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Unformatted text preview: puting the F Test Statistic We will see how to get MS Groups and MSE and perform the F test. These two mean squares will be a sum of squares (SS) divided by a corresponding degrees of freedom (DF). The data can be generically represented below, where X ij j observation from the i th population. However we really don’t have to worry too much about these subscripts, as we will go through the steps using words! th Data from Population1 X 11 … Data from Population k X k1 X 12 X 22 X k2 X 2 n2 X 1n1 Data from Population 2 X 21 X kn k The details leading to the F statistic are presented in six steps, ending with an ANOVA table. 2 Step 1: Calculate the mean and variance for each sample: xi , si Step 2: Calculate the overall sample mean (using all N n1 n 2 ... n k observations): x Step 3: Calculate the sum of squares between groups: SS Groups groups ni xi x 2 Step 4: Calculate the sum of squares within groups (due to error): SSE groups ni 1si 2 Step 5: OPTIONAL: Calculate the total sum of squares: SS Total values xij x 2 Step 6: Fill in the ANOVA table: Source Groups Sum of Squares SS Groups DF Mean Square F k-1 SSE N-k F=
MSG/MSE Error (Within) MSG =
SSE/(N-k) Total SS Total N-1 173 If H0 is true, then the F statistic, F MS Groups
MSE , has an F(k – 1, N – k) distribution. Below are a few pictures of some F distributions. Table A.4 provides percentiles of an F distribution. However, standard computer output also provides the exact p‐value and completed ANOVA table. We will rely on SPSS output to provide the p‐value, but you should know how the ANOVA table is constructed and be able to sketch a picture of the p‐value for an F‐test. Stat 250 Formula Card Summary of ANOVA: One‐Way ANOVA
SS Groups = SSG = ni ( x i x ) 2 MS Groups = MSG = SSG k 1 ANOVA Table Source
Error MS Error = SS Error = SSE = (ni 1) s i 2 SSE N k
MS Groups F
MS Error MSE = s 2 p groups S...
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