12/1/2017. Chapter. Copyright 2009 by The McGraw-Hill Companies, Inc. 8B-2

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1 Sampling Distributions and Estimation (Part ) 8 Chapter Proportion C.I. for the Difference of Two s, m 1 -m C.I. for the Difference of Two Proportions, p 1 -p Population Variance, s McGraw-Hill/Irwin Copyright 009 by The McGraw-Hill Companies, Inc. 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- 1

2 How to Estimate s? Method 1: Take a Preliminary Sample Take a small preliminary sample and use the sample s in place of sin the sample size formula. Method : Assume Uniform Population Estimate rough upper and lower limits a and b and set s= = [(b-a)/1] ½. 8B-3 How to Estimate s? 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+ s so the range is 4s. Method 4: Poisson Arrivals In the special case when mis a Poisson arrival rate, then s= m 8B-4

3 Using LearningStats There is a sample size calculator in LearningStats for E = 1 and E =.05. 8B-5 Using MegaStat There is a sample size calculator in MegaStat. The Preview button lets you change the setup and see results immediately. 8B-6 3

4 8B-7 Caution 1: Units of Measure When estimating a mean, the allowable error E is expressed in the same units as X and s. Caution : Using z Using z in the sample size formula for a mean is not conservative. Caution 3: Larger n is Better 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 n = z E p(1-p) Since pis a number between 0 and 1, the allowable error E is also between 0 and 1. 8B-8 4

5 Sample Size Determination for a Proportion How to Estimate p? Method 1: Take a Preliminary Sample Take a small preliminary sample and use the sample p in place of pin the sample size formula. Method : Use a Prior Sample or Historical Data How often are such samples available? pmight 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. 8B-9 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 = 0.0. Figure 8.8 8B-10 5

6 Sample Size Determination for a Proportion Caution 1: Units of Measure For a proportion, E is always between 0 and 1. For example, a % error is E = 0.0. Caution : Finite Population For a finite population, to ensure that the sample size never exceeds the population size, use the following adjustment: n' = nn n + (N-1) 8B-11 ` Confidence Interval for the Difference of Two s m 1 m 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 m 1 m depends on our assumption about the unknown variances. 8B-1 6

7 Confidence Interval for the Difference of Two s m 1 m Assuming equal variances: (x 1 x ) + t (n 1 1)s 1 + (n )s n 1 + n n with n= = (n 1 1) + (n 1) degrees of freedom n 1 8B-13 Confidence Interval for the Difference of Two s m 1 m Assuming unequal variances: (x 1 x ) + t s 1 s + n 1 n [s 1 /n 1 + s /n ] with n' = (s1 /n 1 ) + (s /n ) n 1 1 n 1 (Welch s formula for degrees of freedom) Or you can use a conservative quick rule for the degrees of freedom: n* * = min (n 1 1, n 1). 8B-14 7

8 Confidence Interval for the Difference of Two Proportions p 1 p 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 (p 1 p ) + z p 1 (1 - p 1 ) + p (1 - p ) n 1 n 8B-15 Chi-Square Distribution If the population is normal, then the sample variance s follows the chi-square distribution (c ) with degrees of freedom n= n 1. Lower (c L) and upper (c U) tail percentiles for the chi-square distribution can be found using Appendix E. Using the sample variance s, the confidence interval is (n 1)s (n 1)s < s c < U c L 8B-16 8

9 8B-17 Confidence Interval for s To obtain a confidence interval for the standard deviation, just take the square root of the interval bounds. (n 1)s c U < s < (n 1)s c L 8B-18 9

10 Using MINITAB MINITAB gives confidence intervals for the mean, median, and standard deviation. Figure B-19 Using LearningStats Here is an example for n= 39. Because the sample size is large, the distribution is somewhat bell-shaped. 8B-0 Figure

11 Caution: Assumption of Normality 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. 8B-1 Applied Statistics in Business & Economics End of Chapter 8B 8B- McGraw-Hill/Irwin Copyright 009 by The McGraw-Hill Companies, Inc. 11