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
predict the grade point average of a student who plays video games for 16 hours each week

Explanation:

Step1: Calculate means of x and y

Let \(x\) be hours playing per week and \(y\) be grade - point average.
\(\bar{x}=\frac{0 + 0+2+3+3+5+8+8+10+12}{10}=\frac{51}{10} = 5.1\)
\(\bar{y}=\frac{3.49+3.05+3.24+2.82+3.19+2.78+2.31+2.54+2.03+2.51}{10}=\frac{27.96}{10}=2.796\)

Step2: Calculate the slope \(b_1\)

\[

$$\begin{align*} \sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})&=(0 - 5.1)(3.49-2.796)+(0 - 5.1)(3.05 - 2.796)+(2-5.1)(3.24 - 2.796)+(3 - 5.1)(2.82-2.796)+(3-5.1)(3.19 - 2.796)+(5-5.1)(2.78 - 2.796)+(8-5.1)(2.31 - 2.796)+(8-5.1)(2.54 - 2.796)+(10-5.1)(2.03 - 2.796)+(12-5.1)(2.51 - 2.796)\\ &=(- 5.1)\times0.694+(-5.1)\times0.254+(-3.1)\times0.444+(-2.1)\times0.026+(-2.1)\times0.394+(-0.1)\times(-0.016)+2.9\times(-0.486)+2.9\times(-0.256)+4.9\times(-0.766)+6.9\times(-0.286)\\ &=-3.5394-1.2954 - 1.3764-0.0546-0.8274 + 0.0016-1.4094-0.7424-3.7534-1.9734\\ &=-14.9604 \end{align*}$$

\]
\[

$$\begin{align*} \sum_{i=1}^{n}(x_i-\bar{x})^2&=(0 - 5.1)^2+(0 - 5.1)^2+(2-5.1)^2+(3 - 5.1)^2+(3-5.1)^2+(5-5.1)^2+(8-5.1)^2+(8-5.1)^2+(10-5.1)^2+(12-5.1)^2\\ &=(-5.1)^2+(-5.1)^2+(-3.1)^2+(-2.1)^2+(-2.1)^2+(-0.1)^2+2.9^2+2.9^2+4.9^2+6.9^2\\ &=26.01+26.01 + 9.61+4.41+4.41+0.01+8.41+8.41+24.01+47.61\\ &=158.9 \end{align*}$$

\]
\(b_1=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}=\frac{-14.9604}{158.9}\approx - 0.0941\)

Step3: Calculate the intercept \(b_0\)

\(b_0=\bar{y}-b_1\bar{x}=2.796-(-0.0941)\times5.1=2.796 + 0.47991=3.27591\)

Step4: Find the regression equation

The regression equation is \(\hat{y}=b_0 + b_1x=3.27591-0.0941x\)

Step5: Predict for \(x = 16\)

\(\hat{y}=3.27591-0.0941\times16=3.27591 - 1.5056=1.77031\approx1.77\)

Answer:

1.77