1. Economics 173 Business Statistics Lecture 3 Fall, 2001 Professor J. Petry http://www.cba.uiuc.edu/jpetry/Econ_173_fa01/

2. Introduction to Estimation Chapter 9

3. 3 9.1 Introduction Statistical inference is the process by which we acquire information about populations from samples. There are two procedures for making inferences: –Estimation. –Hypotheses testing.

4. 4 9.2 Concepts of Estimation The objective of estimation is to determine the value of a population parameter on the basis of a sample statistic. There are two types of estimators –Point Estimator –Interval estimator

5. 5 –A point estimator draws inference about a population by estimating the value of an unknown parameter using a single value or a point. Point estimator Point Estimator

6. 6 Sample distribution Point estimator Population distribution Parameter ? Point Estimator –A point estimator draws inference about a population by estimating the value of an unknown parameter using a single value or a point.

7. 7 –An interval estimator draws inferences about a population by estimating the value of an unknown parameter using an interval. –The interval estimator is affected by the sample size. Interval estimator Population distribution Sample distribution Parameter Interval Estimator

8. 8 9.3 Estimating the Population Mean when the Population Standard Deviation is Known How is an interval estimator produced from a sampling distribution? –To estimate , a sample of size n is drawn from the population, and its mean is calculated. –Under certain conditions, is normally distributed (or approximately normally distributed.), thus

9. 9 –We know that –This leads to the relationship 1 -  of all the values of obtained in repeated sampling from this distribution, construct an interval that includes (covers) the expected value of the population. 1 -  of all the values of obtained in repeated sampling from this distribution, construct an interval that includes (covers) the expected value of the population.

10. 10 Lower confidence limit Upper confidence limit 1 -  Confidence level See simulation results demonstrating this point

11. 11 Not all the confidence intervals cover the real expected value of 100. 1000 LCL UCL The selected confidence level is 90%, and 10 out of 100 intervals do not cover the real  The confidence interval are correct most, but not all, of the time.

12. 12 Four commonly used confidence levels z  Estimate the mean value of the distribution resulting from the throw of a fair die. It is known that  = 1.71. Use 90% confidence level, and 100 repeated throws of the die. Solution: The confidence interval is The mean values obtained in repeated draws of samples of size 100 result in interval estimators of the form [sample mean -.28, Sample mean +.28] 90% of which cover the real mean of the distribution. The mean values obtained in repeated draws of samples of size 100 result in interval estimators of the form [sample mean -.28, Sample mean +.28] 90% of which cover the real mean of the distribution.

13. 13 The width of the 90% confidence interval = 2(.28) =.56 The width of the 95% confidence interval = 2(.34) =.68 Because the 95% confidence interval is wider, it is more likely to include the value of  The width of the 90% confidence interval = 2(.28) =.56 The width of the 95% confidence interval = 2(.34) =.68 Because the 95% confidence interval is wider, it is more likely to include the value of  Recalculate the confidence interval for 95% confidence level. Solution:.95.90

14. 14 Example 9.1 –The number and the types of television programs and commercials targeted at children is affected by the amount of time children watch TV. –A survey was conducted among 100 North American children, in which they were asked to record the number of hours they watched TV per week. –The population standard deviation of TV watch was known to be  = 8.0 –Estimate the watch time with 95% confidence level.

15. 15 –The parameter to be estimated is  the mean time of TV watch per week per child (of all American Children). –We need to compute the interval estimator for  –From the data in the book, the sample mean is: Since 1 -  =.95,  =.05. Thus  /2 =.025. Z.025 = 1.96 Solution

16. 16 Interpreting the interval estimate wrong –It is wrong to state that the interval estimator is an interval for which there is 1 -  chance that the population mean lies between the LCL and the UCL. –This is so because the  is a parameter, not a random variable. Interpreting the interval estimate wrong –It is wrong to state that the interval estimator is an interval for which there is 1 -  chance that the population mean lies between the LCL and the UCL. –This is so because the  is a parameter, not a random variable.

17. 17 –LCL, UCL and the sample mean are the random variables,  is a parameter, NOT a random variable. –Thus, it is correct to state that there is 1 -  chance that LCL will be less than  and UCL will be greater than 

18. 18 Example 9.2 –To lower inventory costs, the Doll Computer company wants to employ an inventory model. –Lead time demand is normally distributed with standard deviation of 50 computers. –It is required to know the mean in order to calculate optimum inventory levels. –Estimate the mean demand during lead time with 95% confidence.

19. 19 Solution –The parameter to be estimated is  The interval estimator is Demand during 60 lead times is recorded 514, 525, …., 476. The sample mean is calculated The 95% confidence interval is:

20. 20 –Wide interval estimator provides little information. Where is  ?????????????????????????????? Ahaaa! Here is a much smaller interval. If the confidence level remains unchanged, the smaller interval provides more meaningful information. Here is a much smaller interval. If the confidence level remains unchanged, the smaller interval provides more meaningful information. Information and the Width of the Interval

21. 21 The width of the interval estimate is a function of: the population standard deviation the confidence level the sample size.

22. 22 90% Confidence level To maintain a certain level of confidence, changing to a larger standard deviation requires a longer confidence interval. To maintain a certain level of confidence, changing to a larger standard deviation requires a longer confidence interval.  /2 =.05 Suppose the standard deviation has increased by 50%.

23. 23 90% Confidence level 95% Let us increase the confidence level from 90% to 95%.  /2 = 2.5% Increasing the sample size decreases the width of the interval estimate while the confidence level can remain unchanged. Increasing the sample size decreases the width of the interval estimate while the confidence level can remain unchanged. There is an inverse relationship between the width of the interval and the sample size Increasing the confidence level produces a wider interval Increasing the confidence level produces a wider interval  /2 = 5%

24. 24 9.4 Selecting the Sample size We can control the width of the interval estimate by changing the sample size. Thus, we determine the interval width first, and derive the required sample size. The phrase “estimate the mean to within W units”, translates to an interval estimate of the form

25. 25 The required sample size to estimate the mean is Example 9.3 –To estimate the amount of lumber that can be harvested in a tract of land, the mean diameter of trees in the tract must be estimated to within one inch with 99% confidence. –What sample size should be taken? (assume diameters are normally distributed with  = 6 inches.

26. 26 Solution –The estimate accuracy is +/-1 inch. That is w = 1. –The confidence level 99% leads to  =.01, thus z  /2 = z.005 = 2.575. –We compute