Extra Exercise 17
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(a) The degrees of freedom are 3-1=2 for the groups and 30-1=29 for total, as
expected. It is also seen that the Mean Square values are calculated by dividing
the Sum of Squares by the df. Finally, F equals the ratio between the two Mean
Square values.

(b) When the blue population mean goes from 19.99 to 20, the P-value switches
from P<0.05 to P>0.05 as F decreases from 3.367 to 3.333. The 95%-percentile
of the F(2,27) distribution was determined (in Minitab) to be 3.354, in-between
those two values, as expected.

(c) When moving one group away from the others, the between-group variation 
increases. This is because the between-group variation is computed from the 
group means as the sum of squared deviations from the overall mean. As the 
numerator of the F-statistic is a constant times this between-group variation, 
the F-statistic increases and eventually becomes significant when the groups 
get far apart. Intuitively, the larger group differences, the more indicative the
data are of group (population) means being unequal.

(d) When increasing the variation within the groups the F-statistic
decreases and the P-value increases. Mathematically, the F-statistic is
computed as a ratio with the estimated within-group variation in the 
denominator; thus it is no surprise that F decreases when the
within-group variation increases. Intuitively, the more variation in the data
(within the groups), the more difficult it should be to detect
differences between groups. Note that the between-group variation
is constant because the group means are unchanged. The F-test is one-tailed 
and the P-value includes only larger values, therefore a smaller value 
of F also implies a larger P-value.