Question

playing video games makes gwen so happy that she thinks it even helps her complete more schoolwork. gwens roommate jonah is skeptical, so over the next few days, jonah notes the number of minutes gwen spends playing video games, x. he also takes the number of pages assigned, y. for each day, jonah notes the number of minutes gwen spends playing video games, x. he also takes the number of pages assigned, y. gwen reads and divides it by the total number of pages gwen reads and divides it by the total number of pages assigned, y. minutes playing video games: 35, 42, 47, 57, 73, 78. percentage of reading assignment: 38, 56, 32, 46, 30, 45. find the correlation coefficient, r, of the data described below. 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 minutes playing video - games and \(y\) be the percentage of reading assignment.
We have \(n = 6\) data points.
First, calculate the necessary sums:

\(x\)	\(y\)	\(xy\)	\(x^{2}\)	\(y^{2}\)
42	56	\(42\times56 = 2352\)	\(42^{2}=1764\)	\(56^{2}=3136\)
47	32	\(47\times32 = 1504\)	\(47^{2}=2209\)	\(32^{2}=1024\)
57	46	\(57\times46 = 2622\)	\(57^{2}=3249\)	\(46^{2}=2116\)
73	30	\(73\times30 = 2190\)	\(73^{2}=5329\)	\(30^{2}=900\)
78	45	\(78\times45 = 3510\)	\(78^{2}=6084\)	\(45^{2}=2025\)

\(\sum x=35 + 42+47+57+73+78=332\)
\(\sum y=38 + 56+32+46+30+45=247\)
\(\sum xy=1330 + 2352+1504+2622+2190+3510 = 13508\)
\(\sum x^{2}=1225+1764+2209+3249+5329+6084 = 19860\)
\(\sum y^{2}=1444+3136+1024+2116+900+2025 = 10645\)

Step2: Substitute values into the formula

\[

$$\begin{align*} n(\sum xy)&=6\times13508 = 81048\\ (\sum x)(\sum y)&=332\times247=82004\\ n\sum x^{2}&=6\times19860 = 119160\\ (\sum x)^{2}&=332^{2}=110224\\ n\sum y^{2}&=6\times10645 = 63870\\ (\sum y)^{2}&=247^{2}=61009 \end{align*}$$

\]

\[

$$\begin{align*} r&=\frac{6\times13508-332\times247}{\sqrt{(6\times19860 - 332^{2})(6\times10645-247^{2})}}\\ &=\frac{81048 - 82004}{\sqrt{(119160 - 110224)(63870 - 61009)}}\\ &=\frac{- 956}{\sqrt{(8936)(2861)}}\\ &=\frac{-956}{\sqrt{25566896}}\\ &=\frac{-956}{5056.37}\\ &\approx - 0.19 \end{align*}$$

\]

Answer:

\(-0.19\)