Question

Source	DF	SS	MS	F	P-value
Treatments	2	25.75	12.875	14.420	.000
IR1 vs. DR1, DR2	1	22.6875	22.6875	25. 41
Error	21	18.75	.893
Total	23	44.5

What can you conclude from the null hypothesis test on the linear contrast between the IR group and the other two groups (DR1, DR2)?

Select ALL that APPLY and Explain

A- The linear contrast was statistically significant at the .10 level of significance.

B- There was no statistically significant difference, at the .05 alpha level, in the population between the IR group and the other two groups.

C-The linear contrast was statistically significant at the .05 alpha level of significance.

D- The difference in average ratings in the population, for the IR group was significantly different at the .05 alpha level, from the other two groups combined.

E- The difference in average ratings, in the population, for the IR group was significantly different, at the .05 alpha level, from either one of the other two groups.

Answer 1

A- The linear contrast was statistically significant at the .10 level of significance.

For this, we need to get the value of F, 2,21 where =0.10 and the tabulated value is 2.96096.

The calculated value of the F statistic is 25. 41 > 2.96096, Hence, we can reject the null hypothesis at the .10 level of significance and similarly we can say the linear contrast was statistically significant at the .10 level of significance.

Option A is correct.

B- There was no statistically significant difference, at the .05 alpha level, in the population between the IR group and the other two groups.

This option is stating about the difference in the population ie, in the treatments. As the p-value for comparing the treatment means is 0 and less than 0.05, then we can reject the null hypothesis that the treatments are not statistically different.

Option B is not correct.

C-The linear contrast was statistically significant at the .05 alpha level of significance.

Similarly, in the case of option A, here we need to get the value of F, 2,21 where =0.05 and the tabulated value is 4.3248.

The calculated value of the F statistic is 25. 41 > 4.3248, Hence, we can reject the null hypothesis at the .05 level of significance and similarly we can say the linear contrast was statistically significant at the .05 level of significance as well.

Option C is correct.

D- The difference in average ratings in the population, for the IR group was significantly different at the .05 alpha level, from the other two groups combined.

Linear contrast between the IR group and the other two groups (DR1, DR2) can be expressed as l11+l22+l33.

where sum(l1,l2,l3) =0 and i's are average ratings for IR, DR1 and DR2 respectively, i=1,2,3.

Test for any linear contrast between IR group and the other two groups are statistically significant at 0.05 and as well as 0.10 level of significance.

The difference in average ratings in the population, for the IR group and the other two groups, is a specific value of l1=1, l2=-1 and l3=-1

So this linear contrasts is also statistically significant at 0.05 level of significance.

Option D is correct.

E- The difference in average ratings, in the population, for the IR group was significantly different, at the .05 alpha level, from either one of the other two groups.

This option holds as for the similar reason stated in option D, it is a specific case of linear contrasts where l1=1 and either l2=1 & l3=0 or l2=0 & l3=1.

Option E is correct.