Question

find the correlation coefficient, r, of the data described below. playing video games makes nick so happy that he thinks it even helps him complete more schoolwork. nicks roommate eli is skeptical, so over the next few days, eli asks nick about his progress on his daily reading. for each day, eli notes the number of minutes nick spends playing video games, x. he also takes the number of pages nick reads and divides it by the total number of pages assigned, y.

minutes playing video games	percentage of reading assignment
63	59
87	37
88	54
95	30

round your answer to the nearest thousandth.
r =

Explanation:

Step1: Definir variables

Sean $x_i$ los minutos jugando videojuegos y $y_i$ el porcentaje de la asignación de lectura. Tenemos $n = 5$ pares de datos.

Step2: Calcular sumatorias

Calcular $\sum_{i = 1}^{n}x_i$, $\sum_{i = 1}^{n}y_i$, $\sum_{i = 1}^{n}x_i^2$, $\sum_{i = 1}^{n}y_i^2$ y $\sum_{i = 1}^{n}x_iy_i$.
$\sum_{i = 1}^{5}x_i=49 + 63+87+88+95 = 382$
$\sum_{i = 1}^{5}y_i=54 + 59+37+54+30 = 234$
$\sum_{i = 1}^{5}x_i^2=49^2+63^2 + 87^2+88^2+95^2=49^2+3969+7569+7744+9025=31058$
$\sum_{i = 1}^{5}y_i^2=54^2+59^2+37^2+54^2+30^2=2916+3481+1369+2916+900 = 11582$
$\sum_{i = 1}^{5}x_iy_i=49\times54+63\times59+87\times37+88\times54+95\times30$
$=2646+3717+3219+4752+2850 = 17184$

Step3: Aplicar fórmula de coeficiente de correlación

La fórmula para el coeficiente de correlación de Pearson es $r=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{\sqrt{n\sum_{i = 1}^{n}x_i^2 - (\sum_{i = 1}^{n}x_i)^2}\sqrt{n\sum_{i = 1}^{n}y_i^2-(\sum_{i = 1}^{n}y_i)^2}}$
Sustituir $n = 5$, $\sum_{i = 1}^{5}x_i = 382$, $\sum_{i = 1}^{5}y_i = 234$, $\sum_{i = 1}^{5}x_i^2=31058$, $\sum_{i = 1}^{5}y_i^2 = 11582$ y $\sum_{i = 1}^{5}x_iy_i=17184$
$n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i=5\times17184-382\times234$
$=85920 - 89388=- 3468$
$n\sum_{i = 1}^{n}x_i^2 - (\sum_{i = 1}^{n}x_i)^2=5\times31058-382^2$
$=155290 - 145924 = 9366$
$n\sum_{i = 1}^{n}y_i^2-(\sum_{i = 1}^{n}y_i)^2=5\times11582-234^2$
$=57910-54756 = 3154$
$\sqrt{n\sum_{i = 1}^{n}x_i^2 - (\sum_{i = 1}^{n}x_i)^2}\sqrt{n\sum_{i = 1}^{n}y_i^2-(\sum_{i = 1}^{n}y_i)^2}=\sqrt{9366}\sqrt{3154}$
$\approx96.77\times56.16\approx5437.5$
$r=\frac{-3468}{5437.5}\approx - 0.638$

Answer:

$-0.638$