IEEM 3201 One and Two-Sample Estimation Problems

IEEM 320IEEM151 Notes 17, Page 2 Outline  background  point estimate: unbiased; most efficient  interval estimate of mean for known and unknown variance  prediction interval for known and unknown variance

IEEM 320IEEM151 Notes 17, Page 3 Statistical Inference  sample values x 1,..., x n : realizations of i.i.d. r.v.’s X 1, …, X n drawn from the population  distribution of X i characterized by parameter  ? (23.51,..., 47.39) from Normal ( ,  2 ) ? statistical inference on parameters ? estimation (e.g., values of , ,  2 ) ? hypothesis testing (e.g., H 0 :  = 0.5 vs. H 1 :   0.5)

IEEM 320IEEM151 Notes 17, Page 4 Arrangement of Material  Chapter 9: parameter estimation ? Chapter 10: hypothesis testing ? underlying theory: normal distribution and  2, t, F distributions

IEEM 320IEEM151 Notes 17, Page 5 Unbiased estimator: statistic is an unbiased estimator of parameter  if Point Estimation  Point estimate of a population parameter  : a single value of a statistic that estimates   is the sample value of statistic that estimates mean 

IEEM 320IEEM151 Notes 17, Page 6 Point Estimation Example: E(S 2 ) =  2, where V(X i ) =  2 and E(X i ) = 

IEEM 320IEEM151 Notes 17, Page 7 Point Estimation  there can be many point estimators of a statistic  the most efficient estimator of a parameter: an unbiased estimator of a parameter that has the smallest variance among all possible ones

IEEM 320IEEM151 Notes 17, Page 8 Interval Estimation ? interval estimates are useful in real life ? compare two statements: “The mean life of our TVs is 5 years” and “The lives of our TVs are between 4 to 6 years”. ? want to find an interval (  L,  U ) from x 1,..., x n such that  L     U ? a matter of believe ? e.g., Is 0.4 < P(head) < 0.6 if you get all heads on 10 flips? ? always bear statistical risk, making wrong estimation, accepting a wrong “believe”, or rejecting a true “believe”

IEEM 320IEEM151 Notes 17, Page 9 Procedure for Interval Estimation ? determine , “the among 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

IEEM 320IEEM151 Notes 17, Page 10 Known Results z 0 z  /2 -z  /2 1 –   /2 ~ standard normal if X i ~ normal, and, by CLT, approximately so for any distribution

IEEM 320IEEM151 Notes 17, Page 11 Known Results ~ t-distribution of n-1 degrees of freedom if X i ~ normal; and is approximately so for any distribution

IEEM 320IEEM151 Notes 17, Page 12 where z  /2 is the z-value leaving an area of  /2 to the right ? a (1-  )100% confidence interval of  ;  known Single Sample: Estimating the Mean for known 

IEEM 320IEEM151 Notes 17, Page 13 Example: n = 36; population standard deviation = 0.3; sample mean = 2.6; 95% c.i. for the population mean = ? Solution: n = 36,  = 0.3 For 95% confidence interval, 1-  = 0.95,  = 0.05,  /2 = 0.025, z  /2 = z =1.96. The 95% c.i. is Single Sample: Estimating the Mean

IEEM 320IEEM151 Notes 17, Page 14 Single Sample: Estimating the Mean for unknown  ? a (1-  )100% confidence interval of  ;  unknown where t  /2 is the t-value of v = n-1 degrees of freedom, leaving an area of  /2 to the right

IEEM 320IEEM151 Notes 17, Page 15 Solution: Example: The contents of 7 similar containers of sulfuric acid are 9.8,10.2,10.4,9.8,10.0,10.2, and 9.6 liters. Find a 95% confidence interval for the mean, assuming an normal distribution? Single Sample: Estimating the Mean Form Table A.4 for

IEEM 320IEEM151 Notes 17, Page 16 ? Prediction Interval: the confidence interval for a new observation x 0 Prediction Interval ? X 0 independent of ? variance of X 0 - =  2 (n+1)/n

IEEM 320IEEM151 Notes 17, Page 17 (1-  )100% prediction interval of a future observation, x 0 Prediction Interval:  known

IEEM 320IEEM151 Notes 17, Page 18 Prediction Interval:  unknown (1-  )100% prediction interval of a future observation, x 0