AGENDA Estimation of variances Confidence interval of variances Hypotheses concerning one variance Hypotheses concerning two variances ESTIMATION OF VARIANCES Sample variances Sample ranges SAMPLE VARIANCES Commonly used to estimate population variance SAMPLE VARIANCE CONFIDENCE INTERVAL OF VARIANCES Using sample distribution chi-square, a confidence interval can be estimated for the population variance CONFIDENCE INTERVAL OF VARIANCES FORMULA EXAMPLE Refractive indices of 20 pieces of glass Variance is 1.2 x10-4 Construct a 95% confidence interval of sigma Assume independence and normality CONFIDENCE INTERVAL OF VARIANCES SAMPLE RANGES Commonly used in industry Easier to calculate ranges than variances on the shop floor Used in statistical process control (SPC) charts Describes spread of data Highest value minus lowest value in sample Sigma can be estimated with sample range using d2 d2 is a constant that depends on the sample size ESTIMATING SIGMA WITH SAMPLE RANGE HYPOTHESES CONCERNING ONE VARIANCE Used to compare the variance of a sample against a known standard Important in quality applications Example Comparison of the variance of a production lot to a known specification Is the variability unacceptable HYPOTHESES TEST PROCEDURE Identify Ho and Ha Determine level of significance (generally 0.05 or 0.01) Determine “critical value” criterion from level of significance Calculate “test statistic” Make decision Fail to reject Ho Reject Ho HYPOTHESES Null sigma squared = sigma squared 0 Alternate sigma squared < sigma squared 0 (1 sided) sigma squared > sigma squared 0 (1 sided) sigma squared not equal to sigma squared 0 (2 sided) CHI-SQUARE for a particular number of degrees of freedom HYPOTHESES CONCERNING ONE VARIANCE TEST STATISTIC EXAMPLE Lapping process for grinding silicone wafers Acceptable only if sigma is at most 0.50 mil At 0.05 test Ho: sigma = 0.50 (satisfactory) Ha: sigma > 0.50 (unsatisfactory) Sample of 15 s=0.64 CHI-SQUARE for 14 degrees of freedom TEST STATISTIC DECISION Test statistic of 22.94 is less than the critical value of 23.685 Cannot reject the Ho English Not sufficient evidence to conclude that the lapping process is unsatisfactory HYPOTHESES CONCERNING TWO VARIANCES Used to compare the differences in variance of two populations Which hypotheses test concerning two means to use Used in quality applications Example Comparison of two different lots or methods for consistency Want the most consistent lot or method HYPOTHESES TEST PROCEDURE Identify Ho and Ha Determine level of significance (generally 0.05 or 0.01) Determine “critical value” criterion from level of significance Calculate “test statistic” Make decision Fail to reject Ho Reject Ho HYPOTHESES CONCERNING TWO VARIANCES Null sigma squared 1= sigma squared 2 Alternate 1 sigma squared 1< sigma squared 2 (1 sided) F=variance 2 / variance 1 Alternate 2 sigma squared 1> sigma squared 2 (1 sided) F=variance 1 / variance 2 Alternate 2 sigma squared 1 not equal to sigma squared 2 (2 sided) F =variance larger / variance smaller HYPOTHESES CONCERNING TWO VARIANCES CRITICAL VALUE HYPOTHESES CONCERNING TWO VARIANCES TEST STATISTIC EXAMPLE ONE SIDED TEST Determine whether there is less variability in the silver plating between company 1 and company 2 Sample sizes of 12 Company 1, s1=0.035 Company 2, s2=0.062 At 0.05 test Ho: sigma squared 1 = sigma squared 2 Ha: sigma squared 1 < sigma squared 2 CRITICAL VALUE FOR V1=11, V2=11 HYPOTHESES CONCERNING TWO VARIANCES TEST STATISTIC DECISION Test statistic of 3.14 is > critical value of 2.82 Reject Ho – Plating by company 1 is less variable EXAMPLE TWO SIDED TEST Is it reasonable to assume that the variances in the heat producing capacity of the coal between the two mines populations sampled are equal Samples Mine 1, var1=15,750, n1=5 Mine 2, var2=10,920, n2=6 At 0.02 test Ho: sigma squared 1 = sigma squared 2 Ha: sigma squared 1 not equal to sigma squared 2 CRITICAL VALUE FOR VM=4, Vm=5 HYPOTHESES CONCERNING TWO VARIANCES TEST STATISTIC DECISION Test statistic of 1.44 is < critical value of 11.39 Cannot Reject Ho – No difference in variability