Question

It is argued that the temperature condition is highly correlated with crop harvest in the tomatoes business. The information from north eastern Ghana is summarized in the table below. Use the table to answer the questions that follow
Yield (crop)
40
50
70
65
80
45
90
100
120

Temperature
10
15
20
25
21
28
30
31
35

Calculate the pearsons correlation coefficient and interpret it.
Did the result support past argument?

Answer 1

$$\begin{matrix} & x & y & x^2 & y^2 & xy \\ & 10 & 40 & 100 & 1600 & 400 \\ & 15 & 50 & 225 & 2500 & 750 \\ & 20 & 70 & 400 & 4900 & 1400 \\ & 25 & 65 & 625 & 4225 & 1625 \\ & 21 & 80 & 441 & 6400 & 1680 \\ & 28 & 45 & 784 & 2025 & 1260 \\ & 30 & 90 & 900 & 8100 & 2700 \\ & 31 & 100 & 961 & 10000 & 3100 \\ & 35 & 120 & 1225& 14400 & 4200 \\ Sum= & 215 & 660 & 5661 & 54150 & 17115 \end{matrix}$$

$$\bar{x}={1 \over 9}\displaystyle\sum_{i=1}^9x_i={215 \over 9}\approx23.88888889$$ $$\bar{y}={1 \over 9}\displaystyle\sum_{i=1}^9y_i={660 \over 9}\approx73.33333333$$

$$SS_{xx}=\displaystyle\sum_{i=1}^9x_i^2-{1 \over 9}(\displaystyle\sum_{i=1}^9x_i)^2=$$ $$=5661-{215^2 \over 9}\approx524.88888889$$

$$SS_{yy}=\displaystyle\sum_{i=1}^9y_i^2-{1 \over 9}(\displaystyle\sum_{i=1}^9y_i)^2=$$ $$=54150-{660^2 \over 9}=5750$$

$$SS_{xy}=\displaystyle\sum_{i=1}^9x_iy_i-{1 \over 9}(\displaystyle\sum_{i=1}^9y_i)(\displaystyle\sum_{i=1}^9x_i)=$$ $$=17115-{215(660)\over 9}\approx1348.33333333$$

$$B={SS_{xy} \over SS_{xx}}={1348.33333333 \over 524.88888889}\approx2.5688$$

$$A=\bar{y}-B\bar{x}=73.33333333-2.5688(23.88888889)\approx$$ $$\approx11.9676$$

We find that the regression equation is:

$$y=11.9676+2.5688x$$

Calculate the pearsons correlation coefficient

$$r={SS_{xy} \over \sqrt{S_{xx}}\sqrt{SS_{yy}}}\approx{1348.33333333\over \sqrt{524.88888889}\sqrt{5750}}\approx0.776121$$

$|r|=0.776121>0.7,$ strong correlation

The temperature condition is highly correlated with crop harvest in the tomatoes business.