 # Applied Statistics and Agricultural Experiments

## Question

Three different soil treatments are compared in an agricultural experiment.
Each treatment is used in twenty yields (60 yields in all). The response variable is Y = crop yield. A useful predictor is X = crop yield in the yield last year. Define the indicator variables:
T1 = 1 if Treatment 1, 0 otherwise
T2 = 1 if Treatment 2, 0 otherwise.
(a)Consider this regression equation for the mean of Y : E(Y) = β0 +β1X1 +β2T1 +β3T2. For the fields in which Treatment 1 is used, what are the values of T1 and T2? Substitute these values into the regression model to determine the equation for E(Y) if Treatment 1 is used.
(b) Repeat part (a) for Treatment 2.
(c) Repeat part (a) for Treatment 3 (= Neither Treatment 1 nor Treatment 2).
(d) Using the answers to parts (a) through (c), explain what each of the following parameters measures.
i. β1
ii. β2
iii. β3
(e) Suppose that we will test H0 : β2 = β3 = 0. In the context of this situation,explain what it would mean if we fail to reject this null hypothesis. Explain what it would mean if we reject this null hypothesis.
(f) Briefly explain how you would carry out the test of the null hypothesis given in the previous part.
Describe the reduced and full models and give numerator and denominator df for the test.
(g) Suppose that we think there might be interaction between treatments and the variable X (last year's crop yield). Write the appropriate equation for E(Y).
(h) For the interaction model that you wrote in part (g), describe how you would test whether there is an interaction between treatments and last year's crop yield. Give a mathematical statement of the null hypothesis, describe the reduced and full models, and give numerator and denominator df for the general linear F-test.

## Solution Preview

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Substitute these values into the regression model to determine the equation for E(Y) if Treatment 1 is used.
E(Y) = β0 +β1X1 +β2*1 +β3*0.
E(Y) = β0 +β1 X1 +β2
(a)Consider this regression equation for the mean of Y: E(Y ) = β0 +β1X1 +β2T1 +β3T2. For the fields in which Treatment 1 is used, what are the values of T1 and T2?
The values for T1 and T2 are 1 and 0; respectively.

Substitute these values into the regression model to determine the equation for E(Y ) if Treatment 1 is used.
E(Y ) = β0 +β1X1 +β2*1 +β3*0.
E(Y ) = β0 +β1 X1 +β2...
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