=78

a) mean of the sampling distribution is np = 78*0.26=20.28;

b) standard error of the sampling distribution of the sample proportion =Sqrt (p*q)/n) where q=1-p

Therefore SE = Sqrt ( 0.26*0.74/78)=0.05

c)one SE above the populaiton proportion is = 0.26+0.05 = 0.31 i.e 31%

d)

Here mean = 20.28 and standard deviation = sqrt ( 78*0.26*0.74)=3.874

The probability that the among the sample of 78 students less than 13 of themare members of a fraternity of sorority is

= P(X<13)= P[z< (13-20.28)/3.874]= P(z<-1.879)=0.03