A modern use of regression is to predict crop yields of grap

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.

Solution

****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