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8 Chapter Sampling Distributions and Estimation (Part 2) Sample Size Determination for a Mean Sample Size Determination for a Proportion C.I. for the Difference of Two Means, m1-m2 C.I. for the Difference of Two Proportions, p1-p2 Confidence Interval for a Population Variance, s2 McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. 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 8B-2 2 Sample Size Determination for a Mean How to Estimate s? 8B-3 • Method 1: Take a Preliminary Sample Take a small preliminary sample and use the sample s in place of s in the sample size formula. • Method 2: Assume Uniform Population Estimate rough upper and lower limits a and b and set s = [(b-a)/12]½. Sample Size Determination for a Mean How to Estimate s? • • 8B-4 Method 3: Assume Normal Population Estimate rough upper and lower limits a and b and set s = (b-a)/4. This assumes normality with most of the data with m + 2s so the range is 4s. Method 4: Poisson Arrivals In the special case when m is a Poisson arrival rate, then s = m Sample Size Determination for a Mean Using LearningStats • 8B-5 There is a sample size calculator in LearningStats for E = 1 and E = .05. Sample Size Determination for a Mean Using MegaStat • 8B-6 There is a sample size calculator in MegaStat. The Preview button lets you change the setup and see results immediately. Sample Size Determination for a Mean Caution 1: Units of Measure • When estimating a mean, the allowable error E is expressed in the same units as X and s. Caution 2: Using z • Using z in the sample size formula for a mean is not conservative. Caution 3: Larger n is Better • 8B-7 The sample size formulas for a mean tend to underestimate the required sample size. These formulas are only minimum guidelines. 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 • 8B-8 2 p(1-p) Since p is a number between 0 and 1, the allowable error E is also between 0 and 1. Sample Size Determination for a Proportion How to Estimate p? • • • 8B-9 Method 1: Take a Preliminary Sample Take a small preliminary sample and use the sample p in place of p in the sample size formula. Method 2: Use a Prior Sample or Historical Data How often are such samples available? p might be different enough to make it a questionable assumption. Method 3: Assume that p = .50 This conservative method ensures the desired precision. However, the sample may end up being larger than necessary. Sample Size Determination for a Proportion Using LearningStats • The sample size calculator in LearningStats makes these calculations easy. Here are some calculations for p = .5 and E = 0.02. Figure 8.28 8B-10 Sample Size Determination for a Proportion Caution 1: Units of Measure • For a proportion, E is always between 0 and 1. For example, a 2% error is E = 0.02. Caution 2: Finite Population • 8B-11 For a finite population, to ensure that the sample size never exceeds the population size, use the following adjustment: nN n' = n + (N-1) ` Confidence Interval for the Difference of Two Means m1 – m2 • • 8B-12 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 m1 – m2 depends on our assumption about the unknown variances. Confidence Interval for the Difference of Two Means m1 – m2 Assuming equal variances: 2 + (n – 2)s 2 (n – 1)s 1 1 2 2 1 +1 (x1 – x2) + t n1 + n2 - 2 n1 n2 with n = (n1 – 1) + (n2 – 1) degrees of freedom 8B-13 Confidence Interval for the Difference of Two Means m1 – m2 Assuming unequal variances: 2 s 2 s (x1 – x2) + t 1+ 2 n1 n2 [s12/n1 + s22/n2]2 (Welch’s formula for with n' = 2 (s1 /n1)2 + (s22/n2)2 degrees of freedom) n1 – 1 n2 – 1 Or you can use a conservative quick rule for the degrees of freedom: n* = min (n1 – 1, n2 – 1). 8B-14 Confidence Interval for the Difference of Two Proportions p1 – p2 • 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 by (p1 – p2) + z 8B-15 p1(1 - p1) + p2(1 - p2) n1 n2 Confidence Interval for a Population Variance s2 Chi-Square Distribution • • • If the population is normal, then the sample variance s2 follows the chi-square distribution (c2) with degrees of freedom n = 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 2 (n – 1)s2 (n – 1)s 2 < < s c 2U c 2L 8B-16 Confidence Interval for a Population Variance s2 8B-17 Confidence Interval for a Population Variance s2 Confidence Interval for s • To obtain a confidence interval for the standard deviation, just take the square root of the interval bounds. (n – 1)s2 (n – 1)s2 < s < 2 cU c 2L 8B-18 Confidence Interval for a Population Variance s2 Using MINITAB • MINITAB gives confidence intervals for the mean, median, and standard deviation. Figure 8.31 8B-19 Confidence Interval for a Population Variance s2 Using LearningStats • Here is an example for n = 39. Because the sample size is large, the distribution is somewhat bell-shaped. Figure 8.32 8B-20 Confidence Interval for a Population Variance s2 Caution: Assumption of Normality • • 8B-21 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. Applied Statistics in Business & Economics End of Chapter 8B 8B-22 McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc.