Question

devin thinks that text messaging is causing him to talk less on the phone. for each month, he examined his text message and call logs with his closest friends. for each friend, devin checked the number of text messages he sent to that friend, x, and the number of minutes they spoke on the phone, y. text messages sent: 0, 11, 15, 82, 84. minutes on the phone: 98, 89, 82, 54, 66. round your answer to the nearest thousandth.

Explanation:

Step1: Recall correlation - coefficient formula

The formula for the correlation coefficient $r$ is $r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$. Let $x$ be the number of text - messages sent and $y$ be the minutes on the phone. First, calculate the necessary sums:
Let the data points be $(x_1,y_1),(x_2,y_2),\cdots,(x_n,y_n)$. Here $n = 5$.
$\sum x=0 + 11+15 + 82+84=192$
$\sum y=98 + 89+82+54+66=389$
$\sum xy=(0\times98)+(11\times89)+(15\times82)+(82\times54)+(84\times66)=0 + 979+1230+4428+5544=12181$
$\sum x^{2}=0^{2}+11^{2}+15^{2}+82^{2}+84^{2}=0 + 121+225+6724+7056=14126$
$\sum y^{2}=98^{2}+89^{2}+82^{2}+54^{2}+66^{2}=9604+7921+6724+2916+4356=31521$

Step2: Substitute into the formula

$n(\sum xy)=5\times12181 = 60905$
$(\sum x)(\sum y)=192\times389 = 74688$
$n\sum x^{2}=5\times14126=70630$
$(\sum x)^{2}=192^{2}=36864$
$n\sum y^{2}=5\times31521 = 157605$
$(\sum y)^{2}=389^{2}=151321$

The denominator:
$\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}=\sqrt{(70630 - 36864)(157605-151321)}=\sqrt{33766\times6284}$
$=\sqrt{212274544}\approx14569.6$

The numerator: $n(\sum xy)-(\sum x)(\sum y)=60905 - 74688=-13783$

$r=\frac{-13783}{14569.6}\approx - 0.946$

Answer:

$-0.946$