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. $11.561.27.625.811.250.61878.58.640.910.542.314.770.413.154\begin{array} { c | c } \begin{array} { c } \text { Rainfall } \\( \mathrm { mm } ) \end{array} & \begin{array} { c } \text { Wheat Yield } \\\text { (bushels per acre) }\end{array} \\\hline 11.5 & 61.2 \\7.6 & 25.8 \\11.2 & 50.6 \\18 & 78.5 \\8.6 & 40.9 \\10.5 & 42.3 \\14.7 & 70.4 \\13.1 & 54\end{array}$ 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.

11.561.27.625.811.250.61878.58.640.910.542.314.770.413.154\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

Shiri Shaheid · Accepted Answer

Final Answer : C, Explanation :   Spearman's rank correlation coefficient is given by the formula: r s = 1 − 6 ∑ d 2 n ( n 2 − 1 ) r_s=1-\frac{6\sum{d^2}}{n(n^2-1)} r s ​ = 1 − n ( n 2 − 1 ) 6 ∑ d 2 ​ 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:  Rank   Rainfall   Rainfall  ( m m )  Rank   Yield   Yield   ( bushels per acre )  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 \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}  Rank   Rainfall  ​ 1 2 3 4 5 6 7 8 ​  Rainfall  ( mm ) ​ 7.6 8.6 10.5 11.2 11.5 13.1 14.7 18 ​  Rank   Yield  ​ 1 4 5 2 7 6 8 9 ​  Yield   ( bushels per acre )  ​ 25.8 40.9 42.3 50.6 61.2 54 70.4 78.5 ​ ​ Then we calculate the differences in ranks, $d$, and $d^2$:  Rank   Rainfall   Rainfall  ( m m )  Rank   Yield   Yield   (bushels per acre)  d 2 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 \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}  Rank   Rainfall  ​ 1 2 3 4 5 6 7 8 ​  Rainfall  ( mm ) ​ 7.6 8.6 10.5 11.2 11.5 13.1 14.7 18 ​  Rank   Yield  ​ 1 4 5 2 7 6 8 9 ​  Yield   (bushels per acre)  ​ 25.8 40.9 42.3 50.6 61.2 54 70.4 78.5 ​ d 2 0 4 4 4 4 0 1 9 ​ ​ 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).

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