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Question: For the given data, the calculation corresponding to all value of pairs \((x,y)\)is following \({\su...

For the given data, the calculation corresponding to all value of pairs (x,y)(x,y)is following ∑(x−x‾)2=36,∑(y−y‾)2=25,∑(x−x‾)∑(y−y‾)=20{\sum {(x - \overline x )} ^2} = 36,{\sum {(y - \overline y )} ^2} = 25,\sum {(x - \overline x )\sum {(y - \overline y )} } = 20 Then the Karl Pearson’s correlation coefficient is
A)0.2A)0.2
B)0.5B)0.5
C)0.66C)0.66
D)0.33D)0.33

Explanation

Solution

First, we will need to know about the concept of the correlation coefficient.
The coefficient of the correlation is used to measure the relationship extent between 22 separate intervals or variables.
Denoted by the symbol rr. Where r is the value of positive or negative. Thus, this will further be generalized into the form of Pearson’s correlation coefficient. The formula for Pearson’s correlation is given below.
Formula used:
r=∑(x−x‾)∑(y−y‾)∑(x−x‾)2∑(y−y‾)2r = \dfrac{{\sum {(x - \overline x )\sum {(y - \overline y )} } }}{{\sqrt {{{\sum {(x - \overline x )} }^2}} \sqrt {{{\sum {(y - \overline y )} }^2}} }} is the Pearson’s correlation coefficient for the particularly given value.

Complete step-by-step solution:
Since from the given that we have, ∑(x−x‾)2=36,∑(y−y‾)2=25,∑(x−x‾)∑(y−y‾)=20{\sum {(x - \overline x )} ^2} = 36,{\sum {(y - \overline y )} ^2} = 25,\sum {(x - \overline x )\sum {(y - \overline y )} } = 20where these are the calculation corresponding all value of pairs (x,y)(x,y)
Let us find the square root of the first two terms, which are ∑(x−x‾)2=36⇒∑(x−x‾)2=6,∑(y−y‾)2=25⇒∑(y−y‾)2=5{\sum {(x - \overline x )} ^2} = 36 \Rightarrow \sqrt {\sum {(x - \overline x )}^2 } = 6, {\sum {(y - \overline y )} ^2} = 25 \Rightarrow \sqrt {\sum {(y - \overline y )}^2 } = 5 where the square root of 36=6,\sqrt {36} = 6, and the square root of 25=5\sqrt {25} = 5
Now substitute the values into the given formula, we get r=∑(x−x‾)∑(y−y‾)∑(x−x‾)2∑(y−y‾)2=206×5r = \dfrac{{\sum {(x - \overline x )\sum {(y - \overline y )} } }}{{\sqrt {{{\sum {(x - \overline x )} }^2}} \sqrt {{{\sum {(y - \overline y )} }^2}} }} = \dfrac{{20}}{{6 \times 5}}
Further solving we get, r=206×5=2030=23=0.66r = \dfrac{{20}}{{6 \times 5}} = \dfrac{{20}}{{30}} = \dfrac{2}{3} = 0.66
Hence, the option C)0.66C)0.66 is correct.
Additional information:
The standard formula for the correlation coefficient:
Let us consider two different variables x and y that are related commonly, to find the extent of the link between the given numbers x and y, we will choose Pearson's coefficient r method.
In that process, the formula given is used to identify the extent or range of the two variables' equality.
Which is r=n∑xy−∑x∑y[n∑(y)2−(∑x)2][n∑(y)2−(∑y)2]r = \dfrac{{n\sum {xy} - \sum x \sum y }}{{\sqrt {[n{{\sum {(y)} }^2} - (\sum {x{)^2}} ][n{{\sum {(y)} }^2} - (\sum {y{)^2}} ]} }}.

Note: In this formula r=n∑xy−∑x∑y[n∑(y)2−(∑x)2][n∑(y)2−(∑y)2]r = \dfrac{{n\sum {xy} - \sum x \sum y }}{{\sqrt {[n{{\sum {(y)} }^2} - (\sum {x{)^2}} ][n{{\sum {(y)} }^2} - (\sum {y{)^2}} ]} }}
∑x\sum x denotes the number of first variable values.
∑y\sum y denotes the count of the second variable values.
∑x2{\sum x ^2} denotes the addition of a square for the first value.
  ∑y2\;{\sum y ^2} denotes the sum of the second values. And n denotes the total count data quantity.