Question

According to the National Center for Health Statistics (2014), p = 0.21 of adults are smokers. Assume the population is the U.S. adult population of N = 235,224,016. A random sample of n = 450 adults is obtained.
a. Describe the Distribution of the Sample Proportion, ?̂ , in terms of shape, center and spread, for the proportion of U.S. adults who smoke (3 pts):
• Shape: np(1-p) ≥ 10
• Center: ??̂
• Spread: ??̂ = √?̂(1−?̂)?
b. In a random sample of n = 450 adults, what is the probability of obtaining a sample proportion of smokers, ?̂ ≤ 0.18 (2 pts)?
c. In a random sample of n = 450 adults, what is the probability of obtaining a sample proportion of smokers, ?̂ ≥ 0.22 (3 pts)?

Answer 1

a)

Shape: np(1-p) ≥ 10 , normally distributed
• Center: ??̂ = 0.21
• Spread: ??̂ = √?̂(1−?̂)? = 0.019

b)

population proportion ,p=   0.21                          
n=   450   94.5                      
                              
std error , SE = √( p(1-p)/n ) =    0.019                          
                              
sample proportion , p̂ =   0.18                          
Z=( p̂ - p )/SE= (   0.180   -   0.21   ) /    0.019   =   -1.562  
P ( p̂ <    0.180   ) =P(Z<( p̂ - p )/SE) =                      
                              
=P(Z <    -1.562   ) =    0.0591                   (answer)

excel formula for probability from z score is =NORMSDIST(Z)                              

c)

population proportion ,p=   0.21                      
n=   450                      
                          
std error , SE = √( p(1-p)/n ) =    0.0192
                          
sample proportion , p̂ =   0.22                      
Z=( p̂ - p )/SE= (   0.22   -   0.21   ) /    0.0192   =   0.521
P ( p̂ >    0.22   ) =P(Z > ( p̂ - p )/SE) =                  
                          
=P(Z >   0.521   ) =    0.3012