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Interval Estimationfor MeansNotes of STAT6205 by Dr. Fan
Overview• Sections 6.2 and 6.3
• 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
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The selection of sample is random.
But nothing is random after
we take the sample!
(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 µWhen the population distribution is normal
Case 1: the SD σ is known � Z interval
Case 2: the SD is σ unknown � t interval
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nzX
σ*±
n
stX *±
Estimation of One Mean µWhen the population distribution is not normal but
sample size is larger (n> = 30)
Case 1: the SD σ is known
Z interval
Case 2: the SD is σ unknown
Z interval, replacing σ 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 MeansWhen: Two independent random samples from two normal populations
Case 1: variances are known
Z interval
Case 2: variances are unknownwithout equal variance assumption
Approximate t intervalwith equal variance assumption
Pooled t interval
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Difference of Two MeansWhen: 2 independent random samples from two non-normal populations but large samples (n1, n2 >= 30)
Case 1: variances are known
Z interval
Case 2: variances are unknown
Z interval, replacing σi 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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