Question

shifty trends
find the correlation coefficient for the given data. round to the nearest hundredth.
% remaining minutes
41 90
63 65
88 22
82 30
79 45
38 99
93 15
50 75
35 111
47 80

Explanation:

Step1: Recall correlation - coefficient formula

The formula for the Pearson correlation coefficient $r$ is $r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$

Let $x$ be the "% REMAINING" values and $y$ be the "MINUTES" values. First, calculate the necessary sums:
Let's assume we have $n = 10$ data - points.

1. Calculate $\sum x$:

• $x_1 = 41,x_2 = 63,x_3 = 88,x_4 = 82,x_5 = 79,x_6 = 38,x_7 = 93,x_8 = 50,x_9 = 35,x_{10}=47$
• $\sum x=41 + 63+88+82+79+38+93+50+35+47=615$

1. Calculate $\sum y$:

• $y_1 = 90,y_2 = 65,y_3 = 22,y_4 = 30,y_5 = 45,y_6 = 99,y_7 = 15,y_8 = 75,y_9 = 111,y_{10}=80$
• $\sum y=90 + 65+22+30+45+99+15+75+111+80=632$

1. Calculate $\sum xy$:

• $x_1y_1=41\times90 = 3690,x_2y_2=63\times65 = 4095,x_3y_3=88\times22 = 1936,x_4y_4=82\times30 = 2460,x_5y_5=79\times45 = 3555,x_6y_6=38\times99 = 3762,x_7y_7=93\times15 = 1395,x_8y_8=50\times75 = 3750,x_9y_9=35\times111 = 3885,x_{10}y_{10}=47\times80 = 3760$
• $\sum xy=3690+4095+1936+2460+3555+3762+1395+3750+3885+3760=32388$

1. Calculate $\sum x^{2}$:

• $x_1^{2}=41^{2}=1681,x_2^{2}=63^{2}=3969,x_3^{2}=88^{2}=7744,x_4^{2}=82^{2}=6724,x_5^{2}=79^{2}=6241,x_6^{2}=38^{2}=1444,x_7^{2}=93^{2}=8649,x_8^{2}=50^{2}=2500,x_9^{2}=35^{2}=1225,x_{10}^{2}=47^{2}=2209$
• $\sum x^{2}=1681+3969+7744+6724+6241+1444+8649+2500+1225+2209=42391$

1. Calculate $\sum y^{2}$:

• $y_1^{2}=90^{2}=8100,y_2^{2}=65^{2}=4225,y_3^{2}=22^{2}=484,y_4^{2}=30^{2}=900,y_5^{2}=45^{2}=2025,y_6^{2}=99^{2}=9801,y_7^{2}=15^{2}=225,y_8^{2}=75^{2}=5625,y_9^{2}=111^{2}=12321,y_{10}^{2}=80^{2}=6400$
• $\sum y^{2}=8100+4225+484+900+2025+9801+225+5625+12321+6400=50006$

Step2: Substitute into the formula

$n = 10$
$r=\frac{10\times32388-615\times632}{\sqrt{[10\times42391 - 615^{2}][10\times50006-632^{2}]}}$
First, calculate the numerator:
$10\times32388-615\times632=323880 - 388680=-64800$
Second, calculate the denominator:
$10\times42391-615^{2}=423910 - 378225 = 45685$
$10\times50006-632^{2}=500060 - 399424 = 100636$
$\sqrt{(45685\times100636)}=\sqrt{4599301260}\approx67825.5$
$r=\frac{-64800}{67825.5}\approx - 0.96$

Answer:

$-0.96$