Questions: Consider a two-factor factorial design with three levels in factor A, three levels in factor B, and four replicates in each of the nine cells. That is, there are two degrees of freedom in determining the factor A variation, two degrees of freedom in determining the factor B variation, four degrees of freedom in determining the interaction variation, and 27 degrees of freedom in determining the random variation. Assume that SSA=110, SSB=120, SSE=270, and SST=620. Answer parts (a) through (d). a. What is SSAB? SSAB=120 b. What are MSA and MSB? MSA =

Consider a two-factor factorial design with three levels in factor A, three levels in factor B, and four replicates in each of the nine cells. That is, there are two degrees of freedom in determining the factor A variation, two degrees of freedom in determining the factor B variation, four degrees of freedom in determining the interaction variation, and 27 degrees of freedom in determining the random variation. Assume that SSA=110, SSB=120, SSE=270, and SST=620. Answer parts (a) through (d). a. What is SSAB? SSAB=120 b. What are MSA and MSB? MSA =
Transcript text: Consider a two-factor factorial design with three levels in factor A, three levels in factor B, and four replicates in each of the nine cells. That is, there are two degrees of freedom in determining the factor A variation, two degrees of freedom in determining the factor B variation, four degrees of freedom in determining the interaction variation, and 27 degrees of freedom in determining the random variation. Assume that $\operatorname{SSA}=110, \mathrm{SSB}=120, \mathrm{SSE}=270$, and SST=620. Answer parts (a) through (d). a. What is SSAB? \[ S S A B=120 \] b. What are MSA and MSB? MSA = $\square$
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Solution

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Solution Steps

Step 1: Calculate \( SS_{AB} \)

To find the interaction sum of squares \( SS_{AB} \), we use the formula:

\[ SS_{AB} = SS_T - (SS_A + SS_B + SS_E) \]

Substituting the given values:

\[ SS_{AB} = 620 - (110 + 120 + 270) = 620 - 500 = 120 \]

Thus, we have:

\[ \boxed{SS_{AB} = 120} \]

Step 2: Calculate Mean Square for Factor A (\( MS_A \))

The mean square for factor A is calculated using the formula:

\[ MS_A = \frac{SS_A}{df_A} \]

Where \( df_A = 2 \) (degrees of freedom for factor A). Substituting the values:

\[ MS_A = \frac{110}{2} = 55.0 \]

Thus, we have:

\[ \boxed{MS_A = 55.0} \]

Step 3: Calculate Mean Square for Factor B (\( MS_B \))

The mean square for factor B is calculated using the formula:

\[ MS_B = \frac{SS_B}{df_B} \]

Where \( df_B = 2 \) (degrees of freedom for factor B). Substituting the values:

\[ MS_B = \frac{120}{2} = 60.0 \]

Thus, we have:

\[ \boxed{MS_B = 60.0} \]

Final Answer

  • \( SS_{AB} = 120 \)
  • \( MS_A = 55.0 \)
  • \( MS_B = 60.0 \)

The final boxed answers are:

\[ \boxed{SS_{AB} = 120} \] \[ \boxed{MS_A = 55.0} \] \[ \boxed{MS_B = 60.0} \]

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