DO YOU BELIEVE IN FAIRY TALES?
1.
2.
Yes
No
50%
50%
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1
2
UPCOMING WORK

Part 3 of Data Project due today

Quiz #5 in class this Wednesday

HW #10 due Sunday
CHAPTER 20
Comparing Two Proportions
COMPARING TWO PROPORTIONS


Comparisons between two percentages are much
more common than questions about isolated
percentages. And they are more interesting.
We often want to know how two groups differ,
a treatment is better than a placebo control,
 this year’s results are better than last year’s.
 Team A is better than Team B

OTHER EXAMPLES

Compare the proportion of women who have an
abortion that need mental health treatment to the
proportion of women who have a small infant that need
mental health treatment.

http://www.huffingtonpost.com/2011/01/26/abortion-mentalhealth_n_814582.html
THE HYPOTHESIS


The typical hypothesis test for the difference in
two proportions is the one of no difference. In
symbols, H0: p1 – p2 = 0.
The alternatives:



Ha: p1 –p2 > 0
Ha: p1 –p2 < 0
Ha: p1 –p2 ≠ 0
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THE SAMPLING DISTRIBUTION THEORY

Provided that the sampled values are independent, the
samples are independent, and the samples sizes are
large enough, the sampling distribution of pˆ1  pˆ 2 is
modeled by a Normal model with
  p1  p2

Mean:

Standard deviation:
SD  pˆ1  pˆ 2  
p1q1 p2 q2

n1
n2
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1- 7
SE FOR CONFIDENCE INTERVALS


When the conditions are met, we are ready to find the
confidence interval for the difference of two proportions:
The confidence interval is
 pˆ1  pˆ 2   z

 SE  pˆ1  pˆ 2 
where
SE  pˆ1  pˆ 2  

pˆ1qˆ1 pˆ 2 qˆ2

n1
n2
The critical value z* depends on the particular confidence
level, C, that you specify.
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1- 8
PROBLEM

Company X invents a new drug called NoZits that they
believe cures acne problems.
To see if NoZits works they run a test. They have 100
people (Group A) wash their face with NoZits, and 120
people (Group B) wash their face with the other leading
brand.
 Group A – 45 people’s skin clears up the next day
 Group B – 52 people’s skin clears up the next day


Create a 95% confidence interval for the difference
between the two groups.
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1- 9
WHAT DOES PA REPRESENT?
1.
2.
3.
The proportion of people with better skin.
The proportion of people with better skin who
used NoZits.
The proportion of people with better skin who
used the leading brand.
0%
0%
0% Slide
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1
2
3
COMPANY X HOPES TO SHOW THAT NOZITS IS A
BETTER DRUG THAN THE LEADING BRAND.
WHAT HYPOTHESES TEST SHOULD IT RUN?
1.
2.
3.
Ho: pA = pB Ha: pA ≠ pB
Ho: pA = pB Ha: pA < pB
Ho: pA = pB Ha: pA > pB
0%
0%
0% Slide
1- 11
1
2
3
CRITICAL Z* AND SIGNIFICANCE LEVEL
TWO-SIDED
α= .20  CI = 80%  z*=1.282
 α= .10  CI = 90% z*=1.645
 α= .05  CI = 95% z*=1.96
 α= .02  CI = 98%z*=2.326
 α= .01  CI = 99% z*=2.576

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INTERPRETATION OF A CONFIDENCE INTERVAL

We are 95% confidence that the true difference
between the two test groups falls within
(-0.11172, 0.15172)
 Or
 (-11.17%, 15.17%)

Our data fail to reject the null hypothesis. We do NOT
have enough evidence to suggest a difference between
success rates.
  We have no evidence that NoZits is better at getting
Slide
rid of acne.
1- 13

ESTIMATES FOR THE SD

Confidence Intervals


Use each groups individual success rate to calculate the SE
P-value and Z* testing

Use pooled proportion to calculate the SE
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SE FOR P-VALUE AND Z* HYPOTHESIS TESTING


We use the pooled value to estimate the standard error:
pˆ pooled qˆ pooled pˆ pooled qˆ pooled
SE pooled  pˆ1  pˆ 2  

n1
n2
Now we find the test statistic:
pˆ1  pˆ 2   0

z
SE pooled  pˆ1  pˆ 2 

When the conditions are met and the null hypothesis is
true, this statistic follows the standard Normal model,
so we can use that model to obtain a P-value.
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CALCULATING THE POOLED PROPORTION

The pooled proportion is
pˆ pooled
where

Success1  Success2

n1  n2
Success1  n1 pˆ1 and Success2  n2 pˆ 2
If the numbers of successes are not whole numbers, round
them first. (This is the only time you should round values in
the middle of a calculation.)
HOMEWORK PROBLEM
A clinic reported the following statistics
 For women under the age of 38



For women over the age of 38



47 live births to 165 women
4 live births to 78 women
Is there a difference in the effectiveness of the
clinic’s methods for older women?
Use a both a z* test and a 95% confidence
interval to test your hypothesis
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WHAT DOES P1 REPRESENT?
1.
2.
3.
The proportion of live
births
The proportion of
women over 38 with
live births
The proportion of
women under 38 with
live births
0%
0%
0% Slide
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1
2
3
IS THERE EVIDENCE OF A DIFFERENCE
BETWEEN THE TWO GROUPS?
1.
2.
3.
Ho: p1 = p2 Ha: p1 ≠ p2
Ho: p1 = p2 Ha: p1 < p2
Ho: p1 = p2 Ha: p1 > p2
0%
0%
0% Slide
1- 19
1
2
3
WHAT IS OUR TEST ‘RULE’ WE FOLLOW?
1.
2.
3.
4.
If z > critical z* , then reject null hypothesis.
If z > critical z* , then accept null hypothesis.
If z < critical z* , then reject null hypothesis.
If z < critical z* , then accept null hypothesis.
0%
0%
0%
0%Slide
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1
2
3
4
WHAT IS THE RESULT OF THIS HYPOTHESIS
TEST AT A SIGNIFICANCE LEVEL OF 0.05?
1.
0%
2.
0%
3.
0%
Do not reject the null hypothesis b/c there IS
NOT sufficient evidence to make the claim of a
difference
Accept the alternative hypothesis b/c there IS
sufficient evidence to support claim of a
difference
Reject the null hypothesis b/c there IS sufficient
evidence to support the claim of a difference
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WOULD WE GET A DIFFERENT RESULT
WITH A CI?

Let’s calculate the CI at 95%
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OUR CI (14.9%, 31.7%). HOW WOULD WE
INTERPRET THIS CI?
0%
0%
1.
2.
3.
0%
There is 95% confidence the prop. of live births
for clients of this clinic is greater for women
under 38
There is 95% confidence the prop. of live births
is greater for women under 38
There is 95% confidence the prop. of live births
for a sample of clients of this clinic is greater
for women under 38
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HOMEWORK PROBLEM
A survey of older Americans reveals,
 420 out of 1002 men suffer from arthritis
 543 out of 1070 women suffer from arthritis


Create a 95% CI to test the hypothesis that the
proportion of adults suffering from arthritis is
greater for women than for men.
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DOES THIS SUGGEST THAT ARTHRITIS IS MORE
LIKELY TO AFFLICT WOMEN THAN MEN?
1.
2.
3.
4.
No. No conclusion can be made
25%based
25%on the
25%
confidence interval.
No. The interval is too close to 0.
Yes. The entire interval lies above 0.
Yes, we are 95% confident, based on these
samples, that about 13.1% of senior women
suffer from arthritis, while only 4.6% of senior
men suffer from arthritis.
25%
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1
2
3
4
HW - PROBLEM 8

One country reported
84 out of 3157 white women had multiple births
 20 out of 625 black women had multiple births


Does this indicate any racial difference of the
likelihood of multiple births?
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DEFINING THE PROPORTIONS AFFECTS THE
HYPOTHESIS
P1 = proportion of multiple births from white
women
 P2 = proportion of multiple births from black
women

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DOES THIS INDICATE ANY RACIAL
DIFFERENCE?
1.
2.
3.
Ho: p1 – p2 =0
Ha: p1 – p2>0
Ho: p1 – p2 =0
Ha: p1 – p2<0
Ho: p1 – p2 =0
Ha: p1 – p2≠0
0%
0%
0% Slide
1- 28
1
2
3
DO WHITE WOMEN TYPICALLY HAVE MORE
MULTIPLE BIRTHS?
1.
2.
3.
Ho: p1 – p2 =0
Ha: p1 – p2>0
Ho: p1 – p2 =0
Ha: p1 – p2<0
Ho: p1 – p2 =0
Ha: p1 – p2≠0
0%
0%
0% Slide
1- 29
1
2
3
DO BLACK WOMEN TYPICALLY HAVE MORE
MULTIPLE BIRTHS?
1.
2.
3.
Ho: p1 – p2 =0
Ha: p1 – p2>0
Ho: p1 – p2 =0
Ha: p1 – p2<0
Ho: p1 – p2 =0
Ha: p1 – p2≠0
0%
0%
0% Slide
1- 30
1
2
3
SUPPOSE OUR QUESTION IS ‘IS THERE A
DIFFERENCE BETWEEN BLACKS AND WHITE?’
WHAT IS OUR TEST ‘RULE’?
1.
2.
3.
4.
If z > critical z* , then
reject null hypothesis.
If z > critical z* , then
accept null hypothesis.
If z < critical z* , then
reject null hypothesis.
If z > critical z* , then
accept null hypothesis.
0%
0%
0%
0%Slide
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1
2
3
4
WHAT LEVEL OF SIGNIFICANCE DO YOU
WANT TO CHOOSE ?
1.
2.
3.
4.
5.
α= .20  z*=1.282
α= .10  z*=1.645
α= .05  z*=1.96
α= .02  z*=2.326
α= .01  z*=2.576
20%
20%
20%
20%
20%
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1
2
3
4
5
WHAT IS THE RESULT OF THE HYPOTHESIS
TEST?
1.
2.
3.
Reject the null hypothesis b/c there is sufficient
evidence to support the claim of a difference.
Do not reject the null hypothesis b/c there is
NOT sufficient evidence to support the claim of
a difference
Accept the null hypothesis b/c there is NOT
sufficient evidence to support the claim of a
difference
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SUPPOSE OUR LAST TEST WAS INCORRECT,
WHAT TYPE OF ERROR DID WE MAKE?
1.
2.
3.
Type 1 – Rejecting when
Null is True
Type 2 - Failing to
Reject when the Null is
False
This cannot be
determined
0%
0%
0% Slide
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1
2
3
COMING UP…

Quiz #5 on Wednesday