## Anova

ANOVA is an acronym for Analysis Of Variance. It is a method for comparing 3 or more means by using multiple F Tests. For ANOVA we must develop a Null and Alternative Hypothesis, and once again the " = " goes with Ho.

The Hypothesis will be . (All means are equal)
Ha: at least one mean is different from the others.

The Test Statistic F = the variance between samples / the variance within samples.
= MS treatment / MS error.
(sample is also called treatment or factor)

To look up the F critical value we need: the numerator degrees of freedom (k-1)
the denominator degrees of freedom k(n-1)
Where "k" is the number of different samples (also called treatments, or factors)
"n" is the number of items within each sample.
In a One Way ANOVA we are comparing only one variable from each sample.

One Way ANOVA
Hot peppers are officially measured in laboratory units called scovill units. On a common scale they are "taste tested" and ranked on a scale from 1 to 10 with 1 the mildest, 10 hottest. Four different peppers were considered for the "Salsa in a jar" from section #7. The taste test rankings from 12 of each type of pepper were as follows:

Find: a) the variance between the different types of peppers, b) variance within each type of pepper, c) the F Test Statistic, d) the F critical value. Develop the Null and Alternative Hypothesis, and using a = 0.05 test your hypothesis. State your conclusion.

We will begin by entering the data into the appropriate LISTS. Press (you will see this screen).

Press [ always start from a clear list ]

Next enter each piece of data from sample1 into LIST1, sample2 into LIST2, etc.
(When you have finished all four LIST you should be looking at this screen)

3 Now press (you will again see this screen)

Arrow right to TESTS and Arrow up to F : ANOVA (
(you should see this screen.)

Press . (this screen appears)

Here we must enter all factors we want to test. For our problem it will be L1, L2, L3, L4.

so...press

This screen (the one on the right) can be seen by pressing the Arrow down key

As you can see from the screen the MS Factor = 10.05 (this is the amount of variation between pepper types).
If we Arrow down we see that the MS Error = 1.44 ( this is the amount of variance within each type of pepper).

The F Test Statistic is seen to be = 6.96. The p-value = 0.00062
The appropriate Hypothesis and Test :
The Null Hypothesis is :
The Alternative Hypothesis is : Ha : at least one mean is different.
Recall .
The p-value = 0.00062 < 0.05 Therefore we should reject Ho.

From the tables in a Textbook we see that the F c.v. = 2.83 [k=3, n(k-1)=44]
Since our F Statistic = 6.96 > 2.83. Again, we would reject Ho.

In conclusion we say that at the 0.05 level of significance there is sufficient evidence to reject the claim that all four pepper types have the same mean "taste test" ranking.