AP STATISTICS
LESSON 12 – 2
( DAY 2 )
SIGNIFICANCE TESTS FOR p1 – p2
ESSENTIAL QUESTION:
How are significance tests created for
two sample population proportions?
Objectives:
 To learn the procedures for finding
significance tests.
 To be able to pool sample proportions.
Significance Tests for p1 – p2
The null hypothesis says that there is no
difference between the two populations:
Ho: p1 = p2 or Ho: p1 – p2 = 0
The alternative hypothesis says what kind
of difference we expect.
Example 12.12 Page 707
Cholesterol and Heart Attacks
To do a test, standardize p^1 – p^2 to get a z
statistic.
If H is true, all the observations in both samples
really come from a single population of men of
whom a single unknown proportion p will have a
heart attack in five-year period.
So instead of estimating p1 and p2 separately, we
pool the two samples and use the overall sample
proportion to estimate the single population
parameter p.
Call this the pooled sample proportion.
Pooled Sample Proportion
^p = count of success in both samples combined
count of observations in both samples combined
=
X1 + X2
n 1 + n2
Significance Test for Comparing
Two Proportions
To test the hypothesis:
H0 : p1 = p2
first find the pooled proportion p of successes
in both samples combine. Then compute the
z statistic
^p1 – ^p2
z=
√ ^p( 1 – ^p ) ( 1/n1 + 1/ n2 )
P-values
In terms of a variable Z having the standard normal
distribution, the P-value for a test Ho against
Ha: p1 > p2 is P(Z ≥ z )
Ha: p1 < p2 is P(Z ≤ z )
Ha: p1 ≠ p2 is 2P(Z ≥ lzl )
Conditions: Use these tests when the populations are at
least 10 times as large as the samples and n 1^
p, n 1(1 – ^
p),
n 2^
p and n 2( 1 – ^
p ) are all 5 or more.
Example 12.13 Page 709
Cholesterol and Heart Attacks
(continued…)
^
p = count of heart attacks in both samples combined
count of observations in both samples combined
Since P < 0.01, the results are statistically significant at the
a = 0.01 level.
There is strong evidence that the Gemfibrozil reduced the
rate of heart attacks.
Example 12.14 Page 710
Don’t Drink the Water


The P-value, 0.0344 tells us that it is unlikely that we
would obtain a difference in sample proportions as
large as we did if the null hypothesis is true.
Judges have generally adopted a 5% significance
level as their standard for convincing evidence.
The P-value
for the onesided test