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.