AGENDA Exam 3 Review Course Evaluation EXAM 3 REVIEW Variances Ranges Hypotheses tests involving one variance Hypotheses tests involving two variances Chi-squared goodness of fit test U Rank Sum Test Linear Regression SAMPLE VARIANCE CONFIDENCE INTERVAL OF VARIANCES Using sample distribution chi-square, a confidence interval can be estimated for the population 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 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 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) HYPOTHESES CONCERNING ONE VARIANCE TEST STATISTIC 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 for consistency 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 TEST STATISTIC GOODNESS OF FIT TESTS Based on a comparison of observations between Observed data Theoretical data The comparison utilizes a set of intervals or cells Each cell has a lower and upper boundary values The determination of the boundaries are a function of Theoretical distribution Number of observations in the sample 2 different approaches Equi-interval Equi-probable CELLS AND BOUNDARIES Maximum number of cells not to exceed 100 Expected number of observations needs to be at least 5 To calculate observed values in each cell, we must determine the actual x cell boundaries from the equiprobable cells TEST STATISTIC RANK SUM TEST Alternative to: Independent t test Smith-Satherwaithe test Used when the sample data is either Non-normal To small to determine normality U test Based on the rank order of sorted data sample rather than mean and standard deviation Equivalent sets of data with have similar ranks PERFORM CALCULATIONS Calculate rank sums W1 is the sum of the ranks of data from set 1 W2 is the sum of the ranks of data from set 2 Calculate U’s U1=W1-n1(n1+1)/2 U2=W2-n2(n2+1)/2 Next… NEXT Calculate the mean of the ranks u = n1*n2/2 Calculate the variance of the ranks Sigma1 squared = n1*n2*(n1+n2+1)/12 Calculate the test statistic Z Z = (U1-u) / sigma1 LINEAR REGRESSION Statistical method for determining a linear relationship between two variables assuming normality and independence Two types of variables Independent Dependent Method of least squares PREDICTION LINE DETERMINING PREDICTION LINE Calculate the total squared error between the Actual value Predicted value according to the prediction line NORMAL EQUATIONS With some rearrangement Resulting linear equations are called normal equations INTERCEPT AND SLOPE Substitute x, y, xy, and x squared values in normal equations Solve for a Intercept with y axis at x=0 Solve for b Coefficient of slope for x values