Question

find the equation for the least squares regression line of the data described below.
brittany isnt getting enough sleep and suspects its due to playing too many video games before bed. to find out, she decided to track how much time she spends gaming.
over several days, brittany recorded how many hours of sleep she had gotten each night, x, as well as how many hours she had spent playing video games before bed, y.
round your answers to the nearest thousandth.

Explanation:

Step1: Calculate the means of \(x\) and \(y\)

Let \(x\) be hours of video games and \(y\) be hours of sleep.
\(\bar{x}=\frac{0.6 + 1.0+3.0+3.6+3.9}{5}=\frac{12.1}{5} = 2.42\)
\(\bar{y}=\frac{9 + 6+4+9+7}{5}=\frac{35}{5}=7\)

Step2: Calculate the numerator and denominator for the slope \(b\)

The formula for \(b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)
\((x_1-\bar{x})(y_1 - \bar{y})=(0.6 - 2.42)(9 - 7)=(- 1.82)\times2=-3.64\)
\((x_2-\bar{x})(y_2 - \bar{y})=(1.0 - 2.42)(6 - 7)=(-1.42)\times(-1) = 1.42\)
\((x_3-\bar{x})(y_3 - \bar{y})=(3.0 - 2.42)(4 - 7)=(0.58)\times(-3)=-1.74\)
\((x_4-\bar{x})(y_4 - \bar{y})=(3.6 - 2.42)(9 - 7)=(1.18)\times2 = 2.36\)
\((x_5-\bar{x})(y_5 - \bar{y})=(3.9 - 2.42)(7 - 7)=(1.48)\times0=0\)
\(\sum_{i = 1}^{5}(x_{i}-\bar{x})(y_{i}-\bar{y})=-3.64 + 1.42-1.74 + 2.36+0=-1.6\)

\((x_1-\bar{x})^{2}=(0.6 - 2.42)^{2}=(-1.82)^{2}=3.3124\)
\((x_2-\bar{x})^{2}=(1.0 - 2.42)^{2}=(-1.42)^{2}=2.0164\)
\((x_3-\bar{x})^{2}=(3.0 - 2.42)^{2}=(0.58)^{2}=0.3364\)
\((x_4-\bar{x})^{2}=(3.6 - 2.42)^{2}=(1.18)^{2}=1.3924\)
\((x_5-\bar{x})^{2}=(3.9 - 2.42)^{2}=(1.48)^{2}=2.1904\)
\(\sum_{i = 1}^{5}(x_{i}-\bar{x})^{2}=3.3124+2.0164 + 0.3364+1.3924+2.1904=9.248\)

\(b=\frac{-1.6}{9.248}\approx - 0.173\)

Step3: Calculate the intercept \(a\)

The formula for \(a=\bar{y}-b\bar{x}\)
\(a = 7-(-0.173)\times2.42=7 + 0.41866\approx7.419\)

Answer:

\(y=-0.173x + 7.419\)