Question

The following sample data show the average annual yield of wheat in bushels per acre in a given county and the annual rainfall in inches.

Rainfall	Wheat Yield	X^2	XY	Y^2
9	40	81	360	1600
10	43	100	430	1849
16	69	256	1104	4761
13	52	169	676	2704
13	61	169	793	3721
7	27	49	189	729
11	50	121	550	2500
19	79	361	1501	6241
98	421	1306	5603	24105

a. Determine the regression equation from which we can predict the yield of wheat in the county given the rainfall.  Narrate your equation in a sentence or two.

 

b. Use the regression equation obtained in (a) to predict the average yield of wheat when the rainfall is 9 inches.

 

c. Plot the scatter diagram of raw data and the regression line for the equation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

d. What percentage of the total variation of wheat yield is accounted for by differences in rainfall.

 

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e. Calculate the correlation coefficient for this regression. 

Answer 1

I can answer some of the question.

To work out the regression equation we use the table to find the slope and intercept of the line y=a+bx, where a is the intercept and b the slope. b=(N(sum of)XY-(sum of)X*(sum of)Y)/(N(sum of)X^2-((sum of)X)^2). To abbreviate I'll use S() to mean (sum of). S(XY)=46861 (98*421=41258 not 5603). S(X)=196, S(Y)=842. N=9. S(X^2)=10910 (98*98=9604 not 1306). Therefore b=(9*46861-196*842)/(9*10910-196*196)=4.2948. And a=(S(Y)-bS(X))/N=(842-4.2948*196)/9=0.0245. The regression equation is y=0.0245+4.2948x. When x=9, y=38.68, close to actual 40.

The mean rainfall is 196/9=21.778in and the mean yield is 842/9=93.556 bushels per acre. To calculate the correlation coefficient, we need to subtract the means from the data values.

X-Mx	Y-My	(X-Mx)^2	(Y-My)^2	(X-Mx)(Y-My)
-12.78	-53.56	163.27	2868.20	684.50
-11.78	-50.56	138.72	2555.86	595.60
-5.78	-24.56	33.38	602.98	41.96
-8.78	-41.56	77.05	1726.86	364.90
-8.78	-32.56	77.05	1059.86	285.88
-14.78	-66.56	218.38	4429.64	983.76
-10.78	-43.56	116.16	1897.09	469.58
-2.78	-14.56	7.72	211.86	40.48
76.22	327.44	5809.83	107219.86	24957.48

The Pearson correlation coefficient is the sum of the figures in the last column divided by the square root of the product of the separate sums of the third and fourth columns. The sums of the last three columns are respectively 6641.56, 122572.21 and 28424.14.

The coefficient is 28424.14/sqrt(814070687)=28424.14/28531.92=0.9962.