Question

find the equation for the least squares regression line of the data described below. playing video games makes garrett so happy that he thinks it even helps him complete more schoolwork. garretts roommate carla is skeptical, so over the next few days, carla asks garrett about his progress on his daily reading. for each day, carla notes the number of minutes garrett spends playing video games, x. she also takes the number of pages garrett reads and divides it by the total number of pages assigned, y. minutes playing video games: 35, 38, 41, 72, 75. percentage of reading assignment: 45, 45, 37, 35, 31. round your answers to the nearest thousandth. y = x +

Explanation:

Step1: Calcular sumatorias

Sean $x_i$ los minutos jugando y $y_i$ el porcentaje de lectura.
Tenemos $n = 5$ datos.
$\sum_{i = 1}^{n}x_i=35 + 38+41+72+75 = 261$
$\sum_{i = 1}^{n}y_i=45 + 45+37+35+31 = 193$
$\sum_{i = 1}^{n}x_i^2=35^2 + 38^2+41^2+72^2+75^2=1225+1444+1681+5184+5625 = 15169$
$\sum_{i = 1}^{n}x_iy_i=35\times45+38\times45+41\times37+72\times35+75\times31=1575+1710+1517+2520+2325 = 9647$

Step2: Calcular pendiente $m$

La fórmula para la pendiente $m$ de la recta de regresión lineal es $m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^2 - (\sum_{i = 1}^{n}x_i)^2}$
Sustituyendo los valores:
$n = 5$, $\sum_{i = 1}^{n}x_i = 261$, $\sum_{i = 1}^{n}y_i = 193$, $\sum_{i = 1}^{n}x_i^2=15169$ y $\sum_{i = 1}^{n}x_iy_i = 9647$
$m=\frac{5\times9647-261\times193}{5\times15169-(261)^2}$
$m=\frac{48235 - 50373}{75845 - 68121}=\frac{- 2138}{7724}\approx - 0.277$

Step3: Calcular intersección $b$

La fórmula para la intersección $b$ es $b=\overline{y}-m\overline{x}$, donde $\overline{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$ y $\overline{y}=\frac{\sum_{i = 1}^{n}y_i}{n}$
$\overline{x}=\frac{261}{5}=52.2$
$\overline{y}=\frac{193}{5}=38.6$
$b = 38.6-(-0.277)\times52.2$
$b = 38.6 + 0.277\times52.2$
$b = 38.6+14.4594\approx53.060$

Answer:

$y=-0.277x + 53.060$