Questions: A factorial experiment involving two levels of factor A and three levels of factor B resulted in the following data. Factor B ------:--------:------ Level 1 Level 2 Level 3 Level 1 115 70 55 175 86 103 Factor A Level 2 125 137 120 75 105 116 Test for any significant main effects and any interaction. Use α=0.05. Round Sum of Squares to the whole number, F value, Mean Square to two decimals, if necessary, and p-value to four decimals. Source of Variation Sum of Squares Degrees of Freedom Mean Square F value p-value Conclusion ------------------------------------------------------------------------------------------------------------ Factor A Not significant ✓ Factor B Not significant Interaction Not significant Error Total

A factorial experiment involving two levels of factor A and three levels of factor B resulted in the following data.

Factor B
------:--------:------
Level 1 Level 2 Level 3
Level 1 115 70 55
175 86 103
Factor A
Level 2 125 137 120
75 105 116

Test for any significant main effects and any interaction. Use α=0.05. Round Sum of Squares to the whole number, F value, Mean Square to two decimals, if necessary, and p-value to four decimals.

Source of Variation Sum of Squares Degrees of Freedom Mean Square F value p-value Conclusion
------------------------------------------------------------------------------------------------------------
Factor A Not significant ✓
Factor B Not significant
Interaction Not significant
Error
Total

Transcript text: A factorial experiment involving two levels of factor A and three levels of factor B resulted in the following data. \begin{tabular}{|c|c|c|c|c|} \hline \multicolumn{2}{|l|}{\multirow[t]{2}{*}{}} & & Factor B & \\ \hline & & Level 1 & Level 2 & Level 3 \\ \hline & \multirow[b]{2}{*}{Level 1} & 115 & 70 & 55 \\ \hline & & 175 & 86 & 103 \\ \hline \multicolumn{5}{|l|}{Factor A} \\ \hline & Level 2 & 125 & 137 & 120 \\ \hline & & 75 & 105 & 116 \\ \hline \end{tabular} Test for any significant main effects and any interaction. Use $\alpha=0.05$. Round Sum of Squares to the whole number, $F$ value, Mean Square to two decimals, if necessary, and $p$-value to four decimals. \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline Source of Variation & \begin{tabular}{l} Sum of \\ Squares \end{tabular} & Degrees of Freedom & Mean Square & $F$ value & p-value & \multicolumn{2}{|r|}{Conclusion} \\ \hline Factor A & & & & & & Not significant & $\checkmark$ \\ \hline Factor B & & & & & & Not significant & \\ \hline Interaction & & & & & & Not significant & \\ \hline Error & & & & & & & \\ \hline Total & & & & & & & \\ \hline \end{tabular}

Solution

Solution Steps

To test for significant main effects and interaction in a factorial experiment, we perform a two-way ANOVA. The steps include calculating the sum of squares for each source of variation (Factor A, Factor B, Interaction, and Error), determining the degrees of freedom, computing the mean square values, and then calculating the F-values. Finally, we compare the F-values to the critical F-value at the given significance level (α = 0.05) to determine significance. The p-values are also calculated to support the conclusions.

Step 1: Calculate the Means

The overall mean \( \bar{Y} \) is calculated as follows: \[ \bar{Y} = \frac{115 + 70 + 55 + 175 + 86 + 103 + 125 + 137 + 120 + 75 + 105 + 116}{12} = 106.83 \]

The means for Factor A levels are: \[ \bar{Y}_{A1} = \frac{115 + 70 + 55 + 175 + 86 + 103}{6} = 100.67 \] \[ \bar{Y}_{A2} = \frac{125 + 137 + 120 + 75 + 105 + 116}{6} = 113.00 \]

The means for Factor B levels are: \[ \bar{Y}_{B1} = \frac{115 + 175 + 125 + 75}{4} = 122.50 \] \[ \bar{Y}_{B2} = \frac{70 + 86 + 137 + 105}{4} = 99.50 \] \[ \bar{Y}_{B3} = \frac{55 + 103 + 120 + 116}{4} = 98.50 \]

Step 2: Calculate the Sum of Squares

The sum of squares for Factor A \( SS_A \) is: \[ SS_A = 2 \cdot 3 \left( (\bar{Y}_{A1} - \bar{Y})^2 + (\bar{Y}_{A2} - \bar{Y})^2 \right) = 456.33 \]

The sum of squares for Factor B \( SS_B \) is: \[ SS_B = 2 \cdot 2 \left( (\bar{Y}_{B1} - \bar{Y})^2 + (\bar{Y}_{B2} - \bar{Y})^2 + (\bar{Y}_{B3} - \bar{Y})^2 \right) = 1474.67 \]

The total sum of squares \( SS_{total} \) is: \[ SS_{total} = \sum (Y_{ij} - \bar{Y})^2 = 11719.67 \]

The sum of squares for interaction \( SS_{interaction} \) is: \[ SS_{interaction} = SS_{total} - SS_A - SS_B = 9788.67 \]

The sum of squares for error \( SS_{error} \) is: \[ SS_{error} = SS_{total} - SS_A - SS_B - SS_{interaction} = 0 \]

Step 3: Calculate Degrees of Freedom and Mean Squares

The degrees of freedom for each source of variation are:

\( df_A = 1 \)
\( df_B = 2 \)
\( df_{interaction} = 2 \)
\( df_{error} = 6 \)
\( df_{total} = 11 \)

The mean squares are calculated as follows: \[ MS_A = \frac{SS_A}{df_A} = 456.33 \] \[ MS_B = \frac{SS_B}{df_B} = 737.33 \] \[ MS_{interaction} = \frac{SS_{interaction}}{df_{interaction}} = 4894.33 \] \[ MS_{error} = \frac{SS_{error}}{df_{error}} = 0 \]

Step 4: Calculate F-values and p-values

The F-values are calculated as: \[ F_A = \frac{MS_A}{MS_{error}} = \infty \quad (\text{since } MS_{error} = 0) \] \[ F_B = \frac{MS_B}{MS_{error}} = \infty \quad (\text{since } MS_{error} = 0) \] \[ F_{interaction} = \frac{MS_{interaction}}{MS_{error}} = \infty \quad (\text{since } MS_{error} = 0) \]

The p-values are: \[ p_A = 0.0, \quad p_B = 0.0, \quad p_{interaction} = 0.0 \]

Final Answer

The results indicate that all factors and their interaction are statistically significant:

Factor A: \( SS = 456, df = 1, MS = 456.33, F = \infty, p = 0.0 \)
Factor B: \( SS = 1475, df = 2, MS = 737.33, F = \infty, p = 0.0 \)
Interaction: \( SS = 9789, df = 2, MS = 4894.33, F = \infty, p = 0.0 \)

Thus, the conclusion is that all main effects and the interaction are significant.

\[ \boxed{\text{All factors and interaction are significant.}} \]

Was this solution helpful?

Unhelpful

Helpful

Questions asked by other users

< Previous Next >

Thank you for your feedback.

Copied!