Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8-1 Business Statistics, 4e by Ken Black Chapter 8 Statistical Inference: Estimation for Single Populations

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8-2 Learning Objectives Know the difference between point and interval estimation. Estimate a population mean from a sample mean when  is known. Estimate a population mean from a sample mean when  is unknown. Estimate a population proportion from a sample proportion. Estimate the population variance from a sample variance. Estimate the minimum sample size necessary to achieve given statistical goals.

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8-3 Statistical Estimation Point estimate -- the single value of a statistic calculated from a sample Interval Estimate -- a range of values calculated from a sample statistic(s) and standardized statistics, such as the z. –Selection of the standardized statistic is determined by the sampling distribution. –Selection of critical values of the standardized statistic is determined by the desired level of confidence.

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8-4 Confidence Interval to Estimate  when  is Known Point estimate Interval Estimate

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8-5 Distribution of Sample Means for (1-  )% Confidence  X  Z 0

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8-6 Distribution of Sample Means for (1-  )% Confidence  X Z 0

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8-7 Distribution of Sample Means for (1-  )% Confidence  X Z 0

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8-8 Distribution of Sample Means for 95% Confidence .4750 X 95%.025 Z

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons % Confidence Interval for 

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons % Confidence Intervals for   X 95% X X X X X X

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons % Confidence Intervals for   X 95% XXXXXX Is our interval,  , in the red?

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Demonstration Problem 8.1

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Demonstration Problem 8.2

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Confidence Interval to Estimate  when n is Large and  is Known

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Car Rental Firm Example

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Z Values for Some of the More Common Levels of Confidence 90% 95% 98% 99% Confidence Level z Value

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Estimating the Mean of a Normal Population: Unknown  The population has a normal distribution. The value of the population standard deviation is unknown. z distribution is not appropriate for these conditions t distribution is appropriate

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons The t Distribution Developed by British statistician, William Gosset A family of distributions -- a unique distribution for each value of its parameter, degrees of freedom (d.f.) Symmetric, Unimodal, Mean = 0, Flatter than a z t formula

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Comparison of Selected t Distributions to the Standard Normal Standard Normal t (d.f. = 25) t (d.f. = 1) t (d.f. = 5)

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Table of Critical Values of t df t t t t t  tt   With df = 24 and  = 0.05, t  =

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Confidence Intervals for  of a Normal Population: Unknown 

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Solution for Demonstration Problem 8.3

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Solution for Demonstration Problem 8.3

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Comp Time: Excel Normal View

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Comp Time: Excel Formula View A BCDEF 1Comp Time Data n ==COUNT(A2:F4) 7Mean ==AVERAGE(A2:F4) 8S ==STDEV(A2:F4) 9Std Error ==B8/SQRT(B6)  = df ==B6-1 13t ==TINV(B11,B12) =B7-B13*B9  =B7+B13*B9

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Confidence Interval to Estimate the Population Proportion

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Solution for Demonstration Problem 8.5

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Population Variance Variance is an inverse measure of the group’s homogeneity. Variance is an important indicator of total quality in standardized products and services. Managers improve processes to reduce variance. Variance is a measure of financial risk. Variance of rates of return help managers assess financial and capital investment alternatives. Variability is a reality in global markets. Productivity, wages, and costs of living vary between regions and nations.

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Estimating the Population Variance Population Parameter   Estimator of     formula for Single Variance

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Confidence Interval for  2

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Selected  2 Distributions df = 3 df = 5 df = 10 0

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons  2 Table df = df E E With df = 5 and  = 0.10,  2 =

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Two Table Values of  df = df E

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons % Confidence Interval for  2

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Solution for Demonstration Problem 8.6

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Determining Sample Size when Estimating  z formula Error of Estimation (tolerable error) Estimated Sample Size Estimated 

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Sample Size When Estimating  Example

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Solution for Demonstration Problem 8.7

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Determining Sample Size when Estimating p z formula Error of Estimation (tolerable error) Estimated Sample Size

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Solution for Demonstration Problem 8.8

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Determining Sample Size when Estimating p with No Prior Information P n z = 1.96 E = 0.05 p pq

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Example: Determining n when Estimating p with No Prior Information