The 2 nd to last topic this year!!

 ANOVA Testing is similar to a “two sample t- test except” that it compares more than two samples to one another. The main point is that we need to measure variability from one sample to another while accounting for variability within each sample.

 We will use the 5-step process like what we have used for all of our tests of significance.  We will use the “F-Distribution” which is a skewed distribution (like the Chi Squared)  We will look at conditions specific to ANOVA and the other “usual conditions”.  We will use “degrees of freedom” like the other distributions.  Except two different values: DF between groups k k-1 and DF within groups is sample size n, n-1

 Each sample is an independent random sample  The distribution of the response variable follows a normal distribution  The population variances are equal across responses for the group levels. This can be evaluated by using the following rule of thumb: if the largest sample standard deviation divided by the smallest sample standard deviation is not greater than two, then assume that the population variances are equal. Or…if the largest standard deviation is more than twice the smallest we break this condition.

 Given that you are comparing k independent groups, the null and alternative hypotheses are:  H0:μ1=μ2=⋯=μk (where k is number of sample means)  Ha: Not all μ’s are equal The null hypothesis is that all of the groups' population means are essentially equal. The alternative is that they are not all equal; there are at least two population means that are not equal to one another.

 ANOVA uses the F-distribution for the “F” value. The “F” value is measured by looking at the ratio of: (the variability between the groups)/ (the variability within the same groups) The larger the F value, the further out on the distribution it lies, much like a t-score or the chi squared value. And more evidence that the variability is coming from: ___________

 lity/statistics-inferential/anova/v/anova-1- calculating-sst-total-sum-of-squares lity/statistics-inferential/anova/v/anova-1- calculating-sst-total-sum-of-squares

 The F statistic will be calculated by your handheld or by Excel (minitab or other software as well). It produces an “ANOVA Source Table” where all of the computations are listed for us:  k = Number of groups (or g for groups) n = Total sample size (all groups combined) n k = Sample size of group k x¯ k = Sample mean of group k x¯ = Grand mean (i.e., mean for all groups combined) SS = Sum of squares MS = Mean square df = Degrees of freedom F = F-ratio (the test statistic)

 We will use technology and in most cases the alpha level will always be.05. Most tables list only the F-statistic values for.05.  (the table uses the degrees of freedom for each axis)

 As with all of our tests for inference, (Hypothesis Tests), we will use the p-value and the test statistic to make our decision about H 0.  If p-value is lower than.05, we will reject null and accept the alternative. (What type of error is possible)  What if we don’t reject Null?

 Based on your decision, make a statement of conclusion that includes reference to the data and the means that you are analyzing:  Our p-value gives evidence that there is in fact a statistical difference in the means for at least of two our samples, so there is a significant difference between…. Eg- the performance ratings of cars made by different manufacturers.

 We will do a practice problem in class with a spreadsheet set of data using Baseball Wins data.  We will do two more data driven questions for homework to be done along with the on-line assignment. You may put the on-line answers on the back of the spreadsheet.  Quiz will cover one of each test: linear regression t-test, ANOVA, and multi-variable regression.