MOVING ON TO ANOVA (Analysis of Variance)

We have just finished comparing the means of TWO data sets to determine if there is a significant difference between them. We have used the z-Test (proportions) and the t-Test to do this. We FINALLY find a use for the VARIANCE!

We haven't used the variance very much in this course, just its square root, the standard deviation.

There is a LOT of squaring (and maybe swearing) in ANOVA calculations and if you recall from your Week 1 exercise in calculating variance, we squared distances to make any negative ones, positive. This is what we are essentially doing here, but in multiple dimensions.

We are basically comparing the Means and Variances of multiple populations (three or more) to see if ANY are significantly different from the others. (see the Additional Guidance attachment for assistance.) It will NOT tell us which population is different, however.

7) (made-up data) WORK THIS PROBLEM:

Plant 1	Plant 2	Plant 3	Plant 4	Plant 5
1.2	16.4	12.1	11.5	24
10.1	-6	9.7	10.2	-3.7
-2	-11.6	7.4	3.8	8.2
1.5	-1.3	-2.1	8.3	9.2
-3	4	10.1	6.6	-9.3
-0.7	17	4.7	10.2	8
3.2	3.8	4.6	8.8	15.8
2.7	4.3	3.9	2.7	22.3
-3.2	10.4	3.6	5.1	3.1
-1.7	4.2	9.6	11.2	16.8
2.4	8.5	9.8	5.9	11.3
0.3	6.3	6.5	13	12.3
3.5	9	5.7	6.8	16.9
-0.8	7.1	5.1	14.5
19.4	4.3	3.4	5.2
2.8	19.7	-0.8	7.3
13	3	-3.9	7.1
42.7	7.6	0.9	3.4
1.4	70.2	1.5	0.7
3	8.5
2.4	6
1.3	2.9

What is the NULL Hypothesis, Ho:

What is the Alternate Hypothesis, Ha:

In the formula: "k" is simply the number of groups and capital "N" is the TOTAL of ALL the data values involved (all groups together) and small "n" is the AVERAGE number of data points per group = (N/k)

The 5 groups involved have DIFFERENT numbers of samples (n), but they are relatively close, so I used the AVERAGE "n" and got an F-Test close to the text value. This is NOT technically the official way to calculate F, but the official way is quite complex for an introductory overview)

	Plant 1	Plant 2	Plant 3	Plant 4	Plant 5
n
Mean
Variance

1. Fill in the Table above ^
2. Determine k = _____
3. Calculate N = ______
4. Calculate the average n = ____
5. Calculate the VARIANCE of the 5 sample means = ______ *average-n = _______ = _________
6. Calculate the AVERAGEVARIANCE of the 5-groups' variances = _____________

F-Test = (e) / (f) = _________(This is by using the average n)

(The Text software gets an F-Test of 1.16)

To find the F-Critical value:

1. Calculate the first degrees of freedom: df1 = k - 1 = ____
2. Calculate the second degrees of freedom:df2 = N - k = ____
3. Find the F-Critical from the F-Table below for these degrees of freedom (df) and = 1% : ___________
4. Compare your F-Test to the F-critical and do you accept or reject Ho?____________

You can use this website:https://stattrek.com/online-calculator/f-distribution.aspx to find the p-value (probability) for a given F-Test statistic. What is the p-value for the F-Test you calculated? ___________

Comparing these to the = 0.01 or 1%, would you draw the same Reject/Fail to reject Ho conclusion?