Question 7

The following data represent the results from an independent-measures experiment comparing three treatment conditions. Use a spreadsheet to conduct an analysis of variance with $\alpha=0.05$ to determine whether these data are sufficient to conclude that there are significant differences between the treatments.
\begin{tabular}{|c|c|c|}
\hline Treatment A & Treatment B & Treatment C \\
\hline 7 & 8 & 10 \\
\hline 5 & 5 & 13 \\
\hline 5 & 6 & 12 \\
\hline 7 & 7 & 11 \\
\hline 6 & 9 & 9 \\
\hline
\end{tabular}

F-ratio $=$ critical value $=$

Conclusion:
These data do not provide evidence of a difference between the treatments
There is a significant difference between treatments

Step 1 ：Calculate the means of each treatment: \(\frac{7+5+5+7+6}{5} = 6\) for Treatment A, \(\frac{8+5+6+7+9}{5} = 7\) for Treatment B, and \(\frac{10+13+12+11+9}{5} = 11\) for Treatment C.

Step 2 ：Calculate the overall mean: \(\frac{6+7+11}{3} = 8\).

Step 3 ：Calculate the Sum of Squares Between (SSB): \(5*((6-8)^2 + (7-8)^2 + (11-8)^2) = 30\).

Step 4 ：Calculate the Sum of Squares Within (SSW): \((7-6)^2 + (5-6)^2 + (5-6)^2 + (7-6)^2 + (6-6)^2 + (8-7)^2 + (5-7)^2 + (6-7)^2 + (7-7)^2 + (9-7)^2 + (10-11)^2 + (13-11)^2 + (12-11)^2 + (11-11)^2 + (9-11)^2 = 20\).

Step 5 ：Calculate the degrees of freedom between (dfB) and within (dfW): \(dfB = 3 - 1 = 2\) and \(dfW = 15 - 3 = 12\).

Step 6 ：Calculate the Mean Square Between (MSB) and Mean Square Within (MSW): \(MSB = \frac{SSB}{dfB} = \frac{30}{2} = 15\) and \(MSW = \frac{SSW}{dfW} = \frac{20}{12} = 1.67\).

Step 7 ：Calculate the F-ratio: \(F-ratio = \frac{MSB}{MSW} = \frac{15}{1.67} = 8.98\).

Step 8 ：The critical value for F(2, 12) at \(\alpha=0.05\) is approximately 3.89. Since our calculated F-ratio (8.98) is greater than the critical value (3.89), we reject the null hypothesis and conclude that there is a significant difference between the treatments. The final answer is \(\boxed{8.98}\).