Sum2 2011
NCSU
ST512
QUIZ 1
NAME
We are interested in studying the relationship, if any, between yield (Y, in bushels/acre), and
rainfall (X, inches/yr) on corn raised in farms of similar acreage and management located in
the midwest, during the period from 1890 to 1927. Our main goal is to find out an equation
that will help in the prediction of corn yield for a given rainfall.
We decided to run a multiple regression analysis after looking at a representation of the relationship
between Year, Rainfall and Yield
Model YIELD = RAINFALL YEAR
YEAR*RAIN
yi   o  1 x1i   2 x2i   3 x3i  ei
X 1  Rainfall
Regression model fitted
X 2  Year from 1890
X 3  X 1 *X2
Y= Yield

e ~ N 0, 2
i

Next we present the Analysis of variance table and parameter estimates.
1. Please fill in the missing entries.
Analysis of Variance
DF
Sum of
Squares
Mean
Square
F Value
Pr > F
Model
3
340.60856
113.53619
10.61
<.0001
Error
34
363.94197
10.70418
Corrected Total
37
704.55053
Source
2.
Give the null (and alternative) hypothesis tested by the overall F-test for the model .
1
July 07, 2011
Sum2 2011
NCSU
ST512
QUIZ 1
Next we have the table of Parameter estimates
3. Please fill in the missing entries.
Parameter Estimates
Variable
DF
Parameter
Estimate
Standard
Error
t Value
Pr > |t|
Type I SS
Intercept
1
4.48573
5.48728
0.82
0.4193
38707
X1
1
2.31128
0.50479
4.58
<.0001
114.21474
X2
1
1.08545
0.27376
3.96
0.0004
95.99361
-0.08781
0.02516
-3.49
0.0014
130.40020
X3
4. Write the estimated regression equation.
5. Calculate an estimate for the conditional mean yield of farms raising corn in year 1900 receiving
a rainfall of 9 inch/year.
The variance-covariance matrix for the estimates of the regression coefficients is presented next.
Covariance of Estimates
2
Variable
Intercept
rainfall
Intercept
30.110197368
-2.719640859
rainfall
-2.719640859 0.2548146526 0.1197135611
-0.01120618
YEAR1
-1.320670004 0.1197135611 0.0749470666
-0.006778894
YR_RAIN
0.1195693604
-0.01120618
YEAR1
YR_RAIN
-1.320670004 0.1195693604
-0.006778894 0.0006329419
July 07, 2011
Sum2 2011
NCSU
ST512
QUIZ 1
6. Show how you would calculate the standard error of the conditional mean in 5. You do not need
to do the actual calculations
7. A researcher is analyzing the effect of the year by rainfall interaction, and ask your help. She
needs to draw the regression line between Yield and Rainfall at each of the following years:
1890, 1915 and 1935.
a. Compute the regression equation for each of these three years
Year
X2 = (Year-1890)
1890
0
1915
25
1925
35
Intercept
Regression
coefficient (Slope
for rainfall)
b. Explain changes in these three slopes?
8. I regressed Y on X1 and X2 getting these Type I and Type II sums of
squares from PROC REG.
What would have happened if I had run the
regression in the opposite order (PROC REG; MODEL Y = X2 X1; )
3
July 07, 2011
Sum2 2011
NCSU
Variable
DF
Type I SS
INTERCEP
1
90000
18000
X1
1
800
900
X2
1
700
700
ST512
QUIZ 1
Type II SS
Fill missing entries
Variable
DF
Type I SS
INTERCEP
1
90000
18000
X2
1
______
______
X1
1
______
______
4
Type II SS
July 07, 2011