Chapter 10: CI and HT Based on Two
Samples or Treatments
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Notation
Population
Mean Variance SD
Population 1
μ1
1
𝜎12
Population 2
μ2
2
𝜎22
Sample from
Population 1
Sample from
Population 2
Sample Statistics
Sample size Mean Variance SD
n1
x1̄
s1
𝑠12
n2
x2̄
𝑠22
s2
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Independent and Paired Samples
1. Two samples are independent if the process
of selecting individuals or objects in sample 1
has no effect on, or no relation to, the
selection of individuals or objects in sample 2.
We call this 2 – sample independent.
2. A paired data set is the result of matching
each individual or object in sample 1 with a
similar individual or object in sample 2. We
call this 2 – sample paired.
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10.1/10.2: Comparing Two Population
Means Using Independent Samples - Goals
• Be able to determine when you can perform 2-sample
independent analyses.
• Perform a two-sample hypothesis test and summarize
the results when the two samples are independent.
• Be able to construct a level C confidence interval for
the difference between two means and interpret the
results when the two samples are independent.
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Conditions for Inference: 2 – sample
independent
1. Each group is considered to be a sample from
a distinct population.
• We have an SRS from the population of
interest for each variable.
2. The responses in each group are
independent of those in the other group.
3. The statistic that we measure has a Normal
distribution.
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Two-sample independent (Z):
Hypothesis
Test
• Step 1
• Step 2
H0: µ1 - µ2 = Δ0
• Step 3
𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑜𝑟 − 𝑛𝑢𝑙𝑙 𝑣𝑎𝑙𝑢𝑒
𝑧𝑡𝑠 =
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑜𝑟
𝑥1 − 𝑥2 − Δ0
=
𝜎12 𝜎22
+
𝑛1 𝑛2
• Step 4
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Two-sample independent (Z):
Confidence Interval
The 100 (1 – α)% confidence interval for µ1 - µ2
is
estimator ± (critical value)(standard deviation of
the estimator)
= 𝑥1 − 𝑥2 ± 𝑧𝛼
2
𝜎12 𝜎22
+
𝑛1 𝑛2
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Satterthwaite Approximation
s s 
  
n1 n 2 
df 
2
2
2
2
1 s1 
1 s2 
  
 
n1 1 n1  n 2 1 n 2 
2
1
2
2
2

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Two-sample Test (independent): Summary
Null hypothesis: H0: μ1 – μ2 = Δ0
𝑇𝑒𝑠𝑡 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐:
′
𝑇𝑡𝑠
=
𝑥1 − 𝑥2 − Δ0
𝑠12 𝑠22
+
𝑛1 𝑛2
Upper-tailed
Lower-tailed
two-sided
Alternative
Hypothesis
Ha: μ1 – μ2 > Δ
Ha: μ1 – μ2 < Δ
Ha: μ1 – μ2 ≠ Δ
P-Value
P(T ≥ tts)
P(T ≤ tts)
2P(T ≥ |tts|)
Note: If we are determining if the two populations are equal,
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then Δ0 = 0
Two-sample independent (t):
• Hypothesis test
𝑇′𝑡𝑠 =
𝑥1 − 𝑥2 − Δ0
𝑠12 𝑠22
+
𝑛1 𝑛2
• Confidence interval
𝑥1 − 𝑥2 ± 𝑡𝛼
2,𝜈
𝑠12 𝑠22
+
𝑛1 𝑛2
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Robustness of the 2 sample tprocedure
• The t-procedure is very robust against
normality. Let n = n1 + n2
– n < 15 : population distribution should be
close to normal.
– 15 < n < 40: mild skewedness is acceptable
– n > 40: procedure is usually valid.
• Best when n1  n2
• Best when distributions are similar.
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10.3: Paired Data - Goals
• Be able to determine when you should use 2-sample
paired analyses.
• Be able to construct a level C confidence interval for a
matched pair and interpret the results.
• Perform a matched pair t hypothesis test and
summarize the results.
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Conditions for Inference: 2 – sample
paired
1. Each pair is considered to be a sample from a
population of pairs.
• We have an SRS from the population of
pairs.
2. Each pair is independent of the other pairs.
3. The difference of the each pair that we
measure has a Normal distribution with
mean D and standard deviation σD.
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Two-sample Matched Pair
𝑆𝐷
𝑆𝐸 =
𝑛
𝑆𝐷
𝑑 ± 𝑡𝛼 2,𝑛−1
𝑛
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Two-sample matched pair Test: Summary
Null hypothesis: H0: μD = 0
𝐷 − Δ0
𝑇𝑒𝑠𝑡 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐: 𝑇𝑡𝑠 =
𝑆𝐷 𝑛
Alternative
Hypothesis
One-sided: upper-tailed Ha: μD > 0
One-sided: lower-tailed Ha: μD < 0
two-sided
Ha: μD ≠ 0
P-Value
P(T ≥ tts)
P(T ≤ tts)
2P(T ≥ |tts|)
Note: If we are determining if the two populations
are equal, then Δ0 = 0
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Independent vs. Paired
1. If there is great heterogeneity between
experimental units and a large correlation
within experimental units then a paired
experiment is preferable.
2. If the experimental units are relatively
homogeneous and the correlation within pairs
is not large, then unpaired experiments should
be used
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