1. Sampling Distributions and Estimation Chapter88 Sampling Variation Estimators and Sampling Distributions Sample Mean and the Central Limit Theorem Confidence Interval for a Mean (  ) with Known  Confidence Interval for a Mean (  ) with Unknown  Confidence Interval for a Proportion (  ) Sample Size Determination for a Mean and a Proportion Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

2. 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.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)Sample variation (uncontrollable) Population variation (uncontrollable)Population variation (uncontrollable) Sample size (controllable)Sample size (controllable) 8-2

3. Estimators and Sampling Distributions Estimators and Sampling Distributions Sample statistic – a random variable whose value depends on which population items happen to be included in the random sample.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.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.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 parametersPopulation parameters are represented by Greek letters and the are represented by Greek letters and the corresponding statistic by Roman letters. corresponding statistic by Roman letters. Some Terminology Some Terminology 8-3

4. Estimators and Sampling Distributions Estimators and Sampling Distributions Examples of Estimators Examples of Estimators 8-4

5. Estimators and Sampling Distributions Estimators and 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.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.An estimator is a random variable since samples vary. Sampling Distributions Sampling Distributions Sampling error =  – Sampling error =  – ^ 8-5

6. Estimators and Sampling Distributions Estimators and Sampling Distributions Bias is the difference between the expected value of the estimator and the true parameter.Bias is the difference between the expected value of the estimator and the true parameter. Bias Bias Bias = E(  ) – Bias = E(  ) – ^ An estimator is unbiased if E(  ) = An estimator is unbiased if E(  ) = ^ On average, an unbiased estimator neither overstates nor understates the true parameter.On average, an unbiased estimator neither overstates nor understates the true parameter. 8-6

7. Estimators and Sampling Distributions Estimators and Sampling Distributions 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 Figure 8.5 8-7

8. Estimators and Sampling Distributions Estimators and Sampling Distributions A consistent estimator converges toward the parameter being estimated as the sample sizeA consistent estimator converges toward the parameter being estimated as the sample sizeincreases. Consistency Consistency Figure 8.6 8-8

9. Sample Mean and the Central Limit Theorem Sample Mean and the Central Limit Theorem If a random sample of size n is drawn from a population with mean  and standard deviation , the distribution of the sample mean x approaches a normal distribution with mean  and standard deviation  x =  / n as the sample size increase.If a random sample of size n is drawn from a population with mean  and standard deviation , the distribution of the sample mean x approaches a normal distribution with mean  and standard deviation  x =  / n as the sample size increase. If the population is normal, the distribution of the sample mean is normal regardless of sample size.If the population is normal, the distribution of the sample mean is normal regardless of sample size. Central Limit Theorem (CLT) for a Mean Central Limit Theorem (CLT) for a Mean 8-9

10. Sample Mean and the Central Limit Theorem Sample Mean and the Central Limit Theorem Illustrations of Central Limit Theorem Illustrations of Central Limit Theorem 8-10

11. Confidence Interval for a Mean (  ) with Known  Confidence Interval for a Mean (  ) with Known  A sample mean x is a point estimate of the population mean .A sample mean x is a point estimate of the population mean . 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. 8-11

12. Confidence Interval for a Mean (  ) with Known  Confidence Interval for a Mean (  ) with Known  What is a Confidence Interval? What is a Confidence Interval? The confidence interval for  with known  is:The confidence interval for  with known  is: 8-12

13. Confidence Interval for a Mean (  ) with Unknown  Confidence Interval for a Mean (  ) with Unknown  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 x + tx + tx + tx + t snsnsnsn The confidence interval for  (unknown  ) isThe confidence interval for  (unknown  ) is x - t snsnsnsn x + t snsnsnsn <  < 8-13

14. Confidence Interval for a Proportion (  ) If both n  > 10 and n(1-  ) > 10, then the confidence interval for  isIf both n  > 10 and n(1-  ) > 10, then the confidence interval for  is Confidence Interval for  Confidence Interval for   (1-  )  n  + z + z Since  is unknown, the confidence interval for p = x/n (assuming a large sample) isSince  is unknown, the confidence interval for p = x/n (assuming a large sample) is  p(1-p)  n p + zp + zp + zp + z Where z is based on the desired confidence. 8-14

15. Sample Size Determination for a Mean Sample Size Determination for a Mean To estimate a population mean with a precision of + E (allowable error), you would need a sample of sizeTo estimate a population mean with a precision of + E (allowable error), you would need a sample of size Sample Size to Estimate  Sample Size to Estimate  n =n =n =n = zzEEzzEEE2 8-15

16. Sample Size Determination for a Proportion Sample Size Determination for a Proportion 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 z E n =  (1-  ) 2 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. 8-16