5.15 Can this happen? You conduct an ANOVA to compare three groups,imagine the p

Question

5.15 Can this happen?   You conduct an ANOVA to compare three groups,imagine the possibilities then EXPLAIN why.

a. Is it possible that all of the residuals in one group are positive? Explain how this could happen, or why it cannot happen.
answer: NOT possible, EXPLAIN WHY

b. Is it possible that all but one of the residuals in one group is positive? Explain how this could happen, or why it cannot happen.
answer: possible, but EXPLAIN WHY

c. If you and I are two subjects in the study, and if we are in the same group, and if your score is higher than mine, is it possible that your residual is smaller than mine? Explain how this could happen, or why it cannot happen.
answer: possible, but EXPLAIN WHY

d. If you and I are two subjects in the study, and if we are in different groups, and if your score is higher than mine, is it possible that your residual is smaller than mine? Explain how this could happen, or why it cannot happen.
answer: possible, but EXPLAIN WHY

Explanation / Answer

Answer to part a)

All residuals cannot be positive

this is because the line of best fit is designed in a manner that few points fall above it and few points fall below it so that the sum of residuals is Zero. If all residuals turn out to be positive this will result is a large positive error , thus indicating that the line so obtained is not a best fit for the given data. thus it is not possible to get all residuals positive for the line of best fit.

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Answer to part b)

if all residuals are negative , except one , this situation is completely possible

Say there are five residuals : -1,-1,+4,-1,-1

In this situation above we got 4 negative residuals , and one positive residuals , such that the sum of residuals is : -1-1+4-1-1 = 0

Thus in this case the residuals add up to Zero and hence we find the line is still the best fit despite the fact that only one residual is positive in this case

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Answer to part c)

Yes this is possible. The actual score is compared to the predicted score to find the residual

Say my actual score is 150 , and your score is 110

My predicted score = 140 and your predicted score = 80

Then my residual = 150-140 = 10 , whereas your residual will be : 110-80 = 30

thus though my scores were higher , still your residual is higher.

This can always happen in any set of data.

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Answer to part d)

Again the same concept applies here , though the cause of difference is not the same . In the previous case it was the errors within the group that caused the difference in residuals to arise. In this situation it is mentioned and the two subjects are in two different groups. This implies that the difference caused in the error amount is due to the difference of the two groups. This is called "between group residual"

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