Asked by Joseph Smith on Apr 26, 2024
Verified
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. Rainfall (mm) Wheat Yield (bushels per acre) 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} Rainfall (mm) 11.57.611.2188.610.514.713.1 Wheat Yield (bushels per acre) 61.225.850.678.540.942.370.454 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
Spearman's Rank Correlation
A measure that does not rely on parameters, used to determine the correlation in ranking, and assesses the ability to explain the relationship between two variables with a monotonic function.
Wheat Yield
The amount of wheat produced per unit area, commonly measured in bushels per acre or kilograms per hectare.
Rainfall
The total amount of rain that falls in a specific area over a specific period, often measured in millimeters or inches.
- Develop competencies in utilizing and deciphering Spearman's rank correlation for evaluating the association's intensity between two variables.
Verified Answer
rs=1−6∑d2n(n2−1)r_s=1-\frac{6\sum{d^2}}{n(n^2-1)}rs=1−n(n2−1)6∑d2
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 (mm) Rank Yield Yield ( bushels per acre ) 17.6125.828.6440.9310.5542.3411.2250.6511.5761.2613.1654714.7870.4818978.5\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 12345678 Rainfall (mm)7.68.610.511.211.513.114.718 Rank Yield 14527689 Yield ( bushels per acre ) 25.840.942.350.661.25470.478.5
Then we calculate the differences in ranks, $d$, and $d^2$:
Rank Rainfall Rainfall (mm) Rank Yield Yield (bushels per acre) d217.6125.8028.6440.94310.5542.34411.2250.64511.5761.24613.16540714.7870.41818978.59\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 12345678 Rainfall (mm)7.68.610.511.211.513.114.718 Rank Yield 14527689 Yield (bushels per acre) 25.840.942.350.661.25470.478.5d204444019
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).
Learning Objectives
- Develop competencies in utilizing and deciphering Spearman's rank correlation for evaluating the association's intensity between two variables.
Related questions
A Professor Was Interested in the Relationship Between a Student's ...
The Population Spearman Correlation Coefficient Is Labeled , and ...
The Spearman Rank-Correlation Is a Nonparametric Test That 1) Uses ...
The General Manager of a Chain of Pet Stores Believes ...
Two Movie Critics Wish to See If They Have Similar ...