Question

The following data were recorded as part of a study on sustainable farming techniques that took place in Boone County, IA. (Davis, Adam S. et al. Increasing Cropping System Diversity Balances Productivity, Profitability and Environmental Health. PLOS ONE. October 10, 2012. DOI:10.1371/journal.pone.0047149) Means are for the April-November growing seasons.

(1): Construct a two-way scatter plot for “air temperature” against the “total precipitation” and on a separate graph construct a two-way scatter plot for “air temperature” against “log of total precipitation”. Looking at the two graphs you plotted, explain as to which of these two do you consider to be closest to a linear relationship?

(2): At the 0.05 level of significance, test the null hypothesis that the (“air temperature” and the “total precipitation”) population correlation coefficient [ρ] is equal to 0.

(3): Compute the equation of the linear regression relationship between the “air temperature” and “total precipitation”.

Year	Mean air temperature (centigrade) [X]		Total precipitation (mm) [Y]
2003	14.9	790
2004	15.0	697
2005	15.9	748
2006	15.6	777
2007	16.4	839
2008	15.2	1145
2009	14.8	755
2010	16.5	1165
2011	15.2	701

Answer 1

1.

Converting y to log value we get below data and scatter plot

Mean air temperature (centigrade) [X]	Total precipitation (mm) [Y]
14.9	2.897627091
15	2.843232778
15.9	2.873901598
15.6	2.890421019
16.4	2.923761961
15.2	3.058805487
14.8	2.877946952
16.5	3.066325925
15.2	2.845718018

2.

X Values
∑ = 139.5
Mean = 15.5
∑(X - M_x)² = SS_x = 3.26

Y Values
∑ = 7617
Mean = 846.333
∑(Y - M_y)² = SS_y = 260218

X and Y Combined
N = 9
∑(X - M_x)(Y - M_y) = 392.2

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = 392.2 / √((3.26)(260218)) = 0.4258

The sample size is n=9, so then the number of degrees of freedom is df=n−2=9−2=7

The corresponding critical correlation value rc​ for a significance level of α=0.05, for a two-tailed test is:

rc​=0.666

Observe that in this case, the null hypothesis is rejected if ∣r∣>rc​=0.666.

Here r=0.4258<rc=0.666, so we fail to reject the null hypothesis

Hence test is not significant

c.

Sum of X = 139.5
Sum of Y = 7617
Mean X = 15.5
Mean Y = 846.3333
Sum of squares (SS_X) = 3.26
Sum of products (SP) = 392.2

Regression Equation = ŷ = bX + a

b = SP/SS_X = 392.2/3.26 = 120.3068

a = M_Y - bM_X = 846.33 - (120.31*15.5) = -1018.4213

ŷ = 120.3068X - 1018.4213