McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. A PowerPoint Presentation Package to Accompany Applied Statistics in Business & Economics, 4 th edition David P. Doane and Lori E. Seward Prepared by Lloyd R. Jaisingh

8-2 Sampling Distributions and Estimation Chapter Contents 8.1 Sampling Variation 8.2 Estimators and Sampling Errors 8.3 Sample Mean and the Central Limit Theorem 8.4 Confidence Interval for a Mean (μ) with Known σ 8.5 Confidence Interval for a Mean (μ) with Unknown σ 8.6 Confidence Interval for a Proportion (π) 8.7 Estimating from Finite Populations 8.8 Sample Size Determination for a Mean 8.9 Sample Size Determination for a Proportion 8.10 Confidence Interval for a Population Variance,  2 (Optional) Chapter 8

8-3 Chapter Learning Objectives LO8-1: Define sampling error, parameter, and estimator. LO8-2: Explain the desirable properties of estimators. LO8-3: State the Central Limit Theorem for a mean. LO8-4: Explain how sample size affects the standard error. LO8-5: Construct a 90, 95, or 99 percent confidence interval for μ. LO8-6: Know when to use Student’s t instead of z to estimate μ. LO8-7: Construct a 90, 95, or 99 percent confidence interval for π. LO8-8: Construct confidence intervals for finite populations. LO8-9: Calculate sample size to estimate a mean or proportion. LO8-10: Construct a confidence interval for a variance (optional). Chapter 8 Sampling Distributions and Estimation

8-4 Sample statistic – a random variable whose value depends on which population items are included in the random sample. Depending on the sample size, the sample statistic could either represent the population well or differ greatly from the population. This sampling variation can easily be illustrated. Chapter Sampling Variation

8-5 Chapter Sampling Variation Consider eight random samples of size n = 5 from a large population of GMAT scores for MBA applicants.Consider eight random samples of size n = 5 from a large population of GMAT scores for MBA applicants. The sample means tend to be close to the population mean (  = ).The sample means tend to be close to the population mean (  = ).

8-6 The dot plots show that the sample means have much less variation than the individual sample items.The dot plots show that the sample means have much less variation than the individual sample items. Chapter Sampling Variation

8-7 Estimator – a statistic derived from a sample to infer the value of a population parameter.Estimator – a statistic derived from a sample to infer the value of a population parameter. Estimate – the value of the estimator in a particular sample.Estimate – the value of the estimator in a particular sample. Population parameters are represented by Greek letters and the corresponding statistic by Roman letters.Population parameters are represented by Greek letters and the corresponding statistic by Roman letters. Some Terminology Some Terminology Chapter Estimators and Sampling Errors LO8-1 LO8-1: Define sampling error, parameter and estimator.

8-8 Examples of Estimators Examples of Estimators Chapter 8 Sampling Distributions Sampling Distributions The sampling distribution of an estimator is the probability distribution of all possible values the statistic may assume when a random sample of size n is taken. Note: An estimator is a random variable since samples vary. 8.2 Estimators and Sampling Errors LO8-1

On average, an unbiased estimator neither overstates nor understates the true parameter.On average, an unbiased estimator neither overstates nor understates the true parameter. Chapter Estimators and Sampling Errors LO8-1 Sampling error Sampling error is the difference between an estimate and the corresponding population parameter. Example for the sample mean. 8-9

8-10 Efficiency refers to the variance of the estimator’s sampling distribution.Efficiency refers to the variance of the estimator’s sampling distribution. A more efficient estimator has smaller variance.A more efficient estimator has smaller variance. Efficiency Efficiency Consistency Consistency A consistent estimator converges toward the parameter being estimated as the sample size increases. Figure 8.6 Chapter 8 LO8-2: Explain the desirable properties of estimators. Note: Unbiasness is also a desirable property. 8.2 Estimators and Sampling Errors LO8-2

8-11 Chapter Sample Mean and the Central Limit Theorem LO8-3: State the Central Limit Theorem for a mean. The Central Limit Theorem is a powerful result that allows us to approximate the shape of the sampling distribution of the sample mean even when we don’t know what the population looks like. If the population is exactly normal, then the sample mean follows a normal distribution for any sample size.If the population is exactly normal, then the sample mean follows a normal distribution for any sample size. LO8-3

8-12 Illustrations of Central Limit Theorem Illustrations of Central Limit Theorem Note: Chapter Sample Mean and the Central Limit Theorem LO8-3 Using the uniform and a right skewed distribution.

8-13 Even if the population standard deviation σ is large, the sample means will fall within a narrow interval as long as n is large. The key is the standard error of the mean: The standard error decreases as n increases. Sample Size and Standard Error Sample Size and Standard Error Chapter 8 LO8-4: Explain how sample size affects the standard error. LO8-5: Construct a 90, 95, or 99 percent confidence interval for μ. What is a Confidence Interval? What is a Confidence Interval? A confidence interval for the mean is a range  lower <  <  upperA confidence interval for the mean is a range  lower <  <  upper The confidence level is the probability that the confidence interval contains the true population mean.The confidence level is the probability that the confidence interval contains the true population mean. The confidence level (usually expressed as a %) is the area under the curve of the sampling distribution.The confidence level (usually expressed as a %) is the area under the curve of the sampling distribution. 8.3 Sample Mean and the Central Limit Theorem LO8-4

8-14 What is a Confidence Interval? What is a Confidence Interval? The confidence interval for  with known  is:The confidence interval for  with known  is: Chapter Confidence Interval for a Mean (  ) with known  LO8-5

8-15 Use the Student’s t distribution instead of the normal distribution when the population is normal but the standard deviation  is unknown and the sample size is small.Use the Student’s t distribution instead of the normal distribution when the population is normal but the standard deviation  is unknown and the sample size is small. Student’s t Distribution Student’s t Distribution Chapter 8 LO8-6: Know when to use Student’s t instead of z to estimate . The confidence interval for  (unknown  ) can be rewritten asThe confidence interval for  (unknown  ) can be rewritten as Note: The degrees of freedom for the t distribution is n – Confidence Interval for a Mean (  ) with known  LO8-6

8-16 Chapter Confidence Interval for a Proportion  LO8-7: Construct a 90, 95, or 99 percent confidence interval for π. LO8-7

8-17 Chapter Estimating from Finite Populations LO8-8: Construct Confidence Intervals for Finite Populations LO8-8: Construct Confidence Intervals for Finite Populations. LO8-8 N = population size; n = sample size

8-18 To estimate a population mean with a precision of ± E (allowable error), you would need a sample of size. Now,To estimate a population mean with a precision of ± E (allowable error), you would need a sample of size. Now, Sample Size to Estimate  Sample Size to Estimate  Chapter Sample Size Determination for a Mean LO8-9: Calculate sample size to estimate a mean or proportion LO8-9: Calculate sample size to estimate a mean or proportion. LO8-9

8-19 To estimate a population proportion with a precision of ± E (allowable error), you would need a sample of sizeTo estimate a population proportion with a precision of ± E (allowable error), you would need a sample of size Since  is a number between 0 and 1, the allowable error E is also between 0 and 1. Since  is a number between 0 and 1, the allowable error E is also between 0 and 1. Chapter Sample Size Determination for a Proportion LO8-9

8-20 If the population is normal, then the sample variance s 2 follows the chi-square distribution (  2 ) with degrees of freedom d.f. = n – 1.If the population is normal, then the sample variance s 2 follows the chi-square distribution (  2 ) with degrees of freedom d.f. = n – 1. Lower (  2 L ) and upper (  2 U ) tail percentiles for the chi-square distribution can be found using Appendix E.Lower (  2 L ) and upper (  2 U ) tail percentiles for the chi-square distribution can be found using Appendix E. Using the sample variance s 2, the confidence interval isUsing the sample variance s 2, the confidence interval is Chi-Square Distribution Chi-Square Distribution LO8-10: Construct a confidence interval for a variance (optional). To obtain a confidence interval for the standard deviation , just take the square root of the interval bounds.To obtain a confidence interval for the standard deviation , just take the square root of the interval bounds. LO Confidence Interval for a Population Variance  2