Download 9.1.2 Confidence Intervals

Confidence Intervals about a Population Mean
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Confidence Intervals
The first step in estimating the value of an unknown parameter is to obtain a random sample and
use the data from the sample to obtain a point estimate of the parameter.
A point estimate is the value of a statistic that estimates the value of a parameter.
The sample mean, x , is a point estimate of the population mean, µ.
The sample standard deviation, s, is a point estimate of the population standard deviation σ.
The second step in estimating the value of an unknown parameter is to obtain a confidence
interval in which we have a certain level of confidence that the true value of the parameter lies.
A confidence interval for an unknown parameter consists of an interval of numbers about the
point estimate of the parameter.
The level of confidence represents the expected proportion of intervals that will contain the
parameter if a large number of different samples is obtained. The level of confidence is
denoted (1 − α)·100%.
Confidence Intervals about a Population Mean
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(1 − α)·100%
α
α
2
2
µ
 σ 
zα / 2 

 n
 σ 
zα / 2 

 n
Confidence Intervals about a Population Mean
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Constructing a (1 − α)·100% Confidence Interval about µ, with σ Known
Suppose a simple random sample of size n is taken from a population with unknown mean µ
and known standard deviation σ. A (1 − α)·100% confidence interval for µ is given by
 σ 
 σ 
x − zα / 2 
 < µ < x + zα / 2 

 n
 n
where zα / 2 is the critical Z-value.
Note: The sample size must be large (n ≥ 30) or the population must be normally distributed.
For a 90% confidence interval: zα / 2 = 1.645
For a 95% confidence interval: zα / 2 = 1.960
For a 99% confidence interval: zα / 2 = 2.575
Interpretation of a Confidence Interval
1. A (1 − α)·100% confidence interval indicates that, if we obtain many simple random
samples of size n from the population whose mean, µ, is unknown, then approximately
(1 − α)·100% of the intervals will contain µ.
2. We are (1 − α)·100% confident that the population mean, µ, is between
.
and
Confidence Intervals about a Population Mean
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Interpretation of a Confidence Interval
A (1 − α)·100% confidence interval indicates that, if we obtain many simple random samples
of size n from a population whose mean µ is unknown, then approximately (1 − α)·100% of the
intervals will contain µ.
In other words: We are (1 − α)·100% confident that the population mean is between the
lower bound and upper bound of the confidence interval.
Caution: A 95% confidence interval does not mean that there is a 95% probability that the
interval contains µ.
Confidence Intervals about a Population Mean
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Margin of Error
The margin of error, E, in a (1 − α)·100% confidence interval in which σ is known is given by
E = zα / 2 ⋅
σ
n
where n is the sample size.
Note: The sample size must be large (n ≥ 30) or the population must be normally distributed.