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Section 12.1
Inference for a
Population Proportion
AP Statistics
Parameters vs Statistics

Parameters

 Mean

 Mean
µ
 Standard


Deviation
x-bar
 Standard
σ
 Proportion

Statistics

Deviation
s
 Proportion
p

AP Statistics, Section 12.1
p-hat
2
What we know about inference

We are trying to make sense about what is
happening at the population level by looking at
sample data
 Step
1: “What is the population and the parameter of
interest?”

We make assumptions in the form of H0
 Step

1: “What is H0?”
We need to know about the distribution of the
sample statistic
 Step
2: “Is the distribution of sample means normal?”
AP Statistics, Section 12.1
3
Our inferential work so far…


Has been about the
distribution of sample  x  

means
x 
n
and the distribution of
the difference of
 x1  x2  1   2
sample means.
 12  22
 x1  x2 

n1 n2
AP Statistics, Section 12.1
x 
z
 n
x 
t
s n
t
x1  x2
s12 s22

n1 n2
4
But what about proportions?

We learned in
Chapter 9 about the
distribution of sample
proportions.
 pˆ  p
 pˆ 
AP Statistics, Section 12.1
pq
n
5
But what about proportions?

We know that the
distribution of
sample
statistic-parameter
Test statistic =
proportions is
standard dev. of statistic
approximately
p̂  p
z
normal when
pq
these conditions
n
are met…
 np>10
 nq>10
AP Statistics, Section 12.1
6
Simulation
A recent study concluded that 25% of all
U.S. teenage females have a STD.
 Simulate sampling 500 randomly chosen
teenage females using…

 randBin(500,.25)

Simulate finding the sample proportion by
using…
 randBin(500,.25)/500
AP Statistics, Section 12.1
7
Test of significance
A recent sample of 500 female teenagers
from southeastern Oakland county found
the 22% have an STD.
 (I made these numbers up.)
 Is this strong evidence to suggest that
teenage females from SE Oakland county
have a lower infection rate than the
national average of 25%?

AP Statistics, Section 12.1
8
1: Population, Parameter of Interest,
H0 and Ha
2: Procedure Name & Conditions
3: Calculations
4: Interpret
AP Statistics, Section 12.1
9
Confidence Intervals
CI  statistic  critical value   standard dev. of statistic 
CI  pˆ  z
*
ˆˆ
pq
n
AP Statistics, Section 12.1
10
Calculate the Confidence Interval
1: Population & Parameter of Interest
2: Procedure Name & Conditions
3: Calculations
4: Interpret
AP Statistics, Section 12.1
11
Choosing a sample size



We get p* by using
previous studies.
Or
We use p*=.5
because this is the
most conservative
choice.
ME  z
AP Statistics, Section 12.1
* *
*
pq
n
12
Example



.15  .85
.05  1.96
n
We believe that 15%
.15  .85
of people are left
.05  1.96
n
handed.
How large of a
.15  .85
n  1.96
sample is needed to
.05
get the ME of a 95%
2

.15  .85 
CI below 5%?
n  1.96
  195.92
.05 
Answer: at least 196.

AP Statistics, Section 12.1
13
Exercises

12.5-12.19 odd
AP Statistics, Section 12.1
14