# Inference by yurtgc548

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```									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
Assumptions:           Step 1: check assumptions!
•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 

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!
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
station only if the station can prove that
more than 20% of the residents of the city
have heard the ad and recognize the
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.

```
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