A modern use of regression is to predict crop yields of grapes. In July, the gra
ID: 3435557 • Letter: A
Question
A modern use of regression is to predict crop yields of grapes. In July, the grape vines produce clusters of berries, and a count of these clusters can be used to predict the final crop yield at harvest time. For a typical data set of 12 pair of X and Y values, X denotes the Cluster Count and Y denotes yields (tons/acre). Excel output is provided t for a simple linear regression analysis.
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.913214
R Square
0.83396
Adjusted R Square
0.817356
Standard Error
0.364062
Observations
12
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Intercept
-1.0279
0.783552
0.218887
-2.77376
X Variable 1
0.051381
0.00725
3.35E-05
0.035227
***Find out the coefficient of determination (r2). How much of variation in Y could be explained by the linear relationship between Cluster Count and Yields?
***Decide the value of the test statistic of the slope that could be used to test whether Cluster Count is a useful predictor of Yields (there is linear relationship between X and Y).
***Please identify the slope and the intercept.
***Please write down the regression equation.
***Please interpret the meaning of the slope.
a. as the cluster count increases by one unit, the yield decreases by 1.0279 tons/acre.
b. as the cluster count increases by one unit, the yield increases by 1.0279 tons/acre.
c. as the cluster count increases by one unit, the yield decreases by 0.051381 tons/acre.
d. as the cluster count increases by one unit, the yield increases by 0.051381 tons/acre.
***Please predict the average yield (tons per acre) when the cluster count is 100.
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.913214
R Square
0.83396
Adjusted R Square
0.817356
Standard Error
0.364062
Observations
12
Explanation / Answer
****Coefficient of determination (r2)=0.83396
ie., 83% of variation in Y could be explained by the linear relationship between Cluster Count and Yields.
**** Test statistic of the slope= t = b1 / SE = 0.051381/0.00725= 6.85
***** slope= 0.051381 and intercept= -1.0279
**** Regression equation is
Y=-1.0279+(0.051381 )* Cluster count
***** (d) As the cluster count increases by one unit, the yield increases by 0.051381 tons/acre.
**** when the cluster count is 100 then average yield (tons per acre) is
Y=-1.0279+(0.051381 )*100= 4.0721
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