Question

the effect of regularly playing video games on grades
a high school biology class conducted a study of whether playing video games had any effect on grades. ten volunteers told the class how many hours per week they spent playing video games and what their grade point average was. the results are tabulated below.
hours spent playing per week grade point average
0 3.49
0 3.05
2 3.24
3 2.82
3 3.19
5 2.78
8 2.31
8 2.54
10 2.03
12 2.51
what is the y - intercept, b, of the line of best fit?

Explanation:

Step1: Recall the formula for the y - intercept

The formula for the y - intercept $b$ of the line of best fit in the simple linear regression equation $y = mx + b$ (where $m$ is the slope) can be calculated using the following set of formulas. First, we need to calculate some summary statistics. Let $x$ be the hours spent playing per week and $y$ be the grade - point average. We have $n = 10$ data points. Calculate $\sum x$, $\sum y$, $\sum x^2$, $\sum xy$.
Let $x_i$ and $y_i$ be the individual data points.
$\sum_{i = 1}^{n}x_i=0 + 0+2 + 3+3 + 5+8 + 8+10+12=51$
$\sum_{i = 1}^{n}y_i=3.49+3.05 + 3.24+2.82+3.19+2.78+2.31+2.54+2.03+2.51 = 27.96$
$\sum_{i = 1}^{n}x_i^2=0^2+0^2 + 2^2+3^2+3^2+5^2+8^2+8^2+10^2+12^2=0 + 0+4 + 9+9+25+64+64+100+144 = 429$
$\sum_{i = 1}^{n}x_iy_i=(0\times3.49)+(0\times3.05)+(2\times3.24)+(3\times2.82)+(3\times3.19)+(5\times2.78)+(8\times2.31)+(8\times2.54)+(10\times2.03)+(12\times2.51)=0 + 0+6.48+8.46+9.57+13.9+18.48+20.32+20.3+30.12 = 127.63$

Step2: Calculate the slope $m$ first

The formula for the slope $m$ is $m=\frac{n\sum xy-\sum x\sum y}{n\sum x^2-(\sum x)^2}$
$m=\frac{10\times127.63 - 51\times27.96}{10\times429-51^2}=\frac{1276.3-1425.96}{4290 - 2601}=\frac{- 149.66}{1689}\approx - 0.0886$

Step3: Calculate the y - intercept $b$

The formula for the y - intercept $b=\bar{y}-m\bar{x}$, where $\bar{x}=\frac{\sum x}{n}$ and $\bar{y}=\frac{\sum y}{n}$
$\bar{x}=\frac{51}{10}=5.1$
$\bar{y}=\frac{27.96}{10}=2.796$
$b = 2.796-(-0.0886)\times5.1=2.796 + 0.45186\approx3.25$

Answer:

$3.25$