In Exercises 2534, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 1020020110100101
4.3.2 WILCOXON SIGNED-RANK TEST 1) Used for the differences between two samples. In this case we take the differences (D’s) and test the null hypothesis a. H :0Distribution of D’s is symmetric about the nullval∆e against the alternatives of D ▯ values tending to be smaller than▯ or D values tending to be larger th∆n▯ b. Rank the absolute values of the differences c. Calculate the test statistis+, by adding all off differences 2) Note: 3) The test statistic, denoted by W: 4) The null is rejected if 4.3.3 LEVENES TEST FOR VARIANCES 1) Used for comparing more than 2 population variations a. Deals with variances b. The null is that all of these particular variancesare equal c. The alternative is that they are not 2) Levenes test is used because it is the least restrictive, in that, there are no assumptions regarding sample size or distribution. a. Only requirement is that the populations are independent 3) We will use software to calculate because calculation is tedious a. Where N is grand total of observations b. The null is rejected if the p-value is less than alpha 4.4 CONTINGENCY TABLES TEST FOR INDEPENDENCE 1) They are cross-tabulations of frequency counts a. i.e. 2) Notation for contingency tables: th th a. ▯ =▯▯he number in the i row and j column i. ▯ is representative of rows and ▯ of columns ii. i.e.▯▯ = 18 b. ▯ =▯▯he total of row i i. i.e. ▯▯ = 68 c. ▯ =▯▯e total of column j i. i.e. ▯▯ = 81 d. ▯ ▯▯roportion of that number ▯ ▯▯ ▯▯▯ i. i.e. ▯▯ = ▯ = 18/216 ▯▯▯▯▯ e. ▯ ▯▯proportion of row total f. ▯ = proportion of column total ▯▯ 3) Expected values a. The numbers that are already in the table are the Observed values b. Expected values help us calculate residuals and thetest statistic(which I will go into next) c. How to calculate an expected value: 4) Test statistic a. We use chi squared distribution: b. This tests independence of the variables in the table because the null hypothesis is that they are independent. c. So when the p-value is less than alpha, we reject the null and the variables are dependent. d. When we fail to reject the null, aka when the p-value is greater than alpha, the variables are independent. 5) Residuals a. Residuals are how many standard deviations the observed and expected values are from eachother b. We calculate residuals if the variables are dependent (after doing the test statistic and testing the null) c. If the values are over 2 (deviations) then we can claim that this specific value appears frequently d.