QM-1/2011/Estimation Page 1 Quantitative Methods Estimation

QM-1/2011/Estimation Page 2 Estimation Process Population Mean, , is unknown Random Sample Mean  X  = 50 Are we confident that  is 50? OR we are more confident in saying that  is between 48 & 52?

QM-1/2011/Estimation Page 3 Population Parameters are Estimated Population ParameterSample Statistic  22 xixi n  (x i -x) 2 n-1

QM-1/2011/Estimation Page 4 Estimation Methods Estimation Point EstimationInterval Estimation

QM-1/2011/Estimation Page 5 Estimation Methods Estimation Point Estimation Interval Estimation

QM-1/2011/Estimation Page 6 Point Estimation  Point Estimate  Draw a sample from a population and compute AVERAGE to ESTIMATE the MEAN Value of the population.  The formula used for computing Average is called ‘Statistic’ or ‘Estimator’.  The computed value based on the sample is called ‘Estimate’ for the population parameter (here ‘Mean’).

QM-1/2011/Estimation Page 7 Problem with Point Estimate  Following data is a sample from a population.  13,13,16,19,17,15,14,13,15,15,18.  Point estimate for Mean is simply the average of these data, i.e  If we had collected lesser or larger number of samples the average would have been different.

QM-1/2011/Estimation Page 8 Problem with Point Estimate Population Parameter 1 st group of sample 2 nd group of sample Estimates from different samples will be different, but, within a limit, around the Population parameter. This is applicable for all estimates (i.e. Average, s.d. etc). All estimates from different samples are ‘Point estimates’.

QM-1/2011/Estimation Page 9 Estimation Methods Estimation Point Estimation Interval Estimation

QM-1/2011/Estimation Page 10 Interval Estimate  This estimate gives a lower and upper limit. The population parameter will be in this interval with a certain degree of confidence.  A confidence level is associated with an interval estimate.  Higher the confidence level desired, the spread of the interval will be larger.

QM-1/2011/Estimation Page 11 Error in Estimate Sample statistic (point estimate)  Population Parameter (unknown) x X ~ N( ,  /√n) ~ N(0, 1) X –   /√n

QM-1/2011/Estimation Page 12 Standard Normal & Tail area Z 0.05 P(Z < Z 0.05 ) = 0.95

QM-1/2011/Estimation Page 13 Standard Normal & Tail area (1-  )  Z 

QM-1/2011/Estimation Page 14 Standard Normal & 2-Tail area (1-  )   Z   

QM-1/2011/Estimation Page 15 Estimation of Interval  P(|Z |< Z ) = 0.95  P(|X –  | < Z ) = 0.95  P(|X –  | < Z  /√n) = 0.95  P(-Z  /√n <  - X < Z  /√n) = 0.95  P(X - Z  /√n <  < X + Z  /√n) = 0.95  /√n

QM-1/2011/Estimation Page 16 Confidence Interval for Mean  P(X – Z  /2  /√n <  < X + Z  /2  /√n) = 1-  (1-  )   /2 Z  /

QM-1/2011/Estimation Page 17 Wider interval for higher confidence 99% x   x x   x x   x x   x 95% x   x x   x 90% x   x x   x

QM-1/2011/Estimation Page 18 Factors Affecting Interval Width  Data Dispersion  Measured by   Higher  results wider interval.  Sample Size   X =  n  Higher sample size results narrower interval.  Level of Confidence (1 -  )  Affects Z  Higher level of confidence, wider interval.

QM-1/2011/Estimation Page 19 Confidence Interval Estimates Confidence Interval MeanProportion Known  Unknown 

QM-1/2011/Estimation Page 20 Confidence Interval Estimates Confidence Interval Mean Proportion Known  Unknown 

QM-1/2011/Estimation Page 21 Confidence Interval of Mean -  known  Population Standard Deviation  is Known  Population is Normally Distributed.  n > 30, if Population is Not Normal.  Confidence Interval: n ZX n ZX      2/2/

QM-1/2011/Estimation Page 22 Exercise  The mean of a random sample of n = 36 is  X = 50. Set up a 95% confidence interval estimate for  if  = 12.

QM-1/2011/Estimation Page 23 Exercise  You’re a Q/C inspector of Coke. The  for 2-liter bottles is 0.05 liters. A random sample of 100 bottles showed  X = 1.99 liters. What is the 90% confidence interval estimate of the true mean amount in 2-liter bottles?

QM-1/2011/Estimation Page 24 Confidence Interval Estimates Confidence Interval Mean Proportion Known  Unknown 

QM-1/2011/Estimation Page 25 Confidence Interval of Mean -  Unknown  Normal population.  n > 30  The sample sd s is a good estimator of   Confidence interval is as below. n ZX n ZX s  s   2/2/