Dr. Jerrell T. Stracener

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EMIS 7370 STAT 5340
Department of Engineering Management, Information
and Systems
Probability and Statistics for Scientists and
Engineers
Estimation of Standard Deviation Percentiles
Dr. Jerrell T. Stracener
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Estimation of Standard Deviation Percentiles

• Estimation of Standard Deviation - Normal
  Distribution
• Estimation of Ratio of Standard Deviations -
  Normal Distributions
• Estimation of Percentiles - Tolerance Intervals

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Estimation of Standard Deviations
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Estimation of Standard Deviation - Normal
Distribution

• Point Estimate of ?
• (1 - ?) ? 100 Confidence Interval for ? is,
  
• where
• and

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Example Estimation of ?
The following are the weights, in decagrams, of
10 packages of grass seed 46.4, 46.1, 45.8,
47.0, 46.1, 45.9, 48.8, 46.9, 45.2,
46.0 Estimate ? in terms of a point estimate and
a 95 confidence interval for the standard
deviation of all such packages of grass seed,
assuming a normal population.
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Example - Solution
First we find
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Example - Solution
Then a point estimate of ?
A (1 - ?) ? 100 Confidence Interval for ?
is ,
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Example - Solution
Where
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Example - Solution
and
, and
where and and is the
value of X x2n-1 such that
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Estimation of Ratio of Two Standard
Deviations Normal Distribution

• Let X11, X12, , , and X21, X22, , be
• random samples from N(?1, ?1) and N(?2, ?2),
• respectively
• Point estimation of
• where
• for i 1, 2

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Estimation of Ratio of Two Standard
Deviations Normal Distribution

• (1 - ?) ? 100 Confidence Interval for
• where
• and

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Estimation of Ratio of Two Standard
Deviations Normal Distribution
where is the value of the
F-Distribution with and degrees of
freedom for which
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Example - Estimation of ?1/?2
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Example - solution
and
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Estimation of Percentiles - Tolerance Intervals
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Tolerance Limits
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Example - Tolerance Interval
A machine is producing metal pieces that are
cylindrical in shape. A sample of these pieces
is taken and the diameters are found to be 1.01,
0.97, 1.03, 1.04, 0.99, 0.98, 0.99, 1.01, and
1.03 centimeters. Find tolerance limits that
will contain at least 95, with a 99
confidence, of the metal pieces produced by this
machine, assuming a normal distribution.
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Solution
The sample mean and standard deviation for
the given data are x 1.0056 and s
0.0245 From the Tolerance Factors Table for n
9, 1 - ? 0.99, and 1 - ? 0.95 we find k
4.550 for two-sided limits. Hence the tolerance
limits are and
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Solution
That is, we are 99 confident that the tolerance
interval from 0.894 to 1.117 will contain at
least 95 of the metal pieces produced by this
machine.