South Dakota School of Mines & Technology Estimation Industrial Engineering

Estimation Interval Estimates (variance) Industrial Engineering Estimation Interval Estimates (variance) Industrial Engineering

Estimator for a Variance  Suppose in a sample of 25 light bulds, we compute a sample variance of 10,000. We would now like to make an inference about the true but unknown population variance  2. If the underlying distribution is normal, then the distribution of the sample variance is chi- square.

Estimator for a Variance  /2   2 n-1  2 n-1,  /2  2 n-1,1-  / )1(   n s n  

Estimator for Variance ))1((1 2 2/1, /,1       nn s nP Miracle 21c occurs 2 2/, /1,1 2 )1()1(         nn snsn

Example u Suppose in our sample of 25 light bulbs we compute a sample variance of 10,000. Compute a 90% confidence for the true variance.

Example 2 2/, /1,1 2 )1()1(         nn snsn )000,10( )000,10(24 2 

Example 6,591 <  2 < 17, )000,10( )000,10(24 2 

Example 6,591 <  2 < 17, )000,10( )000,10(24 2  Note that the confidence interval for  2 is not symmetric. 6,591 10,000 17,799

Summary u To make probabilistic statements about  (  known)N(0,1)  (  unknown)t n-1 normal  (  unknown)N(0,1)n >> 30  2 given s 2 normal  1 2 given  2 2 F n1-1,n2-1 normal p N(0,1)n >>  n 