Inference

							Inference on
Proportions
Assumptions:
• SRS
• Normal distribution
   np > 10 & n(1-p) > 10
• Population is at least 10n
Formula for Confidence interval:
CI  statistic  critical valueSD of statistic
                       Normal
                       curve


                                  p 1  p  
       ˆ
       p  z*                    
                                      n
                                              
                                              
                                             
  Note: For confidence intervals, we DO NOT know p –
  so we MUST substitute p-hat for p in both the SD &
  when checking assumptions.
A May 2000 Gallup Poll found that
38% of a random sample of 1012
adults said that they believe in
ghosts. Find a 95% confidence
interval for the true proportion of
adults who believe in ghost.
Assumptions:           Step 1: check assumptions!
•Have an SRS of adults
•np =1012(.38) = 384.56 & n(1-p) = 1012(.62) = 627.44
Since both are greater than 10, the distribution can be
approximated by a normal curve
                        Step 2: make
•Population of adults is at least 10,1012.calculations


 ˆ  z *  p 1  p    .38  1.96 .38(.62)   .35,.41 
                                            
P
              n                      1012 
                                            
                  Step 3: conclusion in context
We are 95% confident that the true proportion of
adults who believe in ghosts is between 35% and
41%.
Another Gallop Poll is taken
in order to measure the
     To find sample size:

                          1  p 
proportion of adultspwho  
                       
               m z *  
                       
approve of attempts to clonen    
                                
     However, since we have not yet is
humans. What sample size taken a
     sample, to be within p-hat (or of the
necessarywe do not know a + 0.04 p) to
     use!
true proportion of adults who
approve of attempts to clone
humans with a 95% Confidence
Interval?
What p-hat (p) do you use when
trying to find the sample size for a
given margin of error?

.1(.9) = .09   By using .5 for p-hat,
.2(.8) = .16   we are using the worst-
.3(.7) = .21   case scenario and
               using the largest SD in
.4(.6) = .24   our calculations.
.5(.5) = .25
Another Gallop Poll is taken in order to measure the
proportion of adults who approve of attempts to clone
humans. What sample size is necessary to be within +
0.04 of the true proportion of adults who approve of
attempts to clone humans with a 95% Confidence
Interval?          p 1  p  
         m z *               
                              n       
                                      
                              .5.5  
                                              Use p-hat = .5
         .04  1.96                  
                               n 
                                     
          .04         .5.5                 Divide by 1.96
              
         1.96             n
                  2
          .04     .25                      Square both sides
                
          1.96     n
         n  600 .25  601
                                           Round up on sample size
Stop & do homework!
Hypotheses for proportions:

H0: p = value
Ha: p > value
                Use >, <, or ≠
 where p is the true proportion
 of context
Formula for hypothesis test:
                  statistic - parameter
 Test statistic 
                     SD of statistic


                     ˆ
                    p p
    z              p 1  p 
                        n
A company is willing to renew its
advertising contract with a local radio
station only if the station can prove that
more than 20% of the residents of the city
have heard the ad and recognize the
company’s product. The radio station
conducts a random sample of 400 people
and finds that 90 have heard the ad and
recognize the product. Is this
sufficient evidence for the
company to renew its contract?
Assumptions:
•Have an SRS of people
•np = 400(.2) = 80 & n(1-p) = 400(.8) = 320 - Since both are greater
than 10, this distribution is approximately normal.
•Population of people is at least 4000.
                                    Use the parameter in the null
 H0: p = .2      where p is the hypothesis to check assumptions!
                                 true proportion of people who
Ha: p > .2         heard the ad

       .225  .2
 z                 1.25   p  value  .1056 α  .05
         .2(.8)                     Use the parameter in the null
          400                     hypothesis to calculate standard
                                             deviation!
Since the p-value >a, I fail to reject the null hypothesis. There is
not sufficient evidence to suggest that the true proportion of people
who heard the ad is greater than .2.