The Following Is a Partial ANOVA Table

The following is a partial ANOVA table. Source Sum of Squares df Mean Square F Treatment 2 Error 20 Total 500 11 Complete the table and answer the following questions. Use the .05 significance level. a. How many treatments are there? b. What is the total sample size? c. What is the critical value of F? d. Write out the null and alternate hypothesis? e. What is your conclusion regarding the null hypothesis?

The ANOVA Table

ANOVA Table Source of Sum of Degrees of Mean Square Variation Squares Freedom F Treatments SST k  1 SST/ ( k  1 ) = MST / MSE MST Error SSE n  k SSE/ ( n  k ) = MSE Total SS Total n  1

There are (k 1) degrees of freedom associated with the numerator of the formula for F, and (n  k) degrees of freedom associated with the denominator, where k is the number of populations and n is the total number of sample observations.

From the given ANOVA table we have,

SS Total = 500, k  1 = 2, n  1 = 11 MSE = SSE/ ( n  k ) = 20

Therefore, k = 2+1 =3, and n = 11 +1 = 12 Also, SSE = MSE*( n  k ) = 20*(12-3) =20*9 = 180 Now, SST =SS Total – SSE = 500 – 180 = 320 Thus, MST = SST/( k  1 ) = 320/2 = 160 Also, F = MST/MSE = 160/20 = 8

Thus the completed ANOVA table is ANOVA Table Source of Sum of Degrees of Mean Square Variation Squares Freedom F Treatments 320 2 160 8 Error 180 9 20 Total 500 11 a. How many treatments are there?

There are three treatments. b. What is the total sample size?

The total sample size is 12 c. What is the critical value of F?

For a = 0.05, from the F distribution with (k-1, n-k) = (2,9) degrees of freedom, the critical value of F is 4.26. d. Write out the null and alternate hypotheses.

Ho: There is no significant difference between the treatment means.

Ha: At least one treatment mean is different from the others. e. What is your conclusion regarding the null hypothesis?

We reject the null hypothesis if F > 4.26. Here F = 8 > 4.26. Since the calculated value is greater than critical value, we reject Ho and conclude that at least one treatment mean is different from the others.