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A PowerPoint Presentation Package to Accompany Applied Statistics in Business & Economics, 4th edition David P. Doane and Lori E. Seward Prepared by Lloyd R. Jaisingh McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 8 Sampling Distributions and Estimation Chapter Contents 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 Sampling Variation Estimators and Sampling Errors Sample Mean and the Central Limit Theorem Confidence Interval for a Mean (μ) with Known σ Confidence Interval for a Mean (μ) with Unknown σ Confidence Interval for a Proportion (π) Estimating from Finite Populations Sample Size Determination for a Mean Sample Size Determination for a Proportion Confidence Interval for a Population Variance, 2 (Optional) 8-2 Chapter 8 Sampling Distributions and Estimation Chapter Learning Objectives LO8-1: LO8-2: LO8-3: LO8-4: LO8-5: LO8-6: LO8-7: LO8-8: LO8-9: LO8-10: Define sampling error, parameter, and estimator. Explain the desirable properties of estimators. State the Central Limit Theorem for a mean. Explain how sample size affects the standard error. Construct a 90, 95, or 99 percent confidence interval for μ. Know when to use Student’s t instead of z to estimate μ. Construct a 90, 95, or 99 percent confidence interval for π. Construct confidence intervals for finite populations. Calculate sample size to estimate a mean or proportion. Construct a confidence interval for a variance (optional). 8-3 Chapter 8 8.1 Sampling Variation • • • 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. 8-4 Chapter 8 8.1 Sampling Variation • • 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 (m = 520.78). 8-5 Chapter 8 8.1 Sampling Variation • The dot plots show that the sample means have much less variation than the individual sample items. 8-6 Chapter 8 LO8-1 8.2 Estimators and Sampling Errors LO8-1: Define sampling error, parameter and estimator. Some Terminology • • • 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. Population parameters are represented by Greek letters and the corresponding statistic by Roman letters. 8-7 Chapter 8 LO8-1 8.2 Estimators and Sampling Errors Examples of Estimators 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. 8-8 Chapter 8 LO8-1 8.2 Estimators and Sampling Errors • Sampling error is the difference between an estimate and the corresponding population parameter. Example for the sample mean. • On average, an unbiased estimator neither overstates nor understates the true parameter. 8-9 Chapter 8 LO8-2 8.2 Estimators and Sampling Errors LO8-2: Explain the desirable properties of estimators. Note: Unbiasness is also a desirable property. Efficiency • • Efficiency refers to the variance of the estimator’s sampling distribution. A more efficient estimator has smaller variance. Consistency Figure 8.6 A consistent estimator converges toward the parameter being estimated as the sample size increases. 8-10 Chapter 8 8.3 Sample Mean and the Central Limit Theorem LO8-3 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. 8-11 Chapter 8 LO8-3 8.3 Sample Mean and the Central Limit Theorem Illustrations of Central Limit Theorem Using the uniform and a right skewed distribution. Note: 8-12 Chapter 8 LO8-4 8.3 Sample Mean and the Central Limit Theorem LO8-4: Explain how sample size affects the standard error. Sample Size and Standard Error 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. LO8-5: Construct a 90, 95, or 99 percent confidence interval for μ. What is a Confidence Interval? • • • A confidence interval for the mean is a range mlower < m < mupper 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. 8-13 Chapter 8 LO8-5 8.4 Confidence Interval for a Mean (m) with known What is a Confidence Interval? • The confidence interval for m with known is: 8-14 Chapter 8 LO8-6 8.4 Confidence Interval for a Mean (m) with known LO8-6: Know when to use Student’s t instead of z to estimate m. Student’s t Distribution • 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. • The confidence interval for m (unknown ) can be rewritten as Note: The degrees of freedom for the t distribution is n – 1. 8-15 Chapter 8 LO8-7 8.6 Confidence Interval for a Proportion LO8-7: Construct a 90, 95, or 99 percent confidence interval for π. 8-16 Chapter 8 LO8-8 8.7 Estimating from Finite Populations LO8-8: Construct Confidence Intervals for Finite Populations. N = population size; n = sample size 8-17 Chapter 8 LO8-9 8.8 Sample Size Determination for a Mean LO8-9: Calculate sample size to estimate a mean or proportion. Sample Size to Estimate m • To estimate a population mean with a precision of ± E (allowable error), you would need a sample of size. Now, 8-18 Chapter 8 LO8-9 8.9 Sample Size Determination for a Proportion • To 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. 8-19 LO8-10 8.10 Confidence Interval for a Population Variance 2 LO8-10: Construct a confidence interval for a variance (optional). Chi-Square Distribution • • • • If the population is normal, then the sample variance s2 follows the chi-square distribution (c2) with degrees of freedom d.f. = n – 1. Lower (c2L) and upper (c2U) tail percentiles for the chi-square distribution can be found using Appendix E. Using the sample variance s2, the confidence interval is To obtain a confidence interval for the standard deviation , just take the square root of the interval bounds. 8-20