Wednesday April 30,
2014

Class Activity --- In Minitab,
bring up the data set calories_schools.mtw, which

contains fat calorie values for several lunches at three schools.

1. Make side-by-side boxplots of calories for the three
schools.

2. Make side-by-side boxplots of demo data for the three
schools.

Looking at the boxplots above, we want to explore the question "Are the data
contradictory to the hypothesis

that the three schools' lunches have the same mean fat calories?"

Discussion

Variation is the KEY

Conclusion --- We must analyze
the variation

- among sample averages, and

- within samples

This is formally called *Analysis of Variance* or ANOVA

Formal Notation

Assumptions

1. each of the k population distributions are normal

2. standard deviation is the same for each of the k
populations

3. the observations within each of the k samples are
independent of one another

4. when comparing populations, the k samples are selected randomly and independently of one
another;

when comparing treatment means, treatments are assigned at random to subjects

Cranking it Out

For the calories and schools data

SSTr = n1(sample avg 1 - grand avg)^{2} + n2(sample
avg 2 - grand avg)^{2} + n3(sample avg 3 - grand
avg)^{2}

= 8(145-141)^{2} +
8(138-141)^{2} + 8(140-141)^{2}

= 8(16) + 8(9) + 8(1)

= 208

**MSTr = SSTr / (k-1) = 208 / 2 = 104**

SSE = (n1-1)S1^{2} + (n2-1)S2^{2} + (n3-1)S3^{2}

= 7(10.88)^{2} + 7(8.73)^{2} + 7(8.00)^{2}

= 1810.11

**MSE = SSE / (N-k) = 1810.11 / (24-3) = 86.196**

**F = MSTr / MSE = 104 / 86.196 = 1.21**

p-value = P(F_{2,21} > 1.21) = 0.318

Class Problem --- Bring up the
Minitab data set solvent_sorption.mtw, which provides the sorption
rates

into a test medium of three organic solvents --- aromatics,
chloroalkanes, and esters. Test to see if there

is significant evidence in the data that the mean sorption rates differ
among these solvents.

Finally, check the assumptions for the validity of the ANOVA F test.