Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-1 Business Statistics, 3e by Ken Black Chapter 8 Statistical Inference: Estimation for Single Populations

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-2 Learning Objectives Know the difference between point and interval estimation. Estimate a population mean from a sample mean for large sample sizes. Estimate a population mean from a sample mean for small sample sizes. 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: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 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: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-4 Confidence Interval to Estimate  when n is Large Point estimate Interval Estimate

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-5 Distribution of Sample Means for (1-  )% Confidence  X  Z 0

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-6 Distribution of Sample Means for (1-  )% Confidence  X Z 0

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-7 Distribution of Sample Means for (1-  )% Confidence  X Z 0

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-8 Probability Interpretation of the Level of Confidence

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-9 Distribution of Sample Means for 95% Confidence .4750 X 95%.025 Z

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning % Confidence Interval for 

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning % Confidence Intervals for   X 95% X X X X X X

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning % Confidence Intervals for   X 95% X X X X X X Is our interval, 3.98  4.54, in the red?

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-13 Demonstration Problem 8.1

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-14 Demonstration Problem 8.2

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-15 Confidence Interval to Estimate  when n is Large and  is Unknown

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-16 Z Values for Some of the More Common Levels of Confidence 90% 95% 98% 99% Confidence Level Z Value

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-17 Estimating the Mean of a Normal Population: Small n and Unknown  The population has a normal distribution. The value of the population standard deviation is unknown. The sample size is small, n < 30. Z distribution is not appropriate for these conditions t distribution is appropriate

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-18 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: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-19 Comparison of Selected t Distributions to the Standard Normal Standard Normal t (d.f. = 25) t (d.f. = 5) t (d.f. = 1)

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-20 Table of Critical Values of t df t t t t t  tt  

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-21 Confidence Intervals for  of a Normal Population: Small n and Unknown 

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-22 Solution for Demonstration Problem 8.3

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-23 Solution for Demonstration Problem 8.3

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-24 Confidence Interval to Estimate the Population Proportion

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-25 Solution for Demonstration Problem 8.5

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-26 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: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-27 Estimating the Population Variance Population Parameter   Estimator of     formula for Single Variance

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-28 Confidence Interval for  2

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-29 Selected  2 Distributions df = 3 df = 5 df = 10 0

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-30  2 Table df E E df =

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-31 Two Table Values of  df = df E

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning % Confidence Interval for  2

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-33 Solution for Demonstration Problem 8.6

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-34 Determining Sample Size when Estimating  Z formula Error of Estimation (tolerable error) Estimated Sample Size Estimated 

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-35 Sample Size When Estimating  Example

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-36 Solution for Demonstration Problem 8.7

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-37 Determining Sample Size when Estimating P Z formula Error of Estimation (tolerable error) Estimated Sample Size

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-38 Solution for Demonstration Problem 8.8

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-39 Determining Sample Size when Estimating P with No Prior Information P n Z = 1.96 E = 0.05 P PQ

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-40 Solution for Demonstration Problem 8.9