Question

an experiment has been conducted for four treatments with eight blocks. complete the following analysis of variance table (to 2 decimals but p - value to 4 decimals, if necessary). if answer is zero enter \0\.
source of variation \tsum of squares \tdegrees of freedom \tmean square \tf \tp - value
treatments \t800 \t\t\t
blocks \t300 \t\t\t
error \t\t\t
total \t1500 \t\t\t
use α = 0.05 to test for any significant differences.
the p - value is \t- select your answer -
what is your conclusion?

• select your answer -

Explanation:

Step1: Degrees of Freedom for Treatments

For treatments, number of treatments \( k = 4 \), so degrees of freedom \( df_{treatments}=k - 1=4 - 1 = 3 \)

Step2: Degrees of Freedom for Blocks

Number of blocks \( b = 8 \), so degrees of freedom \( df_{blocks}=b - 1=8 - 1 = 7 \)

Step3: Degrees of Freedom for Total

Total degrees of freedom \( df_{total}=(k\times b)-1=(4\times8)-1 = 31 \) (Wait, no, in randomized block design, \( df_{total}=(k - 1)+(b - 1)+(k - 1)(b - 1)= (k\times b)-1 \). Let's recalculate: \( df_{treatments}=3 \), \( df_{blocks}=7 \), so \( df_{error}=(k - 1)(b - 1)=3\times7 = 21 \), and \( df_{total}=3 + 7+21 = 31 \), which matches \( 4\times8 - 1=31 \))

Step4: Sum of Squares for Error

\( SS_{total}=SS_{treatments}+SS_{blocks}+SS_{error} \), so \( SS_{error}=SS_{total}-SS_{treatments}-SS_{blocks}=1500 - 800 - 300 = 400 \)

Step5: Mean Square for Treatments

\( MS_{treatments}=\frac{SS_{treatments}}{df_{treatments}}=\frac{800}{3}\approx266.67 \)

Step6: Mean Square for Blocks

\( MS_{blocks}=\frac{SS_{blocks}}{df_{blocks}}=\frac{300}{7}\approx42.86 \)

Step7: Mean Square for Error

\( MS_{error}=\frac{SS_{error}}{df_{error}}=\frac{400}{21}\approx19.05 \)

Step8: F - statistic for Treatments

\( F_{treatments}=\frac{MS_{treatments}}{MS_{error}}=\frac{266.67}{19.05}\approx14.00 \)

Step9: F - statistic for Blocks

\( F_{blocks}=\frac{MS_{blocks}}{MS_{error}}=\frac{42.86}{19.05}\approx2.25 \)

Step10: p - value for Treatments

Using F - distribution with \( df_1 = 3 \), \( df_2 = 21 \), and \( F = 14.00 \). Using F - table or calculator, p - value \( \approx0.0001 \) (for \( F = 14.00 \), \( df1 = 3 \), \( df2 = 21 \))

Step11: p - value for Blocks

Using F - distribution with \( df_1 = 7 \), \( df_2 = 21 \), and \( F = 2.25 \). p - value \( \approx0.0787 \) (using F - calculator)

Now, for the test with \( \alpha = 0.05 \):

• For treatments: p - value \( \approx0.0001<0.05 \), so we reject \( H_0 \) (there are significant differences among treatments)
• For blocks: p - value \( \approx0.0787>0.05 \), so we fail to reject \( H_0 \) (no significant differences among blocks)

Filling the ANOVA table:

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	F	p - value
Blocks	300	7	42.86	2.25	0.0787
Error	400	21	19.05	-	-
Total	1500	31	-	-	-

For the p - value (assuming the question is about treatments, as it's more likely to test treatments in a randomized block design for significant differences):

The p - value for treatments is approximately 0.0001.

Conclusion: Since the p - value (0.0001) is less than \( \alpha = 0.05 \), we reject the null hypothesis. There is sufficient evidence to conclude that there are significant differences among the four treatments.

Answer:

ANOVA Table Filled:

• Treatments: Degrees of Freedom = 3, Mean Square = 266.67, F = 14.00, p - value = 0.0001
• Blocks: Degrees of Freedom = 7, Mean Square = 42.86, F = 2.25, p - value = 0.0787
• Error: Sum of Squares = 400, Degrees of Freedom = 21, Mean Square = 19.05
• Total: Degrees of Freedom = 31

p - value (for treatments): 0.0001

Conclusion: Reject the null hypothesis. There are significant differences among the treatments.