Question

question 5
0 / 1 pts
rounded to three decimal places, what is the value of r for this data set?
variable x 8 15 3 7 2 14 20
variable y 50 75 23 31 18 68 96
hint:
open desmos.
in line 1, use the plus sign to insert the table of values.
in line 2, type y1~mx1 + b to see find value of r.

Explanation:

Step1: Recall correlation - coefficient formula

The formula for the correlation coefficient $r$ is $r=\frac{n\sum xy-\sum x\sum y}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$ where $n$ is the number of data - points. Here $n = 7$.
First, calculate the necessary sums:
$\sum x=8 + 15+3+7+2+14+20=69$
$\sum y=50 + 75+23+31+18+68+96=361$
$\sum xy=8\times50+15\times75+3\times23+7\times31+2\times18+14\times68+20\times96$
$=400+1125+69+217+36+952+1920=4719$
$\sum x^{2}=8^{2}+15^{2}+3^{2}+7^{2}+2^{2}+14^{2}+20^{2}$
$=64 + 225+9+49+4+196+400=947$
$\sum y^{2}=50^{2}+75^{2}+23^{2}+31^{2}+18^{2}+68^{2}+96^{2}$
$=2500+5625+529+961+324+4624+9216=23789$

Step2: Substitute values into the formula

$n = 7$
$n\sum xy=7\times4719 = 33033$
$\sum x\sum y=69\times361=24909$
$n\sum x^{2}=7\times947 = 6629$
$(\sum x)^{2}=69^{2}=4761$
$n\sum y^{2}=7\times23789=166523$
$(\sum y)^{2}=361^{2}=130321$
$r=\frac{33033 - 24909}{\sqrt{(6629 - 4761)(166523-130321)}}$
$=\frac{8124}{\sqrt{1868\times36202}}$
$=\frac{8124}{\sqrt{67625336}}$
$=\frac{8124}{8223.46}$
$\approx0.988$

Answer:

$0.988$