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 the means

Let \(x\) be the hours spent playing per week and \(y\) be the 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 \(m\)

\(n = 10\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=(0 - 5.1)(3.49 - 2.796)+(0 - 5.1)(3.05 - 2.796)+(2 - 5.1)(3.24 - 2.796)+(3 - 5.1)(2.82 - 2.796)+(3 - 5.1)(3.19 - 2.796)+(5 - 5.1)(2.78 - 2.796)+(8 - 5.1)(2.31 - 2.796)+(8 - 5.1)(2.54 - 2.796)+(10 - 5.1)(2.03 - 2.796)+(12 - 5.1)(2.51 - 2.796)\)
\(=-5.1\times0.694-5.1\times0.254-3.1\times0.444-2.1\times0.026-2.1\times0.394 - 0.1\times0.016+2.9\times(- 0.486)+2.9\times(-0.256)+4.9\times(-0.766)+6.9\times(-0.286)\)
\(=-3.5394-1.2954 - 1.3764-0.0546-0.8274-0.0016-1.4094-0.7424-3.7534-1.9734\)
\(=-14.963\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=(0 - 5.1)^{2}+(0 - 5.1)^{2}+(2 - 5.1)^{2}+(3 - 5.1)^{2}+(3 - 5.1)^{2}+(5 - 5.1)^{2}+(8 - 5.1)^{2}+(8 - 5.1)^{2}+(10 - 5.1)^{2}+(12 - 5.1)^{2}\)
\(=26.01+26.01 + 9.61+4.41+4.41+0.01+8.41+8.41+24.01+47.61\)
\(=158.9\)
\(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}=\frac{-14.963}{158.9}\approx - 0.094\)

Step3: Calculate the y - intercept \(b\)

\(b=\bar{y}-m\bar{x}=2.796-(-0.094)\times5.1=2.796 + 0.4794=3.2754\)

Step4: Find the regression equation

The regression equation is \(y=mx + b\), so \(y=-0.094x+3.2754\)

Step5: Make the prediction

When \(x = 16\), \(y=-0.094\times16+3.2754=-1.504+3.2754 = 1.7714\approx1.77\)

Answer:

1.77