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Statistical Methods
Introduction to Estimation
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noha hussein elkhidir
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 ( ).
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) is employed to estimate the
noha hussein elkhidir
Some Concepts
*Estimators: Random variables used to estimate
population parameters.
Example:
p^ is an estimator of p
*Estimates: Specific values of an estimator.
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
16/04/35
noha hussein elkhidir
Point Estimator…
A point estimator draws inferences about a population by
estimating the value of an unknown parameter using a single
value or point.
We saw earlier that point probabilities in continuous
distributions were virtually zero. Likewise, we’d expect that
the point estimator gets closer to the parameter value with an
increased sample size, but point estimators don’t reflect the
effects of larger sample sizes. Hence we will employ the
interval estimator to estimate population parameters…
16/04/35
noha hussein elkhidir
Interval Estimator…
An interval estimator draws inferences about a population
by estimating the value of an unknown parameter using an
interval.
That is we say (with some ___% certainty) that the
population parameter of interest is between some lower and
upper bounds.
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noha hussein elkhidir
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 alternative777 statement is:
The mean income is between 380 and 420 $/week.
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noha hussein elkhidir
Qualities of Estimators…Statisticians have already determined
the “best” way to estimate a population parameter.
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.
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noha hussein elkhidir
Confidence Interval Estimator for
The probability 1–
is called the confidence level.
Usually represented
with a “plus/minus”
( ± ) sign
upper confidence
limit (UCL)
lower confidence
limit (LCL)
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noha hussein elkhidir
Graphically…
…the actual location of the population mean
…may be here…
…or here…
…
…or possibly even here…
The population mean is a fixed but unknown quantity. Its incorrect to interpret the
confidence interval estimate as a probability statement about . The interval acts as the
lower and upper limits of thenoha
interval
estimate of the population mean.
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hussein elkhidir
Four commonly used confidence levels…
Confidence Level
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noha hussein elkhidir
Example:
A computer company samples demand during lead time over
25 time periods:
235
421
394
261
386
374
361
439
374
316
309
514
348
302
296
499
462
344
466
332
253
369
330
535
334
Its is known that the standard deviation of demand over lead
time is 75 computers. We want to estimate the mean demand
over lead time with 95% confidence in order to set inventory
levels…
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noha hussein elkhidir
Example :
“We want to estimate the mean demand over lead time with
95% confidence in order to set inventory levels…”
IDENTIFY
Thus, the parameter to be estimated is the pop’n mean:
And so our confidence interval estimator will be:
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noha hussein elkhidir
Example :
In order to use our confidence interval estimator, we need the
following pieces of data:
370.16
Calculated from the data…
1.96
75
n
Given
25
therefore:
The lower and upper confidence limits are 340.76 and 399.56.
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noha hussein elkhidir
Interval Width…
A wide interval provides little information.
For example, suppose we estimate with 95% confidence that
an accountant’s average starting salary is between $15,000
and $100,000.
Contrast this with: a 95% confidence interval estimate of
starting salaries between $42,000 and $45,000.
The second estimate is much narrower, providing accounting
students more precise information about starting salaries.
16/04/35
noha hussein elkhidir
Interval Width…
The width of the confidence interval estimate is a function of
the confidence level, the population standard deviation, and
the sample size…
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noha hussein elkhidir
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:
16/04/35
noha hussein elkhidir
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
produce w i d e r
confidence intervals
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noha hussein elkhidir
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
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noha hussein elkhidir
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
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Solve for n to get requisite sample size!
noha hussein elkhidir
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).
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noha hussein elkhidir
Sample Size to Estimate a Mean…
The general formula for the sample size needed to estimate a
population mean with an interval estimate of:
Requires a sample size of at least this large:
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noha hussein elkhidir
Example :
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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noha hussein elkhidir
Example 10.2…
Things we know:
Confidence level = 99%, therefore
We want
1 , hence W=1.
We are given that
= 6.
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noha hussein elkhidir
=.01
Example :
We compute…
That is, we will need to sample at least 239 trees to have a
99% confidence interval of
1
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noha hussein elkhidir