Source SS DF MS F p-value ================================================================== Among Samples SSA k-1 SSA/(k-1) MSA/MSW Within Samples SSW N-k SSW/(N-k) ================================================================== Total SST N-1k is the number of populations.
N is the total number of observations in all samples.
The formula for SSA and SSW are in the textbook. A simpler formula for SSW, depending on the sample standard deviations, is
SSW = (n1-1)s12 + ... + (nk-1)sk2If s is the sample standard deviation of all N observations, then
SST = (N-1)s2It is also true that SST = SSA + SSW.
Finding these sums of squares is the computationally tedious part of the computation. The remainder of the computations are straightforward.
The F statistic from the ANOVA table is compared to an F distribution with these degrees of freedom. The p-value is the area to the right of this F statistic. This p-value is interpreted like any other p-value. It is the probability of observing a result at least as extreme as the actual test statistic, assuming the null hypothesis is true. Low p-values are indications of strong evidence against the null hypothesis.
Firstbasemen | 31, 25, 21, 23, 9, 18, 16 Shortstops | 13, 1, 2, 14, 5, 2 Outfielders | 14, 5, 11, 23, 24, 36, 18Test the hypothesis that mean homerun totals are the same.
The ANOVA table is:
Source SS DF MS F p-value ================================================================== Among Samples 764.974 2 382.4869 6.009 0.0106 Within Samples 1081.976 17 63.6457 ================================================================== Total 1846.950 19From the limited tables in the textbook, we may conclude that the p-value is between .01 and .025 since the F statistic is between 4.62 and 6.11.
You are given a partial ANOVA table for a problem in which there are three samples of size 5, 3, and 3.
Source SS DF MS F p-value ================================================================== Among Samples 20 Within Samples ================================================================== Total 50This is completed as follows:
SSW = 50 - 20 = 30 dfA = 3 - 1 = 2 dfW = 11 - 3 = 8 MSA = 20 / 2 = 10 MSW = 30 / 8 = 3.75 F = 10 / 3.75 = 2.667 p = area to right of 2.667 under F dist with 2 and 8 df = 0.1296The tables in the book would allow us to conclude that the p-value is more than .10.
Bret Larget, firstname.lastname@example.org