Chapter 22
Comparing Two Proportions
Comparing 2 Proportions
• How do the two groups differ?
• Did a treatment work better than the placebo
control?
• Are this years results better than last years?
Remember!!!
• Var(X – Y) = Var(X) + Var(Y)
• SD(X – Y) = √[(Var(X) + Var(Y)]
– only when X and Y are independent
Standard Deviation Of The Difference
Between Two Proportions
*proportions observed in independent random
samples are independent*
*be sure to use notation to differ between
variables*
SE( p̂1 - p̂2 ) =
SD( p̂1 - p̂2 ) =
p̂1q̂1 p̂2 q̂2
+
n1
n2
p1q1 p2 q2
+
n1
n2
*We don’t always know what the true p is.
When you don’t find the SE (Standard Error)*
SE( p̂1 - p̂2 ) =
p̂1q̂1 p̂2 q̂2
+
n1
n2
Assumptions and Conditions
• Within each group the data should be based
on results for independent individuals. Check:
– Randomization condition: each data set should be
selected at random
– 10% condition: if sampled without replacement,
then the sample should be less than 10% of the
population
– Independent Groups Assumption: The two groups
need to be independent of each other
And…
• Success/Failure Condition: BOTH groups have
at least 10 successes and 10 failures
the sampling distribution model for a difference
between two independent proportions
Provided that the sampled values are
independent, the samples are independent, and
the sample sizes are large enough, the sampling
distribution of p̂1 - p̂2 is modeled by a Normal
model with
mean = p̂1 - p̂2
standard deviation =
SD( p̂1 - p̂2 ) =
p1q1 p2 q2
+
n1
n2
A two-proportion z-interval
When the conditions are met, we are ready to find the
confidence interval for the difference of two proportions,
p1 – p2. The confidence interval is
*
p̂
p̂
±
z
( 1 2 ) ´ SE ( p̂1 - p̂2 )
where we find the standard error of the difference,
p̂1q̂1 p̂2 q̂2 from the observed proportions
SE( p̂1 - p̂2 ) =
n1
+
n2
The critical value z* depends on the particular confidence
level, C, that you specify