Hypothesis testing

Chapter 9Hypothesis Testing: Chi-Square tests and the one way analysis of varianceChi-square test for two ways tablesHyphothesis test for the difference in the proportion of succeses in two or more groupsRelationship between two categorical variabels in a two-way cross-classification table.Counts of particels found cross classified by wafer conditionWafer conditionGood BadTotalParticlesYes143650Presentno32080400Total 334116450Wafer conditionGood BadTotalParticlesYes37.1112.8950Presentno296.89103.11400Total 334116450The null hyphothesis show that no relationship between column variable and row variabelThe alternative hypothesis show that have difference relationship between column variable and row variabel

The critical value of chi-square is equal to 3.481. Because 62.81 > 3.841. It reject the null hypothesis9.2 One way analysis of variance ( ANOVA )Testing for the differences among the means of more than two groupsEvaluating differences between gropus as a one factor experiment ( a completely randomized design ) in which the variable defining the groups is called the factor of interestthe factor of interest can have several numerical levels and categorical lebels

One-way ANOVAComparing the differences among the population means of more than two groups for a one factor

Sum of square total (SST) is the sum of the squared the differences between each individual alue and the mean of all the valueSST= Sum of( Each value- Mean of all value)2 Sum of squares within groups (SSW) is within-group variation that measure the difference, mean, and sum the squares over all groupsSSW= Sum of [ (Each value in the group Group mean)2 ]

The three variance of ANOVAThere are three different variances : the variance among the groups, the variance within groups and the total varianceThe variance are reffered to in yhe analysis of variance terminology as mean squareThe mean square among groups (MSA) is equal to the sum of squares among groups (SSA) divided by the number of groups minus 1The mean square within group (MSW) is equal to the sum of squares within groups (SSW) divided by the sample size minus the number of groupsThe mean square total (MST) is equal to the sum of squares total (SST) divided by the sample size minus 1

Hypothesis testingZ and t Tests

8.1 Testing for the difference between two porpotionsThe sample statitics need to analyze these diffrences are te propotion of occurrenses in group 1 and group 2Normal distributionWafer conditionGood BadTotalParticlesYes143650Presentno32080400Total 334116450A level of significance of = 0.05The proportion of good wafers without particles is 320/334 = 0.9581The proportion of bad wafers without particles is 80/116 = 0.6897

H0 :Ha :

-1,96+1,96Reject H0 if Z < -1.96 or if Z > +1.96Because Z = 7.93 is greater than the upper critical value of +1.96, you will jecet the null hypothesis

Testing for the difference between the means of two independent groupsPooled-Variance t-Test is test of hypothesis that involved two independent groupsConcept: the sample variances of each group be combined (pooled) into the one estimate of the variance common in the two groups

Pooled variance t Test AssumptionsNormally distribution with equal variancesSo, pooled variance t Test will not sensitive to moderate departures from this assumptionsThe sample sizes largeTo check Pooled variance t Test Assumptions use box and whisker plotIn normally distribution , The lower tail is longer than the upper tailIf data didnt from normally distribution you can use a nonparametic procedure like wilcoxon rank sum and separated-variance t-testExample

The paired t-TestThere are two approaches that inolve releted data between gropsIn first approach, you pair or match items under study according to some other variabels. In second approch, repeated measurements are obtained from the same set of items or individuals that will be have alike if trated alike.In either approach about the difference between the value of the paired item rather rhan the values themselvesDifference= Related value in sample 1 related value in sample 2