Answer to Question #129295 in Mechanics | Relativity for aman

Question

A researcher collected the following information for two variables x and y

No. of pairs = 20, r = 0.5, mean of X is 15, mean of Y is 20, standard deviation of X is 4, 

standard deviation of Y is 5.

Later on it was found that one pair of value as been wrongly taken as (X = 16, Y = 30) whereas 

the correct values were (X = 26, Y = 35). Find the correct value of r.

Answer 1

As per the question,

no. of pairs say n=20

Initially Coefficient of correlation r =0.5

mean X=15

mean Y=20

standard deviation for X say $s_x$ =4

standard deviation for Y say $s_y$ =5

Initially the pair was (15,20) for this pair only lets calculate $z_x$ and $z_y$

$z_x$ =$\frac {x_i-mean(X)}{s_x}$

Similarly,

$z_y$=$\frac{y_i-mean(y)}{s_y}$

so $z_x$=$\frac{16-15}{4}$

=0.25

similarly

$z_y$ =$\frac{30-20}{5}$

=2

let calculate the product of $z_x$ and $z_y$ =say m=0.5

let the sum of the product of $z_x$ and $z_y$ of the remaining products be s.

using formula,

r=$\frac{\sum(z_x\times z_y)}{n-1}$

0.5=$\frac{s+0.5}{20-1}$

on solving we get s=9 .....(equation 1

Now the correct pairs was(26,35)

Similarly calculating the value of$z_x and z_y$ using the above mention formula

we get,

$z_x$ =2.75

Similarly

$z_y$ =3

Now the product of $z_x and z_y$ =2.75$\times$ 3

=8.25

Now again using the formula for coefficient of correlation

r=$\frac{\sum(z_x\times z_y)}{n-1}$

=$\frac{(8.25+s)}{20-1}$

putting the value of s from equation 1 we get

r=$\frac{8.25+9}{19}$

=$\frac{17.25}{19}$

r=0.9078

Hence the correct value of r is 0.9078.