1. McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. Sampling Distributions and Estimation (Part 2) Chapter88 Sample Size Determination for a Mean Sample Size Determination for a Proportion C.I. for the Difference of Two Means,  1 -  2 C.I. for the Difference of Two Proportions,  1 -  2 Confidence Interval for a Population Variance,  2

2. 8B-2 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

3. 8B-3 Sample Size Determination for a Mean Method 1: Take a Preliminary Sample Take a small preliminary sample and use the sample s in place of  in the sample size formula.Method 1: Take a Preliminary Sample Take a small preliminary sample and use the sample s in place of  in the sample size formula. Method 2: Assume Uniform Population Estimate rough upper and lower limits a and b and set  = [(b-a)/12] ½.Method 2: Assume Uniform Population Estimate rough upper and lower limits a and b and set  = [(b-a)/12] ½. How to Estimate  ? How to Estimate  ?

4. 8B-4 Sample Size Determination for a Mean Method 3: Assume Normal Population Estimate rough upper and lower limits a and b and set  = (b-a)/4. This assumes normality with most of the data with  + 2  so the range is 4 .Method 3: Assume Normal Population Estimate rough upper and lower limits a and b and set  = (b-a)/4. This assumes normality with most of the data with  + 2  so the range is 4 . How to Estimate  ? How to Estimate  ? Method 4: Poisson Arrivals In the special case when  is a Poisson arrival rate, then  = Method 4: Poisson Arrivals In the special case when  is a Poisson arrival rate, then  = 

5. 8B-5 Sample Size Determination for a Mean There is a sample size calculator in LearningStats for E = 1 and E =.05. Using LearningStats Using LearningStats

6. 8B-6 Sample Size Determination for a Mean There is a sample size calculator in MegaStat. The Preview button lets you change the setup and see results immediately. Using MegaStat Using MegaStat

7. 8B-7 Sample Size Determination for a Mean When estimating a mean, the allowable error E is expressed in the same units as X and .When estimating a mean, the allowable error E is expressed in the same units as X and . Caution 1: Units of Measure Caution 1: Units of Measure Using z in the sample size formula for a mean is not conservative.Using z in the sample size formula for a mean is not conservative. Caution 2: Using z Caution 2: Using z The sample size formulas for a mean tend to underestimate the required sample size. These formulas are only minimum guidelines.The sample size formulas for a mean tend to underestimate the required sample size. These formulas are only minimum guidelines. Caution 3: Larger n is Better Caution 3: Larger n is Better

8. 8B-8 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.

9. 8B-9 Sample Size Determination for a Proportion Method 1: Take a Preliminary Sample Take a small preliminary sample and use the sample p in place of  in the sample size formula.Method 1: Take a Preliminary Sample Take a small preliminary sample and use the sample p in place of  in the sample size formula. Method 2: Use a Prior Sample or Historical Data How often are such samples available?  might be different enough to make it a questionable assumption.Method 2: Use a Prior Sample or Historical Data How often are such samples available?  might be different enough to make it a questionable assumption. Method 3: Assume that  =.50 This conservative method ensures the desired precision. However, the sample may end up being larger than necessary.Method 3: Assume that  =.50 This conservative method ensures the desired precision. However, the sample may end up being larger than necessary. How to Estimate  ? How to Estimate  ?

10. 8B-10 Sample Size Determination for a Proportion The sample size calculator in LearningStats makes these calculations easy. Here are some calculations for  =.5 and E = 0.02.The sample size calculator in LearningStats makes these calculations easy. Here are some calculations for  =.5 and E = 0.02. Using LearningStats Using LearningStats Figure 8.28

11. 8B-11 Sample Size Determination for a Proportion For a proportion, E is always between 0 and 1. For example, a 2% error is E = 0.02.For a proportion, E is always between 0 and 1. For example, a 2% error is E = 0.02. Caution 1: Units of Measure Caution 1: Units of Measure For a finite population, to ensure that the sample size never exceeds the population size, use the following adjustment:For a finite population, to ensure that the sample size never exceeds the population size, use the following adjustment: Caution 2: Finite Population Caution 2: Finite Population n' = nN n + (N-1)

12. 8B-12 Confidence Interval for the Difference of Two Means  1 –  2 If the confidence interval for the difference of two means includes zero, we could conclude that there is no significant difference in means.If the confidence interval for the difference of two means includes zero, we could conclude that there is no significant difference in means. The procedure for constructing a confidence interval for  1 –  2 depends on our assumption about the unknown variances.The procedure for constructing a confidence interval for  1 –  2 depends on our assumption about the unknown variances. `

13. 8B-13 Confidence Interval for the Difference of Two Means  1 –  2 Assuming equal variances: Assuming equal variances: (x 1 – x 2 ) + t (n 1 – 1)s 1 2 + (n 2 – 2)s 2 2 n 1 + n 2 - 2 n 1 + n 2 - 21 11n1n111n1n1 1 11n2n211n2n2 + with = (n 1 – 1) + (n 2 – 1) degrees of freedom

14. 8B-14 Confidence Interval for the Difference of Two Means  1 –  2 Assuming unequal variances: Assuming unequal variances: (x 1 – x 2 ) + t s12n1s12n1 s22n2s22n2 + [s 1 2 /n 1 + s 2 2 /n 2 ] 2 with ' = (s 1 2 /n 1 ) 2 + (s 2 2 /n 2 ) 2 n 1 – 1 n 2 – 1 (Welch’s formula for degrees of freedom) Or you can use a conservative quick rule for the degrees of freedom: * = min (n 1 – 1, n 2 – 1).

15. 8B-15 Confidence Interval for the Difference of Two Proportions  1 –  2 If both samples are large (i.e., np > 10 and n(1-p) > 10, then a confidence interval for the difference of two sample proportions is given byIf both samples are large (i.e., np > 10 and n(1-p) > 10, then a confidence interval for the difference of two sample proportions is given by (p 1 – p 2 ) + z p 1 (1 - p 1 ) + p 2 (1 - p 2 ) n 1 n 2 n 1 n 2

16. 8B-16 Confidence Interval for a Population Variance  2 If the population is normal, then the sample variance s 2 follows the chi-square distribution (  2 ) with degrees of freedom = n – 1.If the population is normal, then the sample variance s 2 follows the chi-square distribution (  2 ) with degrees of freedom = 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 (n – 1)s 2  2 U (n – 1)s 2  2 L <  2 <

17. 8B-17 Confidence Interval for a Population Variance  2

18. 8B-18 Confidence Interval for a Population Variance  2 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. Confidence Interval for  Confidence Interval for  (n – 1)s 2  2 U  2 U (n – 1)s 2  2 L  2 L <  <

19. 8B-19 Confidence Interval for a Population Variance  2 MINITAB gives confidence intervals for the mean, median, and standard deviation. Using MINITAB Using MINITAB Figure 8.31

20. 8B-20 Confidence Interval for a Population Variance  2 Here is an example for = 39. Because the sample size is large, the distribution is somewhat bell-shaped.Here is an example for = 39. Because the sample size is large, the distribution is somewhat bell-shaped. Using LearningStats Using LearningStats Figure 8.32

21. 8B-21 Confidence Interval for a Population Variance  2 The methods described for confidence interval estimation of the variance and standard deviation depend on the population having a normal distribution.The methods described for confidence interval estimation of the variance and standard deviation depend on the population having a normal distribution. If the population does not have a normal distribution, then the confidence interval should not be considered accurate.If the population does not have a normal distribution, then the confidence interval should not be considered accurate. Caution: Assumption of Normality Caution: Assumption of Normality

22. McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. Applied Statistics in Business & Economics End of Chapter 8B 8B-22