Exhibit 10-11
In a completely randomized experimental design involving five treatments, 13 observations were recorded for each of the five treatments a total of 65 observations). The following information is provided.
SSTR = 200 Sum Square Between Treatments) SST = 800 Total Sum Square)
-Refer to Exhibit 10-11. If at 95% confidence we want to determine whether or not the means of the five populations are equal, the p-value is
A) between 0.05 to 0.10
B) between 0.025 to 0.05
C) between 0.01 to 0.025
D) less than 0.01

Question

Quizplus · Accepted Answer

To test whether or not the means of the five populations are equal, we need to conduct an analysis of variance (ANOVA) test. Using the provided information, we can calculate the mean square between treatments (MSB) by dividing SSTR by the degrees of freedom between treatments (dfB), which is equal to the number of treatments minus one (k-1) in a completely randomized design. Thus, MSB = SSTR/(k-1) = 200/4 = 50. We can also calculate the mean square error (MSE) by dividing the sum of squares error (SSE) by the degrees of freedom error (dfe), which is equal to the total number of observations minus the total number of treatments (N-k). Thus, SSE = SST - SSTR = 800 - 200 = 600, and dfe = N-k = 65-5 = 60. Therefore, MSE = SSE/dfe = 600/60 = 10.

Using the MSB and MSE, we can calculate the F-statistic as MSB/MSE = 50/10 = 5. To find the p-value for this F-statistic, we need to consult an F-distribution table with degrees of freedom for the numerator (dfB) equal to k-1 = 4 and degrees of freedom for the denominator (dfe) equal to N-k = 60. The F-critical value at the 0.05 level of significance is 2.70. Since our calculated F-statistic of 5 is larger than the critical value of 2.70, we reject the null hypothesis that all of the means are equal.

The p-value for a one-way ANOVA with k-1 = 4 and N-k = 60 degrees of freedom and an F-statistic of 5 is less than 0.01. Therefore, the best choice is option D.

Exhibit 10-11 In a Completely Randomized Experimental Design Involving Five Treatments, 13