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Description: The calculations for a factorial experiment involving four levels of factor A, three levels of factor B, and three replications resulted in the following data: SST = 271, SSA = 22, SSB = 24, SSAB = 170. Set up the ANOVA table and test for any signifi

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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The calculations for a factorial experiment involving four levels of factor A, three levels of factor B, and three replications resulted in the following data: SST = 271, SSA = 22, SSB = 24, SSAB = 170. Set up the ANOVA table and test for any significant main effect and may interaction effect. use a= .05.

Expert Solution

Step 1

From the given information, the values of sum of squares is $S S T = 271, S S A = 22, S S B = 24, S S A B = 170$ , four levels of factor A, three levels of factor B, and three replications.

The error sum of squares is,

$\begin{array}{rcl} S S E & = & S S T - S S A - S S B - S S A B \\ = & 271 - 22 - 24 - 170 \\ = & 55 \end{array}$

The degrees of freedom for factor A is,

$\begin{array}{rcl} d f_{A} & = & a - 1 \\ = & 4 - 1 \\ = & 3 \end{array}$

The degrees of freedom for factor B is,

$\begin{array}{rcl} d f_{B} & = & b - 1 \\ = & 3 - 1 \\ = & 2 \end{array}$

The degrees of freedom for interaction is,

$\begin{array}{rcl} d f_{i n t e r a c t i o n} & = & (a - 1) (b - 1) \\ = & 3 \times 2 \\ = & 6 \end{array}$

The error degrees of freedom is,

$\begin{array}{rcl} d f_{E} & = & a b (r - 1) \\ = & 4 \times 3 \times (3 - 1) \\ = & 24 \end{array}$

The total degrees of freedom is,

$\begin{array}{rcl} d f_{T} & = & d f_{A} + d f_{B} + d f_{i n t e r a t i o n} + d f_{E} \\ = & 3 + 2 + 6 + 24 \\ = & 35 \end{array}$

The mean sum of squares for factor A is,

$\begin{array}{rcl} M S A & = & \frac{S S A}{d f_{A}} \\ = & \frac{22}{3} \\ = & 7.3333 \end{array}$

The mean sum of squares for factor B is,

$\begin{array}{rcl} M S B & = & \frac{S S B}{d f_{B}} \\ = & \frac{24}{2} \\ = & 12 \end{array}$

The mean sum of squares for interaction A is,

$\begin{array}{rcl} M S A B & = & \frac{S S A B}{d f_{A B}} \\ = & \frac{170}{6} \\ = & 28.3333 \end{array}$

The error mean sum of squares is,

$\begin{array}{rcl} M S E & = & \frac{S S E}{d f_{E}} \\ = & \frac{55}{24} \\ = & 2.2917 \end{array}$

The test statistic for factor A is,

$\begin{array}{rcl} F_{A} & = & \frac{M S A}{M S E} \\ = & \frac{7.3333}{2.2917} \\ = & 3.1999 \end{array}$

The test statistic for factor B is,

$\begin{array}{rcl} F_{B} & = & \frac{M S B}{M S E} \\ = & \frac{12}{2.2917} \\ = & 5.2363 \end{array}$

The test statistic for interaction is,

$\begin{array}{rcl} F_{A B} & = & \frac{M S A B}{M S E} \\ = & \frac{28.3333}{2.2917} \\ = & 12.3634 \end{array}$

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