Answer:
Step-by-step explaSuppose your instructor administers 3 different forms of a final exam. When scores are posted,
you see the observed mean scores for those 3 different forms—82, 66, and 60—are not the same.
Is this due to chance variation, or do the 3 exams not share the same level of difficulty?
Version 1: 65, 73, 78, 79, 86, 93, 100
Version 2: 39, 58, 63, 67, 69, 74, 92
Version 3: 39, 52, 62, 64, 66, 77
[Note: if there were only 2 different exams, a two-sample t test could be used.]
It would be unrealistic to expect 3 identical sample mean scores, even if the exams were all equally
difficult: in other words, there’s bound to be some variation among means in the 3 groups.
Also, of course there will be variation of scores within each group.
If the ratio of variation among groups to variation within groups is large enough, we will have
evidence that the population mean scores for the 3 groups actually differ. Picture two possible
configurations for the data, both of which represent 3 data sets with means 82, 66, and 60. Thus,
variation among means (from the overall mean of 70) would be the same for both configurations.
1. There could be large within-group variation, such as we see in Exams 1a, 2a, and 3a, in
which case the ratio among to within is small; in this case, the data could be coming from
populations that share the same mean.
2. There could be small within-group variation, such as we see in Exams 1b, 2b, and 3b, in
which case the ratio among to within is large; in this case, the population means probably
differ.nation:
