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 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 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 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 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…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.

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc Confidence 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” ( ± ) sign

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

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc Example 10.1… A 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 Example 10.1… “We 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 pop’n mean: And so our confidence interval estimator will be: IDENTIFY

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc Example 10.1… In order to use our confidence interval estimator, we need the following pieces of data: therefore: The lower and upper confidence limits are and n 25 Given Calculated from the data… CALCULATE

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

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc 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 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 Example 10.2… Things 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 Example 10.2… We compute… That is, we will need to sample at least 239 trees to have a 99% confidence interval of 1