Consider the following table for a single-factor ANOVA.

Level of Factor
Replicates 1 2 3
1 1 2 5
2 2 4 7
3 2 4 7

(a) Find x1,2.


(b) Find x2,1.


(c) Find C1.


(d) Find x.


(e) Find (Ci )2.

To answer these questions, we need to understand the terms used in the context of ANOVA. ANOVA stands for Analysis of Variance and is a statistical method used to compare the means of two or more groups.

In this table, "Level of Factor" refers to the different levels of the variable being studied, and "Replicates" refers to the number of observations within each level.

Now let's address each question one by one:

(a) Find x1,2.
To find x1,2, we need to find the mean (x) for Level 1 in Replicate 2. To do this, we simply sum up the values at each replicate and divide by the number of observations.

x1,2 = (2 + 4 + 7) / 3
x1,2 = 13 / 3
x1,2 ≈ 4.33

(b) Find x2,1.
To find x2,1, we need to find the mean (x) for Level 2 in Replicate 1. Using the same approach as before:

x2,1 = (2 + 4 + 7) / 3
x2,1 = 13 / 3
x2,1 ≈ 4.33

(c) Find C1.
To find C1, we need to find the overall mean (x̄) for all the observations in the table. We sum up all the values and divide by the total number of observations.

C1 = (1 + 2 + 5 + 2 + 4 + 7 + 2 + 4 + 7) / 9
C1 = 34 / 9
C1 ≈ 3.78

(d) Find x.
To find x, we need to find the overall mean (x̄) for each level and each replicate. We calculate the mean for each cell as we did in (a) and (b), and then take the average of those means.

x = (x1,1 + x1,2 + x1,3 + x2,1 + x2,2 + x2,3 + x3,1 + x3,2 + x3,3) / 9

Substituting the values:

x = (1 + 2 + 5 + 2 + 4 + 7 + 2 + 4 + 7) / 9
x = 34 / 9
x ≈ 3.78

(e) Find (Ci)2.
To find (Ci)2, we need to calculate the square of the deviation of each value from the overall mean (x).

(Ci)2 = (1 − x)2 + (2 − x)2 + (5 − x)2 + (2 − x)2 + (4 − x)2 + (7 − x)2 + (2 − x)2 + (4 − x)2 + (7 − x)2

Substituting the value of x approximately 3.78:

(Ci)2 = (1 - 3.78)2 + (2 - 3.78)2 + (5 - 3.78)2 + (2 - 3.78)2 + (4 - 3.78)2 + (7 - 3.78)2 + (2 - 3.78)2 + (4 - 3.78)2 + (7 - 3.78)2

Simplifying the equation will give you the result.