Sampling Distributions and Estimation (Part 2)

1. Sampling Distributions and Estimation (Part 2) 8 Chapter 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

2. 2 zs E n = 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

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

4. Method 4: Poisson ArrivalsIn the special case when m is a Poisson arrival rate, then s = m Sample Size Determination for a Mean How to Estimate s? • Method 3: Assume Normal PopulationEstimate 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.

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

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

7. 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 • The sample size formulas for a mean tend to underestimate the required sample size. These formulas are only minimum guidelines.

8. z E 2 p(1-p) n = 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 • Since p is a number between 0 and 1, the allowable error E is also between 0 and 1.

9. Sample Size Determination for a Proportion How to Estimate p? • Method 1: Take a Preliminary SampleTake 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 DataHow 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.

10. 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

11. nN n + (N-1) n' = 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 • For a finite population, to ensure that the sample size never exceeds the population size, use the following adjustment:

12. ` Confidence Interval for the Difference of Two Means m1 – m2 • 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.

13. 1 n2 1 n1 + (n1 – 1)s12 + (n2 – 2)s22 n1 + n2 - 2 (x1 – x2) +t Confidence Interval for the Difference of Two Means m1 – m2 Assuming equal variances: with n = (n1 – 1) + (n2 – 1) degrees of freedom

14. [s12/n1 + s22/n2]2 (Welch’s formula for degrees of freedom) with n' = s22 n2 s12 n1 + (x1 – x2) +t (s12/n1)2 + (s22/n2)2 n1 – 1 n2 – 1 Confidence Interval for the Difference of Two Means m1 – m2 Assuming unequal variances: Or you can use a conservative quick rule for the degrees of freedom: n* = min (n1 – 1, n2 – 1).

15. (p1 – p2) +z p1(1 - p1) + p2(1 - p2) n1n2 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

16. (n – 1)s2 c2U (n – 1)s2 c2L < s2 < 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

17. Confidence Interval for a Population Variance s2

18. (n – 1)s2 c2U (n – 1)s2 c2L < s < 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.

19. Confidence Interval for a Population Variance s2 Using MINITAB • MINITAB gives confidence intervals for the mean, median, and standard deviation. Figure 8.31

20. 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

21. Confidence Interval for a Population Variance s2 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.

22. Applied Statistics in Business & Economics End of Chapter 8B 8B-22