Question

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 = 254, ssa = 26, ssb = 27, ssab = 155. set up the anova table and test for significance using α = 0.05. show entries to 2 decimals, if necessary. if the answer is zero, enter 0. use unrounded intermediate values for calculation.
source
of variation\tsum
of squares\tdegrees
of freedom\tmean
square\tf value\tp - value
factor a\t\t\t\t\t
factor b\t\t\t\t\t
interaction\t\t\t\t\t
error\t\t\t\t
total\t\t\t
the p - value for factor a is – select your answer – ▼
what is your conclusion with respect to factor a?
– select your answer – ▼
the p - value for factor b is – select your answer – ▼
what is your conclusion with respect to factor b?
– select your answer – ▼
the p - value for the interaction of factors a and b is – select your answer – ▼
what is your conclusion with respect to the interaction of factors a and b?
– select your answer – ▼

Explanation:

Step1: Determine Degrees of Freedom

• For Factor A: \( df_A = a - 1 = 4 - 1 = 3 \) (where \( a = 4 \) levels of A)
• For Factor B: \( df_B = b - 1 = 3 - 1 = 2 \) (where \( b = 3 \) levels of B)
• For Interaction (AB): \( df_{AB} = (a - 1)(b - 1) = 3 \times 2 = 6 \)
• For Total: \( df_{Total} = abr - 1 = 4 \times 3 \times 3 - 1 = 35 \) (where \( r = 3 \) replications)
• For Error: \( df_{Error} = df_{Total} - df_A - df_B - df_{AB} = 35 - 3 - 2 - 6 = 24 \)

Step2: Sum of Squares (SS)

• Given: \( SST = 254 \), \( SSA = 26 \), \( SSB = 27 \), \( SSAB = 155 \)
• \( SSE = SST - SSA - SSB - SSAB = 254 - 26 - 27 - 155 = 46 \)

Step3: Mean Square (MS)

• \( MS_A = \frac{SSA}{df_A} = \frac{26}{3} \approx 8.67 \)
• \( MS_B = \frac{SSB}{df_B} = \frac{27}{2} = 13.50 \)
• \( MS_{AB} = \frac{SSAB}{df_{AB}} = \frac{155}{6} \approx 25.83 \)
• \( MS_{Error} = \frac{SSE}{df_{Error}} = \frac{46}{24} \approx 1.92 \)

Step4: F - Values

• \( F_A = \frac{MS_A}{MS_{Error}} = \frac{8.67}{1.92} \approx 4.52 \)
• \( F_B = \frac{MS_B}{MS_{Error}} = \frac{13.50}{1.92} \approx 7.03 \)
• \( F_{AB} = \frac{MS_{AB}}{MS_{Error}} = \frac{25.83}{1.92} \approx 13.45 \)

Step5: p - Values (using F - distribution tables or software)

• For \( F_A = 4.52 \), \( df_1 = 3 \), \( df_2 = 24 \): \( p - value_A \approx 0.012 \)
• For \( F_B = 7.03 \), \( df_1 = 2 \), \( df_2 = 24 \): \( p - value_B \approx 0.004 \)
• For \( F_{AB} = 13.45 \), \( df_1 = 6 \), \( df_2 = 24 \): \( p - value_{AB} \approx 0.000 \)

ANOVA Table:

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	F - value	p - value
Factor B	27.00	2	13.50	7.03	0.004
Interaction (AB)	155.00	6	25.83	13.45	0.000
Error	46.00	24	1.92	-	-
Total	254.00	35	-	-	-

Conclusions:

• Factor A: Since \( p - value_A = 0.012 < \alpha = 0.05 \), we reject \( H_0 \). There is a significant effect of Factor A.
• Factor B: Since \( p - value_B = 0.004 < \alpha = 0.05 \), we reject \( H_0 \). There is a significant effect of Factor B.
• Interaction (AB): Since \( p - value_{AB} = 0.000 < \alpha = 0.05 \), we reject \( H_0 \). There is a significant interaction between Factor A and Factor B.

Answer:

ANOVA Table Entries:

• Factor A: SS = 26.00, df = 3, MS = 8.67, F = 4.52, p - value = 0.012
• Factor B: SS = 27.00, df = 2, MS = 13.50, F = 7.03, p - value = 0.004
• Interaction: SS = 155.00, df = 6, MS = 25.83, F = 13.45, p - value = 0.000
• Error: SS = 46.00, df = 24, MS = 1.92
• Total: SS = 254.00, df = 35

Conclusions:

• p - value for Factor A: 0.012; Conclusion: Reject \( H_0 \) (significant effect)
• p - value for Factor B: 0.004; Conclusion: Reject \( H_0 \) (significant effect)
• p - value for Interaction: 0.000; Conclusion: Reject \( H_0 \) (significant interaction)