A scientist collects data to predict the wheat yield (in bushels per acre)based on rainfall (in millimetres).The results are recorded in the table below. $$\begin{array} { c | c }
\begin{array} { c }
\text { Rainfall } \
( \mathrm { mm } )
\end{array} & \begin{array} { c }
\text { Wheat Yield } \
\text { (bushels per acre) }
\end{array} \
\hline 11.5 & 61.2 \
7.6 & 25.8 \
11.2 & 50.6 \
18 & 78.5 \
8.6 & 40.9 \
10.5 & 42.3 \
14.7 & 70.4 \
13.1 & 54
\end{array}$$ Compute Spearman's rank correlation.Assume that rank ties will have little influence on your result.
A)0.962
B)0.944
C)0.976
D)0.94
E)0.942

Question

A scientist collects data to predict the wheat yield (in bushels per acre)based on rainfall (in millimetres).The results are recorded in the table below. $$\begin{array} { c | c } 
\begin{array} { c } 
\text { Rainfall } \
( \mathrm { mm } )
\end{array} & \begin{array} { c } 
\text { Wheat Yield } \
\text { (bushels per acre) }
\end{array} \
\hline 11.5 & 61.2 \
7.6 & 25.8 \
11.2 & 50.6 \
18 & 78.5 \
8.6 & 40.9 \
10.5 & 42.3 \
14.7 & 70.4 \
13.1 & 54
\end{array}$$ Compute Spearman's rank correlation.Assume that rank ties will have little influence on your result.
A)0.962
B)0.944
C)0.976
D)0.94
E)0.942

Quizplus · Accepted Answer

Spearman's rank correlation coefficient is given by the formula:
$$r_s=1-\frac{6\sum{d^2}}{n(n^2-1)}$$
where $d$ is the difference in ranks and $n$ is the sample size. We can first rank the data in order of increasing rainfall and wheat yield:
$$\begin{array} { c | c  | c | c } 
\begin{array}{c} 
\text { Rank } \
\text { Rainfall }
\end{array} & 
\begin{array}{c} 
\text { Rainfall } \
( \mathrm { mm } )
\end{array}
&
\begin{array}{c} 
\text { Rank } \
\text { Yield }
\end{array} & 
\begin{array}{c} 
\text { Yield } \
\text { ( bushels per acre ) }
\end{array} \
\hline 1 & 7.6 &1 &25.8 \
2 & 8.6 & 4 &40.9 \
3 & 10.5 & 5 &42.3 \
4 & 11.2 & 2 &50.6 \
5 & 11.5 & 7 &61.2 \
6 & 13.1 & 6 &54 \
7 & 14.7 & 8 &70.4 \
8 & 18 & 9 &78.5 \
\end{array}$$
Then we calculate the differences in ranks, $d$, and $d^2$:
$$\begin{array} { c | c  | c | c | c } 
\begin{array}{c} 
\text { Rank } \
\text { Rainfall }
\end{array} & 
\begin{array}{c} 
\text { Rainfall } \
( \mathrm { mm } )
\end{array}
&
\begin{array}{c} 
\text { Rank } \
\text { Yield }
\end{array} & 
\begin{array}{c} 
\text { Yield } \
\text { (bushels per acre) }
\end{array} &
d^2 \
\hline 1 & 7.6 &1 &25.8 & 0 \
2 & 8.6 & 4 &40.9 & 4 \
3 & 10.5 & 5 &42.3 & 4 \
4 & 11.2 & 2 &50.6 & 4 \
5 & 11.5 & 7 &61.2 & 4 \
6 & 13.1 & 6 &54 & 0 \
7 & 14.7 & 8 &70.4 & 1 \
8 & 18 & 9 &78.5 & 9 \
\end{array}$$
Summing the $d^2$ column, we get $\sum{d^2}=26$. Plugging into the formula for $r_s$, we have $r_s=1-\frac{6(26)}{8(8^2-1)}=0.976$. Therefore, the best choice is (C).

A Scientist Collects Data to Predict the Wheat Yield (In