Quizzes and exercise of multi-factor experiments

These quizzes and exercise help us review and practice what we obtained in this chapter. They are presented in "show-hide" form similar to the sections of a web page as we are already familiar. Quizzes are multiple choice questions. After choosing an option, an announcement of result appears. To return back to this page, we click "OK" on the announcement. Of course we can choose again.

In exercise, we are required to calculate a value then fill it in an empty rectangle. Note that in this rectangle, only value is acceptable, its unit is not required. After filling the result, we click on "Answer". If the answer is correct, the border of the rectangle is green and there is green "V" symbol in the adjacent square. If the answer is wrong, the border of the rectangle is red and there is red "X" symbol in the adjacent square. We can erase the answer and retry by clicking on "Retry".

For the exercise, there is a hidden solution. To show this solution, we click on "Solution" tab. But try to solve exercise by ourself, don't abuse that solutions.

Quiz 1

There is interaction between 2 factors A and B. So

Quiz 2

A full factorial experiment was carried out with 3 levels of factor A and 4 levels of factor B. This experiment consists of:

Quiz 3

In multiple factor experiments, the quantity that characterizes the interaction between two factors A and B is

Quiz 4

`MS_(AB)` is calculated from

Quiz 5

`2^4` experiment is used to

Quiz 6

In analysis of variance of full factorial experiment with two factors A and B, the greatest value is:

Quiz 7

An experiment is carried out to study effect of factor `X` and `Y` on response `Z` by response surface methodology. The most frequently empirical function is:

Quiz 8

An experiment is carried out to study effect of 3 factors by CCD method. This experiment consists of

Exercise

A full factorial experiment was carried out to study the effect of factor A with 3 levels and factor B with 4 levels on response `Y`. Each treatment was realized with 3 runs. Some results of analysis of variance of this experiment are shown in Table 1.

Table 2 Some results of analysis of variance for full factorial experiment with 2 factors
Source of variation	`SS`	`MS`
Factor A		20
Factor B		30
Interaction AB
Error	120
Total	400

Let's continue the analysis of variance of this experiment by calculating missing values and filling in the blank of Table 1.

a. Calculate the sums of squares.

• `SS_A` =

• `SS_B` =

• `SS_(AB)` =

b. Calculate the means of squares.

• `MS_(AB)` =

• `MS_E` =

c. Calculate the test statistics `F_o`.

• `F_(oA)` =

• `F_(oB)` =

• `F_(oAB)` =

d. Determine the critical values `F`* with confidence level 95%.

• `F_A`* =

• `F_B`* =

• `F_(AB)`* =

Quiz e

Which statement is correct ?

Solution

From the context : `a=3` ; `b=4` ; `n=3`

So : `df_A=a-1=2` ; `df_B=b-1=3` ; `df_(AB)=(a-1)(b-1)=6` ;
`df_E=ab(n-1)=24`

a. Determine sums of squares `SS`.

`SS_A=MS_A\ df_A=20xx2=40`

`SS_B=MS_B\ df_B=30xx3=90`

`SS_(AB)=SS_T-SS_A-SS_B-SS_E=400-40-90-120=150`

b. Determine means of squares `MS`.

`MS_(AB)=(SS_(AB))/(df_(AB))=150/6=25`

`MS_E=(SS_E)/(df_E)=120/24=5`

c. Determine test statistics `F_o`.

`F_(oA)=(MS_A)/(MS_E)=20/5=4`

`F_(oB)=(MS_B)/(MS_E)=30/5=6`

`F_(oAB)=(MS_(AB))/(MS_E)=25/5=5`

d. Determining critical value `F`* by using percentage point table of Fisher distribution.

`F_A`*`=3,4028`

`F_B`*`=3,0088`

`F_(AB)`*`=2,5082`

The result of these calculations is synthesized in Table 3.

Table 3 Result of analysis of variance for full factorial experiment with 2 factors
Source of variation	Degree of freedom	`SS`	`MS`	`F_o`	`F`*
Factor A	2	40	20	4	3,4028
Factor B	3	90	30	6	3,0088
Interaction AB	6	150	25	5	2,5082
Error	24	120	5
Total	35	400

e. From this result, we conclude that:

Because `F_(oA)>F_A`* : factor A affects response Y.
Because `F_(oB)>F_B`* : factor B affects response Y.
Because `F_(oAB)>F_(AB)`* : there is interaction of factors A and B on response `Y`.

This web page was last updated on 04 December 2018.