Question

multiple choice 4 points
select the most appropriate response.
in an area of the midwest, records were kept on the relationship between the rainfall (in inches) and the yield of wheat (bushels per acre). which is the best predicted value for y given x = 7.1?
bivariate rainfall and yield
assume that the variables x and y have a significant correlation.

Explanation:

Step1: Calculate the mean of \(x\) and \(y\)

Let \(x\) be the rainfall and \(y\) be the yield.
\(\bar{x}=\frac{10.5 + 8.8+13.4 + 12.5+18.8+10.3+7.0+15.6+16.0}{9}=\frac{112.9}{9}\approx12.54\)
\(\bar{y}=\frac{50.5 + 46.2+58.8 + 59.0+82.4+49.2+31.9+76.0+78.8}{9}=\frac{532.8}{9} = 59.2\)

Step2: Calculate \(b_{1}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)

\(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=(10.5 - 12.54)(50.5 - 59.2)+(8.8 - 12.54)(46.2 - 59.2)+(13.4 - 12.54)(58.8 - 59.2)+(12.5 - 12.54)(59.0 - 59.2)+(18.8 - 12.54)(82.4 - 59.2)+(10.3 - 12.54)(49.2 - 59.2)+(7.0 - 12.54)(31.9 - 59.2)+(15.6 - 12.54)(76.0 - 59.2)+(16.0 - 12.54)(78.8 - 59.2)\)
\(=(- 2.04)\times(-8.7)+(-3.74)\times(-13)+(0.86)\times(-0.4)+(-0.04)\times(-0.2)+(6.26)\times(23.2)+(-2.24)\times(-10)+(-5.54)\times(-27.3)+(3.06)\times(16.8)+(3.46)\times(19.6)\)
\(=17.748+48.62-0.344 + 0.008+145.232+22.4+151.242+51.408+67.816\)
\(=504.13\)

\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=(10.5 - 12.54)^{2}+(8.8 - 12.54)^{2}+(13.4 - 12.54)^{2}+(12.5 - 12.54)^{2}+(18.8 - 12.54)^{2}+(10.3 - 12.54)^{2}+(7.0 - 12.54)^{2}+(15.6 - 12.54)^{2}+(16.0 - 12.54)^{2}\)
\(=(-2.04)^{2}+(-3.74)^{2}+(0.86)^{2}+(-0.04)^{2}+(6.26)^{2}+(-2.24)^{2}+(-5.54)^{2}+(3.06)^{2}+(3.46)^{2}\)
\(=4.1616+13.9876+0.7396+0.0016+39.1876+5.0176+30.6916+9.3636+11.9716\)
\(=115.124\)

\(b_{1}=\frac{504.13}{115.124}\approx4.38\)

Step3: Calculate \(b_{0}=\bar{y}-b_{1}\bar{x}\)

\(b_{0}=59.2-4.38\times12.54\)
\(=59.2 - 54.9252=4.2748\)

The regression equation is \(y = b_{0}+b_{1}x=4.2748 + 4.38x\)

Step4: Predict \(y\) when \(x = 7.1\)

\(y=4.2748+4.38\times7.1\)
\(=4.2748+31.098=35.3728\approx35.4\)

Answer:

35.4