Transcript What is a Confidence Interval?

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8
Chapter
Sampling Distributions and
Estimation
Sampling Variation
Estimators and Sampling Distributions
Sample Mean and the Central Limit Theorem
Confidence Interval for a Mean (m) with Known s
Confidence Interval for a Mean (m) with Unknown s
Confidence Interval for a Proportion (p)
Sample Size Determination for a Mean and a Proportion
McGraw-Hill/Irwin
Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
Sampling Variation
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From figure 8.2, we see that the sample means (red markers) have
much less variation than the individual sample items. This is an
example of sample variation.
Sample variation (uncontrollable)
Population variation (uncontrollable)
Sample size (controllable)
8-2
Estimators and Sampling
Distributions
Some Terminology
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Sample statistic – a random variable whose value depends on
which population items happen to be included in the random
sample.
Sampling variation is the variation for a sample statistic based on
different samples.
Estimator – a statistic derived from a sample to infer the value of
a population parameter.
Estimate – the value of the estimator in a particular sample.
Population parameters
are represented by
Greek letters and the
corresponding statistic
by Roman letters.
8-3
Estimators and Sampling
Distributions
Examples of Estimators
8-4
Estimators and Sampling
Distributions
Sampling Distributions
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The sampling distribution of an estimator is the probability
distribution of all possible values the statistic may assume when a
random sample of size n is taken.
An estimator is a random variable since samples vary.
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Sampling error =  – 
8-5
Estimators and Sampling
Distributions
Bias
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Bias is the difference between the expected
value of the estimator and the true parameter.
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Bias = E(  ) – 
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An estimator is unbiased if E(  ) = 
On average, an unbiased estimator neither
overstates nor understates the true
parameter.
8-6
Estimators and Sampling
Distributions
Efficiency
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Efficiency refers to the variance of the estimator’s sampling
distribution.
A more efficient estimator has smaller variance.
Figure 8.5
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Estimators and Sampling
Distributions
Consistency
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A consistent estimator converges toward the parameter being
estimated as the sample size
increases.
Figure 8.6
8-8
Sample Mean and the
Central Limit Theorem
Central Limit Theorem (CLT) for a Mean
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If a random sample of size n is drawn from a population with
mean m and standard deviation s, the distribution of the sample
mean x approaches a normal distribution with mean m and
standard deviation
sx = s/ n as the sample size increase.
If the population is normal, the distribution of the sample mean is
normal regardless of sample size.
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Sample Mean and the
Central Limit Theorem
Illustrations of Central Limit Theorem
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Confidence Interval for a
Mean (m) with Known s
What is a Confidence Interval?
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A sample mean x is a point estimate of the population
mean m.
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A confidence interval for the mean is a range
mlower < m < mupper
The confidence level is the probability that the
confidence interval contains the true population
mean.
The confidence level (usually expressed as a %)
is the area under the curve of the sampling
distribution.
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8-11
Confidence Interval for a
Mean (m) with Known s
What is a Confidence Interval?
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The confidence interval for m with known s is:
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Confidence Interval for a
Mean (m) with Unknown s
Student’s t Distribution
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Use the Student’s t distribution instead of the normal distribution
when the population is normal but the standard deviation s is
unknown and the sample size is small.
x+t
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s
n
The confidence interval for m (unknown s)
is
s
s
x-t
<m< x+t
n
n
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Confidence Interval for a Proportion (p)
Confidence Interval for p
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If both np > 10 and n(1-p) > 10, then the confidence interval for p is
p+z
p(1-p)
n
Where z is based on the desired confidence.
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Since p is unknown, the confidence
interval for p = x/n (assuming a large
sample) is
p(1-p)
p+z
n
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Sample Size Determination for a Mean
Sample Size to Estimate m
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To estimate a population mean with a precision of + E (allowable
error), you would need a sample of size
n = zs
E
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Sample Size Determination
for a Proportion
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To estimate a population proportion with a precision of + E
(allowable error), you would need a sample of size
z
n=
E
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2
p(1-p)
Since p is a number between 0 and 1, the
allowable error E is also between 0 and 1.
8-16