### The Difference in Proportions

```Testing the Difference
between Proportions
Section 11.3
Assumptions

Randomly Selected Samples

Approximately normal since
1 1 ≥ 10
1 1 − 1 ≥ 10

and
2 2 ≥ 10
2 1 − 2 ≥ 10
Independent (at least 10n in population)
Normally the Standard Deviation of
Statistic is
p1q1 p2 q2

n1
n2
But, Since we claim p1  p2 in the
Ho

We can combine the values to form one
proportion:
x1  x2
pc 
n1  n2

And the Standard Deviation of Statistic becomes
1 1
pc qc   
 n1 n2 
Can you find this on the formula
Sheet?

They use a combined p.
A sample of 50 randomly selected men with high
triglyceride levels consumed 2 tablespoons of oat bran daily
for six weeks. After six weeks, 60% of the men had
lowered their triglyceride level. A sample of 80 men
consumed 2 tablespoons of wheat bran for six weeks. After
six weeks, 25% had lower triglyceride levels. Is there a
significant difference in the two proportions at the 0.01
significance level?
To calculate pc we need to find x1 and x2.
x1  0.60(50)  30
x2  0.25(80)  20
So…..
30  20 50
pc 

 0.385
50  80 130
Parameter:
 p1  proportion of men with high triglyceride levels who ate oat bran.

 p2  proportion of men with high triglyceride levels who ate wheat bran.
 p  p  difference in prop. of men w/high triglycerides between those who ate oat & bran
2
 1
Hypothesis:
 H o : p1  p2

 H A : p1  p2
Assumptions:
* Randomly Selected Samples
* Approximately Normal since
n1 p1  50(0.6)  30  5

n1q1  50(.4)  20  5
and
n2 p2  80(0.25)  20  5
n2 q2  80(0.75)  60  5
* Independent –
(at least 500 men eat oat and 800 eat wheat bran
Name of Test:
2-Proportion Z-Test
)
^ ^ 
 p1  p 2    p1  p2 


z
1 1
pc qc   
 n1 n2 
z
 .6  .25  0
1 
 1
.385 * .615 *   
 50 80 
z  3.99
P  Value  P( z  3.99)* 2
P  Value  ncdf (3.99, )* 2
P  Value  0
Reject the Ho since the P-Value(0) < (0.05)
There is sufficient evidence to support the claim that
there is a difference in the proportion of men who
lowered their triglycerides by eating oat bran and the
proportion who lowered their triglycerides by eating
wheat bran.
In a sample of 100 store customers, 43 used a Mastercard. In another sample
of 100, 58 used a Visa card. Is the proportion of customers who use
Mastercard less than those using Visa?
p1  prop using mastercard
p2  prop using visa
p1  p2  diff in prop using mastercard & visa
 H o : p1  p2

 H A : p1  p2
43  58
100  100
101

 0.505
200
pc 
Assumptions:
1. Randomly Selected Samples
2. Approx. Normal
n1 p1  100(.43)  43  5

n1q1  100(.57)  57  5

n2 p2  100(.58)  58  5
n q  100(.42)  42  5
 2 2
3. Independent (at least 1000 of each)
^ ^ 
 p1  p 2    p1  p2 
.43  .58   0



z

 2.12
1 
1 1 
 1
.
505
*
.
495
*

pc qc   
 100 100 


 n1 n2 
P  Val  P( z  2.12)
 ncdf ( , 2.12)  0.017
Reject the Ho since the p-val(.017) <  (0.05)
There is sufficient evidence to support the claim that the
proportion using mastercard is less than the proportion
using visa.
So how would we find a confidence
interval?
PANIC!
In a sample of 80 Americans, 55% wished that they were rich. In a sample
of 90 Europeans, 45% wished that they were rich. Is there a difference in
the proportions. Find and interpret the 95% confidence interval for the
difference of the two proportions.
p1  prop Americans who wish to be rich
p2  prop Europeans who wish to be rich
p1  p2  diff in prop Am & Eurp who wish to be rich
Assumptions:
1. Randomly Selected Samples
2. Approx Norm n1 p1  80(.55)  44  5

n1q1  80(.45)  36  5

n2 p2  90(.45)  41  5
n q  90(.55)  50  5
 2 2
3. Independent (at least 800 Am and 900 Europeans.
^ ^ 
 p1  p 2   z 2


p1q1 p2 q2

n1
n2
.55(.45) .45(.55)

 .55  .45  1.96
80
90
.10  .150
 .05, .25
We’re 95% confident that the difference in proportion of
Americans who wish to be rich and the proportion of
Europeans who wish to be rich is between -.05 and .25.
In fact, since this interval contains 0, there is no
significant difference.
Homework

Worksheet
```