Question

Around 16% of all ABC college students are declared econ majors. a sample of 75 students is taken.

(1) What is the distribution of the proportion of econ majors in your sample? (sampling distribution)

(2) What is the exact distribution of the number of econ majors in your sample?

(3) What’s the probability that more than a quarter of the students you sample are majoring in econ

(4) What’s the probability that less than 10% of the sample will be econ majors?

Answer 1

1)normal distribution ,X~N(n,p) = N(75,0.16)

2) binomial distribution with n=75 , p = 0.16

3)

population proportion ,p=   0.16                      
n=   75                      
                          
std error , SE = √( p(1-p)/n ) =    0.0423                      
                          
sample proportion , p̂ =   0.25                      
Z=( p̂ - p )/SE= (   0.25   -   0.16   ) /    0.0423   =   2.126
P ( p̂ >    0.25   ) =P(Z > ( p̂ - p )/SE) =                  
                          
=P(Z >   2.126   ) =    0.0167              

-------------

4)

population proportion ,p=   0.16                          
n=   75                          
                              
std error , SE = √( p(1-p)/n ) =    0.0423                          
                              
sample proportion , p̂ =   0.1                          
Z=( p̂ - p )/SE= (   0.100   -   0.16   ) /    0.042   =   -1.417  
P ( p̂ <    0.100   ) =P(Z<( p̂ - p )/SE) =                      
                              
=P(Z <    -1.417   ) =    0.0782                   (answer)