Download Section 12.2

Warm-up
 An experiment on the side effects of
pain relievers assigned arthritis
patients to one of several over-thecounter pain medications. Of the 440
patients who took one brand of pain
reliever, 23 suffered some “adverse
symptom.” Does the experiment
provide strong evidence that fewer
than 10% of patients who take this
medication have adverse symptoms?
Section 10.2
Comparing Two Proportions
We’ve only just begun…
 So far, we’ve studied inference for a
population mean:




One sample z-procedures
One sample t-procedures
Matched pairs t-procedures
Two sample t-procedures
 We’ve studied inference for a population
proportion:
 One sample z-procedures
 Next, we’ll examine two-sample inference for
population proportions.
In my opinion…
 The tricky part about inference is remembering the
assumptions. Let’s review the assumptions for inference
for a proportion.
Let’s look at how these would change for inference
comparing TWO proportions.
We need to check both proportions:
•Independent SRS’s
•n1 and n2 times 10 has to be less than or equal to the
population.
•And… n(p1) and n(1-p1) for both samples have to
be greater than or equal to 10.


Two-Sample z Interval for a Difference Between Proportions
When the Random, Normal, and Independent conditions are met, an
approximate level C confidence interval for ( pˆ1  pˆ 2 ) is
( pˆ1  pˆ 2 )  z *
pˆ1 (1  pˆ1 ) pˆ 2 (1  pˆ 2 )

n1
n2
where z * is the critical value for the standard Normal curve with area C
between  z * and z * .
Random The data are produced by a random sample of size n1 from
Population 1 and a random sample of size n2 from Population 2 or by
two groups of size n1 and n2 in a randomized experiment.
Normal The counts of " successes" and " failures" in each sample or
group - - n1 pˆ1, n1(1  pˆ1 ), n2 pˆ 2 and n2 (1  pˆ 2 ) - - are all at least 10.
Independent Both the samples or groups themselves and the individual
observations in each sample or group are independent. When sampling
without replacement, check that the two populations are at least 10 times
as large as the corresponding samples (the 10% condition).
Comparing Two Proportions

 Two-Sample z Interval for p1 – p2
Give it a shot!
 To study the long-term effects of preschool programs
for poor children, the High/Scope Educational
Research Foundation has followed two groups of
Michigan children since early childhood. One group of
62 children attended preschool as 3- and 4-year-olds.
A control group of 61 children from the same area and
similar backgrounds did not attend preschool.
 The response variable of interest is the need for social
services as adults. In the past ten years, 38 of the
preschool sample and 49 of the control sample have
needed social services. Give a 95% confidence
interval for the difference in proportions between the
two populations.
Pooled
 When doing a proportions significance
test H0: p1 = p2. In these instances
we have to do a pooled sample
proportion. Basically, you just take
the total successes and divide by the
total number in the samples. This is
the pc you use in your standard
deviation AND in your condition
check.
 Significance Tests for p1 – p2
test statistic 
statistic  parameter
standard deviation of statistic
( pˆ1  pˆ 2 )  0
z
standard deviation of statistic
If H0: p1 = p2 is true, the two parameters are the same. We call their
common value p. But now we need a way to estimate p, so it makes sense
to combine the data from the two samples. This pooled (or combined)
sample proportion is:
pˆ C 
count of successes in both samples combined
X  X2
 1
count of individuals in both samples combined
n1  n2
Use pˆ C in place of both p1 and p2 in the expression for the denominator of the test
statistic :
( pˆ1  pˆ 2 )  0

z
pˆ C (1  pˆ C ) pˆ C (1  pˆ C )

n1
n2
Comparing Two Proportions
To do a test, standardize pˆ1  pˆ 2 to get a z statistic :
Let’s go back to the preschool
example.
 Does the data indicate that the
difference between the proportion of
preschool children who need social
services is less than the proportion of
no-preschool children?
High levels of cholesterol in the blood are
associated with higher risk of heart attacks. Will
using a drug to lower blood cholesterol reduce
heart attacks? The Helsinki Heart Study recruited
middle-aged men with high cholesterol but no
history of other serious medical problems to
investigate this question. The volunteer subjects
were assigned at random to one of two
treatments: 2051 men took the drug gemfibrozil
to reduce their cholesterol levels, and a control
group of 2030 men took a placebo. During the
next five years, 56 men in the gemfibrozil group
and 84 men in the placebo group had heart
attacks. Is the apparent benefit of gemfibrozil
statistically significant? Perform an appropriate
test to find out.
Homework
Chapter 10
# 14-18, 22, 23