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Confidence Intervals
Confidence Intervals are estimates of population parameters, a range (or interval) of values that
is likely to contain the true value with X% confidence.
They are created to answer the question,
How good is our point estimate (mean or proportion or variance) at estimating the population
value?
(µ  E to µ + E) is the form for a confidence interval for a mean; (p  E to p + E) is the
form for a confidence interval for a proportion.
The margin of error or the maximum error of
estimate is
E
:
For Means:
1.
For n > 30 and ∂ known or ∂ unknown (use s, the sample standard deviation):
OR
If
= .05, then p = .05 that µ is outside the bounds set.
The term
() is the z score that
leaves .025 area in the right tail so that z = 1.96 is the critical value used in the formula for
= .
05.
There is 95% confidence that the true mean lies inside the interval of µ  E to µ + E, where z
= 1.96 is used in the formula for the zscore.
Example
:
Find a 99% confidence interval for the mean of 84.2 on a mathematics placement test
from a sample of size 40 with a standard deviation of 7.3.
is .01 and z for .005 in the right tail is 2.575, so , approximately.
The 99% confidence interval is: 84.2
3.0 (rounded of from 2.97) gives,
81.2 < µ < 87.2
.
There is a probability of .99 that the true or theoretical mean is inside the interval given.
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 Spring '11
 Bailey
 Normal Distribution, Standard Deviation, 1 degrees, 3 pounds

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