8.4: TWO-PROPORTION INFERENCE
TESTS
Objective: To test claims about inferences for two
proportions, under specific conditions
THE STANDARD DEVIATION OF THE
DIFFERENCE BETWEEN TWO
PROPORTIONS
• Proportions observed in independent random samples are
independent. Thus, we can add their variances. So…
• The standard deviation of the difference between two sample
proportions is
p1q1 p2 q2
SD  pˆ1  pˆ 2  

n1
n2
• Thus, the standard error is
SE  pˆ1  pˆ 2  
Remember it’s always a +
pˆ1qˆ1 pˆ 2 qˆ2

n1
n2
ASSUMPTIONS & CONDITIONS
• Independence Assumptions:
• Randomization Condition: The data in each group should be
drawn independently and at random from a homogeneous
population or generated by a randomized comparative experiment.
• The 10% Condition: If the data are sampled without replacement,
the sample should not exceed 10% of the population.
• Independent Groups Assumption: The two groups we’re
comparing must be independent of each other.
ASSUMPTIONS & CONDITIONS
• Sample Size Assumption:
• Each of the groups must be big enough…
• Success/Failure Condition: Both groups are big enough that at
least 10 successes and at least 10 failures have been observed in each.
TWO-PROPORTION Z-INTERVAL
• 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.
STEPS FOR TWO-PROPORTION ZINTERVAL
1. Check Conditions and show that you have checked these!
• Randomization Condition: The data in each group should be
drawn independently and at random from a homogeneous
population or generated by a randomized comparative experiment.
• The 10% Condition: If the data are sampled without replacement,
the sample should not exceed 10% of the population.
• Independent Groups Assumption: The two groups we’re
comparing must be independent of each other.
• Success/Failure Condition: Both groups are big enough that at
least 10 successes and at least 10 failures have been observed in each.
𝒏𝟏 𝒑𝟏 ≥ 𝟏𝟎
𝒏𝟐 𝒑𝟐 ≥ 𝟏𝟎
𝒏𝟏 𝒒𝟏 ≥ 𝟏𝟎
𝒏𝟐 𝒒𝟐 ≥ 𝟏𝟎
STEPS FOR TWO PROPORTION ZINTERVAL (CONT.)
2. State the test you are about to conduct
Ex) Two-Proportion z-Interval
4. Calculate your z-interval
(𝒑𝟏 − 𝒑𝟐 ) ±
𝒛∗
×
𝒑𝟏 𝒒𝟏 𝒑𝟐 𝒒𝟐
+
𝒏𝟏
𝒏𝟐
5. State your conclusion IN CONTEXT.
We are 95% confident that the support group program could raise the
proportion of smokers who manage to quit using the parch by
between 2 and 22 percentage points.
TWO-PROPORTION Z-INTERVAL
EXAMPLE
The table below describes the effect of preschool on later use of social
services:
Set up a 95% confidences interval. Interpret your results.
TWO-PROPORTION Z-INTERVAL
EXAMPLE (CONT.)
POOLING P
• The typical hypothesis test for the difference in two proportions is the
one of no difference (when they are equal). In symbols, H 0: p1 – p2 =
0.
• Since we are hypothesizing that there is no difference between the two
proportions, that means that the standard deviations for each
proportion are the same.
• Since this is the case, we combine (pool) the counts to get one overall
proportion.
POOLING P (CONT.)
• 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.)
POOLING P (CONT.)
• We then put this pooled value into the formula, substituting it for both sample
proportions in the standard error formula:
SE pooled  pˆ1  pˆ 2  
pˆ pooled qˆ pooled
n1

pˆ pooled qˆ pooled
n2
• We’ll reject our null hypothesis if we see a large enough difference in the two
proportions.
• How can we decide whether the difference we see is large?
• Just compare it with its standard deviation.
• Unlike previous hypothesis testing situations, the null hypothesis doesn’t provide a
standard deviation, so we’ll use a standard error (here, pooled).
TWO-PROPORTION Z-TEST
• The conditions for the two-proportion z-test are the same as for the twoproportion z-interval.
• We are testing the hypothesis H 0: p1 – p2 = 0, or, equivalently, H0: p1 = p2.
• Because we hypothesize that the proportions are equal, we pool them to
find
pˆ pooled
Success1  Success2

n1  n2
TWO-PROPORTION Z-TEST
(CONT.)
• We use the pooled value to estimate the standard error:
SE pooled  pˆ1  pˆ 2  
pˆ pooled qˆ pooled
n1

pˆ pooled qˆ pooled
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.
STEPS FOR TWO-PROPORTION ZTESTS
1. Check Conditions and show that you have checked these!
• Randomization Condition: The data in each group should be
drawn independently and at random from a homogeneous
population or generated by a randomized comparative experiment.
• The 10% Condition: If the data are sampled without replacement,
the sample should not exceed 10% of the population.
• Independent Groups Assumption: The two groups we’re
comparing must be independent of each other.
• Success/Failure Condition: Both groups are big enough that at
least 10 successes and at least 10 failures have been observed in each.
𝒏𝟏 𝒑𝒑𝒐𝒐𝒍𝒆𝒅 ≥ 𝟏𝟎
𝒏𝟐 𝒑𝒑𝒐𝒐𝒍𝒆𝒅 ≥ 𝟏𝟎
𝒏𝟏 𝒒𝒑𝒐𝒐𝒍𝒆𝒅 ≥ 𝟏𝟎
𝒏𝟐 𝒒𝒑𝒐𝒐𝒍𝒆𝒅 ≥ 𝟏𝟎
STEPS FOR TWO-PROPORTION ZTESTS (CONT.)
2. State the test you are about to conduct
Ex) Two-proportion z-test
3. Set up your hypotheses
H 0:
H A:
4. Calculate your test statistic
𝒛=
𝒑𝟏 −𝒑𝟐 −𝟎
𝒑𝒑𝒐𝒐𝒍𝒆𝒅 ∙ 𝒒𝒑𝒐𝒐𝒍𝒆𝒅 𝒑𝒑𝒐𝒐𝒍𝒆𝒅 ∙ 𝒒𝒑𝒐𝒐𝒍𝒆𝒅
+
𝒏𝟏
𝒏𝟐
5. Draw a picture of your desired area under the t-model, and
calculate your P-value.
STEPS FOR TWO-PROPORTION ZTESTS (CONT.)
6. Make your conclusion.
P-Value
Action
Conclusion
Low
Reject H0
The sample
mean is
sufficient
evidence to
conclude HA in
context.
High
Fail to reject H0
The sample
mean does not
provide us with
sufficient
evidence to
conclude HA in
context.
CALCULATOR TIPS
• Stat  TESTS
• 6: 2-PropZTest
• Enter values
• Calculate
TWO-PROPORTION Z-TEST
EXAMPLE
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 looked at this question. Middle-aged men 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. What are the
proportions and is the benefit of the drug statistically significant?
TWO-PROPORTION Z-TEST
EXAMPLE (CONT.)
ASSIGNMENTS
• Day 1: 8.4 Book Page # 1, 7, 9, 18
• Day 2: 8.4 Book Page # 3, 10, 20, 22