Interval
Estimation
for Means
Notes of STAT6205 by Dr. Fan
Overview
• Sections 6.2 and 6.3
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Introduction to interval estimation
Confidence Intervals for One mean
General construction of a confidence interval
Confidence Intervals for difference of two means
Pair Samples
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Interval Estimation
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Confidence vs. Probability
The selection of
sample is random.
But nothing is
random after
we take the sample!
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(Symmetric) Confidence
Interval
• A k% confidence interval (C.I.) for a parameter is an
interval of values computed from sample data that
includes the parameter k% of time:
Point estimate + multiplier x standard error
• K% of time = k% of all possible samples
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Estimation of One Mean m
When the population distribution is normal
Case 1: the SD s is known  Z interval
Case 2: the SD is s unknown  t interval
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X z
*
X t
*

n
s
n
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Estimation of One Mean m
When the population distribution is not normal but
sample size is larger (n> = 30)
Case 1: the SD s is known
Z interval
Case 2: the SD is s unknown
Z interval, replacing s by s.
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Examples/Problems
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Examples/Problems
• Example 1:
We would like to construct a 95% CI for the true mean
weight of a newborn baby. Suppose the weight of a
newborn baby follows a normal distribution. Given a
random sample of 20 babies, with the sample mean
of 8.5 lbs and sample s.d. of 3 lbs, construct such a
interval estimate.
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Can CI be Asymmetric?
• Endpoints can be unequal distance from the
estimate
• Can be one-sided interval
Example: Repeat Example 1 but find its one-sided
interval (lower tailed).
• Why symmetric intervals are the best when dealing
with the normal or t distribution unless otherwise
stated?
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How to Construct Good CIs
• Wish to get a short interval with high degree of
confidence
Tradeoff:
• The wider the interval, the less precise it is
• The wider the interval, the more confidence that it
contains the true parameter value.
Best CI:
For any given confidence level, it has the shortest
interval.
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Difference of Two Means
When: Two independent random samples from two
normal populations
Case 1: variances are known
Z interval
Case 2: variances are unknown
without equal variance assumption
Approximate t interval
with equal variance assumption
Pooled t interval
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Difference of Two Means
When: 2 independent random samples from two nonnormal populations but large samples (n1, n2 >= 30)
Case 1: variances are known
Z interval
Case 2: variances are unknown
Z interval, replacing si by Si.
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Examples/Problems
• Example 2: Do basketball players have bigger feet
than football players?
• Example 3: To compare the performance of two
sections, a test was given to both sections.
• From an estimation point of view (for variances),
why is the pooled method preferred?
• How to check the assumption of equal variance?
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Example
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Example
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Paired Samples
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