Chapter 10 - Introduction to Estimation

10.1Chapter 10Introduction to EstimationCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Keller: Stats for Mgmt & Econ, 7th EdApril 9, 2015Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.110.2EstimationThere 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 ( ) is employed to estimate the population mean ( ).

Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.10.3EstimationThe 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 EstimatorCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.10.4Point EstimatorA 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, wed expect that the point estimator gets closer to the parameter value with an increased sample size, but point estimators dont reflect the effects of larger sample sizes. Hence we will employ the interval estimator to estimate population parametersCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.10.5Interval EstimatorAn 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.10.6Point & Interval EstimationFor 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.10.7Qualities of EstimatorsStatisticians 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.Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.10.8Confidence Interval Estimator for :The probability 1 is called the confidence level.

lower confidence limit (LCL)upper confidence limit (UCL)Usually represented with a plus/minus ( ) signCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.10.9

Graphicallythe actual location of the population mean

may be hereor hereor possibly even hereThe 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.10.10Four commonly used confidence levelsConfidence Level

cut & keep handy!Table 10.1Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.10.11Example 10.1A computer company samples demand during lead time over 25 time periods:

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.10.12Example 10.1We want to estimate the mean demand over lead time with 95% confidence in order to set inventory levels

Thus, the parameter to be estimated is the popn mean:

And so our confidence interval estimator will be:

IDENTIFY

Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.10.13

Example 10.1In order to use our confidence interval estimator, we need the following pieces of data:

therefore:

The lower and upper confidence limits are 340.76 and 399.56.370.161.9675n25

Given Calculated from the data

CALCULATECopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.10.14Interval WidthA wide interval provides little information.For example, suppose we estimate with 95% confidence that an accountants 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.10.15Interval WidthThe 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.10.16Interval WidthThe width of the confidence interval estimate is a function of the confidence level, the population standard deviation, and the sample size

A larger confidence levelproduces a w i d e r confidence interval:

Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.10.17

Interval WidthThe 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.10.18Interval WidthThe 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.10.19Selecting the Sample SizeWe 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.10.20Selecting the Sample SizeSolving 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.10.21Sample Size to Estimate a MeanThe 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.10.22Example 10.2A 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.10.23Example 10.2Things we know:

Confidence level = 99%, therefore =.01

We want , hence W=1.We are given that = 6.

1 Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.10.24Example 10.2We compute

That is, we will need to sample at least 239 trees to have a99% confidence interval of

1 Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.