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 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…
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.5
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.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.6
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.
10.7
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.
10.8
Unbiased Estimators…
An unbiased estimator of a population parameter is an
estimator whose expected value is equal to that parameter.
E.g. the sample mean X is an unbiased estimator of the
population mean
, since:
E(X) =
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.9
Consistency…
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.
E.g. X is a consistent estimator of
because:
V(X) is
That is, as n grows larger, the variance of X grows smaller.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.10
Relative Efficiency…
If there are two unbiased estimators of a parameter, the one
whose variance is smaller is said to be relatively efficient.
E.g. both the the sample median and sample mean are
unbiased estimators of the population mean, however, the
sample median has a greater variance than the sample mean,
so we choose since it is relatively efficient when
compared to the sample median.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.11
Estimating
when
is known…
We can calculate an interval estimator from a sampling
distribution, by:
Drawing a sample of size n from the population
Calculating its mean,
And, by the central limit theorem, we know that X is
normally (or approximately normally) distributed so…
…will have a standard normal (or approximately normal)
distribution.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.12
Estimating
when
is known…
Looking at this in more detail…
Known, i.e. standard
normal distribution
Known, i.e. its
assumed we know
the population
standard deviation…
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
Known, i.e. sample
mean
Unknown, i.e. we
want to estimate
the population mean
Known, i.e. the
number of items
sampled
10.13
Estimating
when
is known…
We established in Chapter 9:
Thus, the probability that the interval:
contains the population mean
is 1–
confidence interval estimator for .
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
the confidence
interval
the sample mean is
in the center of the
interval…
. This is a
10.14
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)
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.15
Graphically…
…here is the confidence interval for
:
width
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.16
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 the interval estimate of the population mean.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.17
Four commonly used confidence levels…
Confidence Level

cut & keep handy!

Table 10.1
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.18
Example 10.1…
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…
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.19
Example 10.1…
“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:
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.20
Example 10.1…
CALCULATE
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.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.21
Using Excel…
CALCULATE
By using the Data Analysis Plus™ toolset, on the Xm10-01
spreadsheet, we get the same answer with less effort…
Tools > Data Analysis Plus > Z-Estimate: Mean
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.22
Example 10.1…
INTERPRET
The estimation for the mean demand during lead time lies
between 340.76 and 399.56 — we can use this as input in
developing an inventory policy.
That is, we estimated that the mean demand during lead time
falls between 340.76 and 399.56, and this type of estimator
is correct 95% of the time. That also means that 5% of the
time the estimator will be incorrect.
Incidentally, the media often refer to the 95% figure as “19
times out of 20,” which emphasizes the long-run aspect of
the confidence level.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.23
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.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.24
Interval Width…
The width of the confidence interval estimate is a function of
the confidence level, the population standard deviation, and
the sample size…
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.25
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.
10.26
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
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.27
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.
10.28
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!
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.29
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).
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.30
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:
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.31
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?
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.32
Example 10.2…
Things we know:
Confidence level = 99%, therefore
=.01
We want
1 , hence W=1.
We are given that
= 6.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.33
Example 10.2…
We compute…
That is, we will need to sample at least 239 trees to have a
99% confidence interval of
1
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
10.34