Transcript Chapter 10 - Introduction to Estimation

Chapter 10
Introduction to Estimation
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.1
Statistical Inference…
Statistical inference is the process by which we acquire
information and draw conclusions about populations from
samples.
Statistics
Information
Data
Population
Sample
Inference
Statistic
Parameter
In order to do inference, we require the skills and knowledge of descriptive statistics,
probability distributions, and sampling distributions.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.2
Estimation…
There are two types of inference: estimation and hypothesis
testing; estimation is introduced first.
The objective of estimation is to determine the approximate
value of a population parameter on the basis of a sample
statistic.
E.g., the sample mean (
population mean ( ).
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
) is employed to estimate the
10.3
Estimation…
The objective of estimation is to determine the approximate
value of a population parameter on the basis of a sample
statistic.
There are two types of estimators:
Point Estimator
Interval Estimator
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.4
Point & Interval Estimation…
For example, suppose we want to estimate the mean summer
income of a class of business students. For n=25 students,
is calculated to be 400 $/week.
point estimate
interval estimate
An alternative statement is:
The mean income is between 380 and 420 $/week.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.5
Qualities of Estimators…
Qualities desirable in estimators include unbiasedness,
consistency, and relative efficiency:
• An unbiased estimator of a population parameter is an
estimator whose expected value is equal to that parameter.
• An unbiased estimator is said to be consistent if the
difference between the estimator and the parameter grows
smaller as the sample size grows larger.
• If there are two unbiased estimators of a parameter, the one
whose variance is smaller is said to be relatively efficient.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.6
Confidence Interval Estimator for
The probability 1–
[Sigma known]
:
is called the confidence level.
Usually represented
with a “plus/minus”
( ± ) sign
upper confidence
limit (UCL)
lower confidence
limit (LCL)
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10.7
Four commonly used confidence levels…
Confidence Level

cut & keep handy!

Table 10.1
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10.8
Interval Width…
The width of the confidence interval estimate is a function of
the confidence level, the population standard deviation, and
the sample size…
A larger confidence level
produces a w i d e r
confidence interval:
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.9
Interval Width…
The width of the confidence interval estimate is a function of
the confidence level, the population standard deviation, and
the sample size…
Larger values of Sigma
produce w i d e r
confidence intervals
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.10
Interval Width…
The width of the confidence interval estimate is a function of
the confidence level, the population standard deviation, and
the sample size…
Increasing the sample size decreases the width of the
confidence interval while the confidence level can remain
unchanged.
Note: this also increases the cost of obtaining additional data
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.11
Selecting the Sample Size…
We can control the width of the interval by determining the
sample size necessary to produce narrow intervals.
Suppose we want to estimate the mean demand “to within 5
units”; i.e. we want to the interval estimate to be:
Since:
It follows that
Solve for n to get requisite sample size!
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10.12
Selecting the Sample Size…
Solving the equation…
that is, to produce a 95% confidence interval estimate of the
mean (±5 units), we need to sample 865 lead time periods
(vs. the 25 data points we have currently).
The general formula for the sample size needed to estimate a
population mean with an interval estimate of:
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10.13
Example 10.2…
A lumber company must estimate the mean diameter of trees
to determine whether or not there is sufficient lumber to
harvest an area of forest. They need to estimate this to within
1 inch at a confidence level of 99%. The tree diameters are
normally distributed with a standard deviation of 6 inches.
How many trees need to be sampled?
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10.14