Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc Chapter 10 Introduction to Estimation

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc Statistical Inference… Statistical inference is the process by which we acquire information and draw conclusions about populations from samples. 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 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 ( ) is employed to estimate the population mean ( ).

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc 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 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 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 Confidence Interval Estimator for : The probability 1– is called the confidence level. [Sigma known] lower confidence limit (LCL) upper confidence limit (UCL) Usually represented with a “plus/minus” ( ± ) sign

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc Four commonly used confidence levels… Confidence Level  cut & keep handy! Table 10.1

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc 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 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 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 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 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:

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc 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?