1. IEEM 3201 Two-Sample Estimation: Paired Observation, Difference

2. IEEM 320IEEM151 Notes 19, Page 2 Outline  confidence intervals of difference  known variance  unknown but equal variance  Unknown, unequal variance  paired observations

3. IEEM 320IEEM151 Notes 19, Page 3 Procedure for Interval Estimation ? determine , “the amount of risk that we want to bear”, usually being 0.05 or 0.01 ? letbe two r.v.’s (statistics) such that : (1-  )100% confidence interval are the lower and upper confidence limits. ? the sample values of

4. IEEM 320IEEM151 Notes 19, Page 4 ? ~ approximately normally distributed; mean and variance ? population means:  1 &  2 ; known variances  1 2 &  2 2 Estimations of the Difference Between Two Means ? Therefore

5. IEEM 320IEEM151 Notes 19, Page 5 where z  /2 is the z-value leaving an area of  /2 to the right. Two Samples: Estimating The Difference for Known  ? a (1-  )100% confidence interval of  1 -  2 ;  1 2 and  2 2 known

6. IEEM 320IEEM151 Notes 19, Page 6 Solution: z  /2 = z 0.02 = 2.05 The 96% confidence interval is Example: Find a 96% confidence interval for the difference of. Two Samples: Estimating the Difference for Known 

7. IEEM 320IEEM151 Notes 19, Page 7 Two Samples: Estimating the Difference for Unknown but Equal  ? a (1-  )100% confidence interval of  1 -  2 ;  1 2 =  2 2 =  2 unknown ~  2 of n i -1 degrees of freedom ~  2 of n 1 +n 2 -2 degrees of freedom

8. IEEM 320IEEM151 Notes 19, Page 8 Two Samples: Estimating the Difference for Unknown but Equal  where ? T statistic: n 1 +n 2 -2 degrees of freedom

9. IEEM 320IEEM151 Notes 19, Page 9 where s p is the pooled estimate of the population standard deviation and t  /2 is the t-value with v = n 1 +n 2 -2 degrees of freedom, leaving an area of  /2 to the right. Two Samples: Estimating the Difference for Unknown but Equal  ? a (1-  )100% confidence interval of  1 -  2 ;  1 2 =  2 2 unknown

10. IEEM 320IEEM151 Notes 19, Page 10 Solutio n: Example: Assume that the two populations have equal variances, find a 96% confidence interval for of. Two Samples: Estimating the Difference for Unknown but Equal  The 96% confidence interval is

11. IEEM 320IEEM151 Notes 19, Page 11 ? use statistic: Two Samples: Estimating the Difference for Unknown, Unequal Variances ? a (1-  )100% confidence interval of  1 -  2 ;  1 2 ≠  2 2 unknown with v (round down to integer) degrees of freedom, where

12. IEEM 320IEEM151 Notes 19, Page 12 Two Samples: Estimating the Difference for Unknown, Unequal Variances ? a (1-  )100% confidence interval of  1 -  2 ;  1 2 ≠  2 2 unknown where t  /2 is the t-value with degrees of freedom, leaving an area of  /2 to the right.

13. IEEM 320IEEM151 Notes 19, Page 13 Paired Observations  observations of two populations come in pairs  e.g., 15 individuals; x i and y i : weight before and after going on a diet  previous methods on two samples give the confidence interval of the difference  to reduce the variance of the statistics, we can ? define d i = x i -y i ? find variance on d i as if a single sample

14. IEEM 320IEEM151 Notes 19, Page 14 Two Samples: Estimating the Difference for Paired Observations ? a (1-  )100% confidence interval of  D =  1 -  2 for paired observations where and s d are the mean and standard deviation of the differences of n random pairs of measurements and t  /2 is the t-value with v = n-1 degrees of freedom, leaving an area of  /2 to the right.

15. IEEM 320IEEM151 Notes 19, Page 15 Example: Find a 95% confidence interval for of. Solutio n: Two Samples: Estimating the difference for Paired Observations The 95% confidence interval is