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8 Chapter Sampling Distributions and Estimation Sampling Variation Estimators and Sampling Distributions Sample Mean and the Central Limit Theorem Confidence Interval for a Mean (m) with Known s Confidence Interval for a Mean (m) with Unknown s Confidence Interval for a Proportion (p) Sample Size Determination for a Mean and a Proportion McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. Sampling Variation • • • • From figure 8.2, we see that the sample means (red markers) have much less variation than the individual sample items. This is an example of sample variation. Sample variation (uncontrollable) Population variation (uncontrollable) Sample size (controllable) 8-2 Estimators and Sampling Distributions Some Terminology • • • • • Sample statistic – a random variable whose value depends on which population items happen to be included in the random sample. Sampling variation is the variation for a sample statistic based on different samples. 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. 8-3 Estimators and Sampling Distributions Examples of Estimators 8-4 Estimators and 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. An estimator is a random variable since samples vary. • ^ Sampling error = – 8-5 Estimators and Sampling Distributions Bias • • • • Bias is the difference between the expected value of the estimator and the true parameter. ^ Bias = E( ) – ^ An estimator is unbiased if E( ) = On average, an unbiased estimator neither overstates nor understates the true parameter. 8-6 Estimators and Sampling Distributions Efficiency • • Efficiency refers to the variance of the estimator’s sampling distribution. A more efficient estimator has smaller variance. Figure 8.5 8-7 Estimators and Sampling Distributions Consistency • A consistent estimator converges toward the parameter being estimated as the sample size increases. Figure 8.6 8-8 Sample Mean and the Central Limit Theorem Central Limit Theorem (CLT) for a Mean • • If a random sample of size n is drawn from a population with mean m and standard deviation s, the distribution of the sample mean x approaches a normal distribution with mean m and standard deviation sx = s/ n as the sample size increase. If the population is normal, the distribution of the sample mean is normal regardless of sample size. 8-9 Sample Mean and the Central Limit Theorem Illustrations of Central Limit Theorem 8-10 Confidence Interval for a Mean (m) with Known s What is a Confidence Interval? • A sample mean x is a point estimate of the population mean m. • 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. • • 8-11 Confidence Interval for a Mean (m) with Known s What is a Confidence Interval? • The confidence interval for m with known s is: 8-12 Confidence Interval for a Mean (m) with Unknown s Student’s t Distribution • Use the Student’s t distribution instead of the normal distribution when the population is normal but the standard deviation s is unknown and the sample size is small. x+t • s n The confidence interval for m (unknown s) is s s x-t <m< x+t n n 8-13 Confidence Interval for a Proportion (p) Confidence Interval for p • If both np > 10 and n(1-p) > 10, then the confidence interval for p is p+z p(1-p) n Where z is based on the desired confidence. • Since p is unknown, the confidence interval for p = x/n (assuming a large sample) is p(1-p) p+z n 8-14 Sample Size Determination for a Mean Sample Size to Estimate m • To estimate a population mean with a precision of + E (allowable error), you would need a sample of size n = zs E 2 8-15 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 z n= E • 2 p(1-p) Since p is a number between 0 and 1, the allowable error E is also between 0 and 1. 8-16