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Unformatted text preview: AGENDA Confidence Intervals Concerning Two Means Inferences Concerning Variances CONFIDENCE INTERVALS CONCERNING TWO MEANS Confidence intervals can also be generated for the difference between two means Confidence intervals are based on corresponding t calculations Provides additional information over two sample hypotheses tests If graphed, visually illustrates how big the difference is If the CI covers 0, the difference is not statistically significant Types of confidence intervals Small sample confidence interval Large sample confidence interval Paired sample confidence interval SMALL SAMPLE CONFIDENCE INTERVAL When n1 or n2 or both are <30 Utilizes the t value for alpha / 2 n1+n2-2 degrees of freedom Sample means and variances SMALL SAMPLE CONFIDENCE INTERVAL SMALL SAMPLE CONFIDENCE INTERVAL Tensile strength measurements of an aluminum alloy in 1000 psi Alloy 1 mean 70.70, std 1.80, 58 samples Alloy 2 mean 76.13, std 2.42, 27 samples 95% confidence interval, alpha=0.05 SMALL SAMPLE CONFIDENCE INTERVAL SMALL SAMPLE CONFIDENCE INTERVAL The 95 percent confidence interval does not cover 0 Statistically significant difference at alpha=0.05 between the alloys LARGE SAMPLE CONFIDENCE INTERVAL When n1 and n2 are >=30 Utilizes the Z value for alpha / 2 Sample means and variances LARGE SAMPLE CONFIDENCE INTERVAL LARGE SAMPLE CONFIDENCE INTERVAL Tensile strength measurements of an aluminum alloy in 1000 psi Alloy 1 mean 70.70, std 1.80, 58 samples Alloy 2 mean 76.13, std 2.42, 27 samples * 95% confidence interval, alpha=0.05 * Shouldnt really do this because of sample size is not quite 30. LARGE SAMPLE CONFIDENCE INTERVAL LARGE SAMPLE CONFIDENCE INTERVAL LARGE SAMPLE CONFIDENCE INTERVAL The 95 percent confidence interval does not cover 0 Statistically significant difference at alpha=0.05 between the alloys PAIRED SAMPLE CONFIDENCE INTERVAL Requires paired type data Utilizes the t value for alpha / 2 n-1 degrees of freedom Mean and variance of difference PAIRED SAMPLE CONFIDENCE INTERVAL PAIRED SAMPLE CONFIDENCE INTERVAL Average weekly losses of worker hours Summary statistics Dbar=5.2 S=4.08 N=10 90% confidence Interval, alpha=0.10 PAIRED SAMPLE CONFIDENCE INTERVAL PAIRED SAMPLE CONFIDENCE INTERVAL 90 percent confidence interval does not cover 0 The difference is statistically significant at alpha=0.10 The safety program is effective INFERENCES CONCERNING VARIANCES Estimation of variances Confidence interval of variances Hypotheses concerning one variance ESTIMATION OF VARIANCES Sample variances Sample ranges SAMPLE VARIANCES Commonly used to estimate population variance SAMPLE VARIANCE 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 CONFIDENCE INTERVAL OF VARIANCES Using sample distribution chi-square, a confidence interval can be estimated for the population variance CONFIDENCE INTERVAL OF VARIANCESFORMULA 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 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-SQUAREfor 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-SQUAREfor 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 ...
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- Fall '08