Download Lecture 14 - UC Davis Statistics

STA 13
Lecture 14
Comparison of Two Independent Populations (continued)
III. Inferences When σ1 and σ2 are Unknown
A Practical Problem: Suppose we are interested in comparison
average systolic blood pressures (SBP) between males and females
45-50 years old. We collected the following data
Male:
Female:
130
127
134
138
137
125
135
134
140
134
Question: Is SBP on average higher for males compared to females?
a. If the two populations are normal with unknown variances, then
b. A 100( 1- α) % confidence interval for ( µ1 - µ2 ) when the
population standard deviations are unknown is:
Example: Systolic Blood Pressure
c. To test
Ho : (µ1 - µ 2) = D0 , we compute t =
and reject Ho in favor of Ha of
Ha : (µ1 - µ 2) < D0 if t < - tα
Ha : (µ1 - µ 2) > D0 if t > tα
Ha : (µ1 - µ 2) ≠ D0 if t < - tα/2 or z > tα/2
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Example: Systolic Blood Pressure
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d. The Pooled Two-Sample Procedure:
If σ1 and σ2 are unknown but σ1 = σ2 , then a pooled estimate of the
common variance is:
Using this estimate, a 100(1- α )% confidence interval for ( µ1 - µ2 ) is
To test Ho : (µ1 - µ 2) = D0 , we compute t =
and reject Ho in favor of Ha
Ha : (µ1 - µ 2) < D0 if t < - tα
Ha : (µ1 - µ 2) > D0 if t > tα
Ha : (µ1 - µ 2) ≠ D0 if t < - tα/2 or z > tα/2
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Example: The Orange Juice Study
We would like to compare the taste of a “new” product to an “old”
product based on samples of size 15 and 14. Suppose the mean of the
first sample is 9.5 with a standard deviation of 1.3, and the mean of
the second sample is 7.1 with a standard deviation 1.5. Is the new
product better than the old product by one unit? Test at 0.05 level of
significance.
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