Question

Name: MAT 152-Central Theorem Project Round all answers to four decimal places unless directed otherwise. SHOW WORK Scotland County has a population of 35,093 people. Of those, 22,429 people are registered to vote. 1. What proportion of Scotland County citizens are registered to vote? 2. If 50 people are randomly selected, what is the probability that at least 75 % of them are registered to vote? 3. If 100 people are randomly selected, what is the probability that at least 75% of them are registered to vote? 4. If 250 people are randomly selected, what is the probability that at least 75 % of them are registered to vote? 5. What happens to the probability of 75% of residents being registered to vote as the sample size increases? In May of 2018, 20% of the registered voters in Scotland County voted in the primary election. 6. About how many people voted? (round to the nearest person). 7. If 100 people are randomly selected, what is the probability that at least 15 % of them voted in the primary?

Answer 1

1) population proportion registered to vote ,p=22429/35093=   0.6391

2)n=   50                      
                          
std error , SE = √( p(1-p)/n ) =    0.0679                      
                          
sample proportion , p̂ =   0.75                      
Z=( p̂ - p )/SE= (   0.75   -   0.6391 ) /    0.0679   =   1.632
P ( p̂ >    0.75   ) =P(Z > ( p̂ - p )/SE) =                  
                          
=P(Z >   1.632   ) =    0.0513             

3)

n=   100                      
                          
std error , SE = √( p(1-p)/n ) =    0.0480                      
                          
sample proportion , p̂ =   0.75                      
Z=( p̂ - p )/SE= (   0.75   -   0.6391 ) /    0.0480   =   2.309
P ( p̂ >    0.75   ) =P(Z > ( p̂ - p )/SE) =                  
                          
=P(Z >   2.309   ) =    0.0105              

4)

n=   250                      
                          
std error , SE = √( p(1-p)/n ) =    0.0304                      
                          
sample proportion , p̂ =   0.75                      
Z=( p̂ - p )/SE= (   0.75   -   0.6391 ) /    0.0304   =   3.650
P ( p̂ >    0.75   ) =P(Z > ( p̂ - p )/SE) =                  
                          
=P(Z >   3.650   ) =    0.0001              

5)

as sample size increases, probability get decreases.