SECTION 11.2
COMPARING TWO MEANS
AP Statistics
Comparing Two Means
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Very useful compare two populations
Two population equates to two distributions,
perhaps of different size
Easier math to work with one distribution
Distribution of the difference of means and one
sample t-procedures when possible.
AP Statistics, Section 11.2
Some formulas:
3
z  test
x
z

t  test
x
z
s
n
n
AP Statistics, Section 11.2
Formulas (continued)
4
x  y  x   y
2
X
 XY    
x 

n

2
Y
s
n
2
 x x
1
2
2
2
2
 s1   s2 
s1 s2
 

 
 
n1 n2
 n1   n2 
AP Statistics, Section 11.2
x1  x2    1  2 

z

2
1
n1


2
2
z
n2
x1  x2    1  2 

t
s12 s22

n1 n2
5
If 1  2 then 1  2  0
 x1  x2 

2
1
n1
t


n2
 x1  x2 
2
1
2
2
s
s

n1 n2
AP Statistics, Section 11.2
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Assumptions for Comparing Two Means
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We have two SRSs, from two distinct populations.
The samples are independent. That is, one sample has
no influence on the other. Matching violates
independence, for example.
We measure the same variable for both samples.
Both populations are normally distributed. The means
and standard deviations of the populations are
unknown.
AP Statistics, Section 11.2
Example
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Does increasing the amount of calcium in our diet
reduce blood pressure? Examination of a large
sample of people revealed a relationship between
calcium intake and blood pressure. The relationship
was strongest for black men. Such observational
studies do not establish causation. Researchers
therefore designed a randomized comparative
experiment.
AP Statistics, Section 11.2
Example
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The subjects in part of the experiment were 21 healthy black
men. A randomly chosen group of 10 of the men received a
calcium supplement for 12 weeks.
The group of 11 men received a placebo pill that looked
identical.
The experiment was double-blind.
The response variable is the decrease in systolic (heart
contracted) blood pressure for a subject after 12 weeks, in
millimeters of mercury. An increase appears as a negative
response.
AP Statistics, Section 11.2
Example
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Group 1 (Calcium) results:
7, -4, 18, 17, -3, -5, 1, 10, 11, -2
 n=10, x-bar=5.000, s=8.743

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Group 2 (Placebo) results:
 -1,
12, -1, -3, 3, -5, 5, 2, -11, -1, -3
 n=11, x-bar=-0.273, s=5.901
AP Statistics, Section 11.2
Inference Tool Box
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Step 1: Identify the populations and the parameters
of interest you want to draw conclusions about. State
hypothesis in words and symbols.
Pop1: Black Men on Calcium;
 Pop2: Black Men on Placebo
 Parameters of interest: mean differences in blood pressure
 H0: µ1= µ2 (There is no difference in the blood pressure
changes)
 Ha: µ1> µ2 (The men taking calcium see a larger decrease in
blood pressure)

AP Statistics, Section 11.2
Inference Tool Box
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Step 2: Choose the appropriate inference procedure,
and verify the conditions for using the selected
procedure.
Test? Because we don’t know the population standard
deviation, we’ll use a t test. Since we’re not comparing a
person with himself, we have two sample.
 Independent? SRSs, therefore independent.
 Normal? Use back-to-back stemplots to check for normality.
The book says “no departures from normality”

AP Statistics, Section 11.2
Inference Tool Box
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Step 3: Compute the test statistic and and the Pvalue.
t
5.000  ( .273)
8.7432 5.9022

10
11
 1.604
p  value : P(t  1.604)  0.0644
AP Statistics, Section 11.2
Notes on p-value
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There are two options for calculating p-value:
Option 1: Use 2 sample t procedures from data
and allow calculator to compute.
Option 2: Use procedures based on t-distribution
with the smaller n to find d.f.
AP Statistics, Section 11.2
Exercises
14
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11.2 HOMEWORK
11.33-11.35 all
11.37 – 11.39, 40, 43, 53, 54, 58, 62, 64
Due Friday, March27
Post Test Chapters 1 - 11 on Wed., April 1st
AP Statistics, Section 11.2