Two-factor ANOVA: A scientist wants to meet with you to discuss the analysis of a
two-factor experiment they have run, using a CRD with an equal number of observations
per treatment combination. They send you an email with the following computer output:
Df Sum Sq Mean Sq F value Pr(>F)
F1 2 2.114 1.057 1.4669 0.24078
F2 3 10.752 3.584 4.9750 0.00437 **
F1:F2 6 7.611 1.268 1.7608 0.12743
Residuals 48 34.580 0.720
1. How many levels of each of the factors were there?
2. How many total observations were there?
3. How many observations per treatment combination were there?
4. Fill in the ANOVA table that would have resulted from anova(lm(y∼F1)):
Df Sum Sq Mean Sq F value
F1 _____ _____ _____ _____
Residuals _____ _____ _____
5. Refer back to the original ANOVA table, and let H0refer to the hypothesis that the
additive model is correct.
(a) Write down a numerical estimate of the experimental error that is valid whether
or not H0is true.
(b) Write down another estimate of the experimental error that is valid if H0is true.
6. Now we want to test H0, using the ratio of your two estimators in (a) and (b) as
your test statistic. On the opposite side of this sheet, sketch a curve representing the
sampling distribution of this statistic under H0and assuming normality, independence,
and constant variance of the data. Your picture should include an x-axis, a y-axis, and
a labeled origin. Your curve doesn’t have to be the correct height or exactly the
correct shape, but you need to mark on your x-axis the observed numerical value of
your statistic, which you should be able to obtain from the table provided above.
(a) Precisely identify the name of this distribution:
(b) What should be the numerical value of the area under the curve to the right of
the observed statistic?
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