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Section 9.4
Inferences About Two Means
(Matched Pairs)
Objective
Compare of two matched-paired means using
two samples from each population.
Hypothesis Tests and Confidence Intervals of
two dependent means use the t-distribution
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Definition
Two samples are dependent if there is some
relationship between the two samples so that
each value in one sample is paired with a
corresponding value in the other sample.
Two samples can be treated as the matched
pairs of values.
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Examples
• Blood pressure of patients before they are
given medicine and after they take it.
• Predicted temperature (by Weather
Forecast) and the actual temperature.
• Heights of selected people in the morning
and their heights by night time.
• Test scores of selected students in Calculus-I
and their scores in Calculus-II.
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Example 1
First sample: weights of 5 students in April
Second sample: their weights in September
These weights make 5 matched pairs
Third line: differences between April weights and
September weights (net change in weight for each
student, separately)
In our calculations we only use differences (d),
not the values in the two samples.
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Notation
d
Individual difference between two matched
paired values
μd
Population mean for the difference of the
two values.
n
Number of paired values in sample
d
Mean value of the differences in sample
sd
Standard deviation of differences in sample
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Requirements
(1) The sample data are dependent
(i.e. they make matched pairs)
(2) Either or both the following holds:
The number of matched pairs is large (n>30)
or
The differences have a normal distribution
All requirements must be satisfied to make a
Hypothesis Test or to find a Confidence Interval
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Tests for Two Dependent Means
Goal: Compare the mean of the differences
H0 : μd = 0
H0 : μd = 0
H0 : μd = 0
H1 : μd ≠ 0
H1 : μd < 0
H1 : μd > 0
Two tailed
Left tailed
Right tailed
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Finding the Test Statistic
t=
d – µd
sd
n
Note: md = 0 according to H0
degrees of freedom: df = n – 1
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Test Statistic
Degrees of freedom
df = n – 1
Note: Hypothesis Tests are done in same way as
in Ch.8-5
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Steps for Performing a Hypothesis
Test on Two Independent Means
•
Write what we know
•
State H0 and H1
•
Draw a diagram
•
Calculate the Sample Stats
•
Find the Test Statistic
•
Find the Critical Value(s)
•
State the Initial Conclusion and Final Conclusion
Note: Same process as in Chapter 8
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Example 1
Assume the differences in weight form a normal
distribution.
Use a 0.05 significance level to test the claim that for the
population of students, the mean change in weight from
September to April is 0 kg
(i.e. on average, there is no change)
Claim: μd = 0
using α = 0.05
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Example 1
d Data: -1 -1 4 -2 1
H0 : µd = 0
H1 : µd ≠ 0
Two-Tailed
H0 = Claim
n=5
d = 0.2
t = 0.186
-tα/2 = -2.78
Sample Stats
t-dist.
df = 4
tα/2 = 2.78
sd = 2.387
Use StatCrunch: Stat – Summary Stats – Columns
Test Statistic
Critical Value
tα/2 = t0.025 = 2.78
(Using StatCrunch, df = 4)
Initial Conclusion: Since t is not in the critical region, accept H0
Final Conclusion: We accept the claim that mean change in weight from
September to April is 0 kg.
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Example 1
d Data: -1 -1 4 -2 1
Sample Stats
H0 : µd = 0
H1 : µd ≠ 0
n=5
Two-Tailed
H0 = Claim
d = 0.2
sd = 2.387
Use StatCrunch: Stat – Summary Stats – Columns
Stat → T statistics→ One sample → With summary
Sample mean:
0.2
Sample std. dev.: 2.387
Sample size:
5
● Hypothesis Test
Null: proportion=
0
Alternative
≠
P-value = 0.8605
Initial Conclusion: Since P-value is greater than α (0.05), accept H0
Final Conclusion: We accept the claim that mean change in weight from
September to April is 0 kg.
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Confidence Interval Estimate
We can observe how the two proportions relate by
looking at the Confidence Interval Estimate of μ1–μ2
CI = ( d – E, d + E )
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Example 2
Sample Stats
n=5
d = 0.2
tα/2 = t0.025 = 2.78
Find the 95% Confidence Interval Estimate
of μd from the data in Example 1
sd = 2.387
(Using StatCrunch, df = 4)
CI = (-2.8, 3.2)
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Example 2
Sample Stats
n=5
d = 0.2
Find the 95% Confidence Interval Estimate
of μd from the data in Example 1
sd = 2.387
Stat → T statistics→ One sample → With summary
Sample mean:
0.2
Sample std. dev.: 2.387
Sample size:
5
● Confidence Interval
Level:
0.95
CI = (-2.8, 3.2)
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