Chapter 3.2, Problem 19AYU

a

To determine

Explain why the number of hours spent playing video games is the independent variable and cumulative grade-point average is the dependent variable.

Answer to Problem 19AYU

variable $G$ is the dependent variable.

Explanation of Solution

Given information:

Professor Grant Alexander wanted to find a linear model that relates the number of hours a student plays video games each week, $h$ , to the cumulative grade-point average, $G$ , of the student. He obtained a random sample of $10$ full-time students at his college and asked each student to disclose the number of hours spent playing video games and the student’s cumulative grade-point average.

  $\begin{array}{cc}\begin{array}{l}Hoursof\\ VideoGames\\ PerWeek,h\end{array}& \begin{array}{l}Grade-Po\mathrm{int}\\ Average,G\end{array}\\ 0& 3.49\\ 0& 3.05\\ 2& 3.24\\ 3& 2.82\\ 3& 3.19\\ 5& 2.78\\ 8& 2.31\\ 8& 2.54\\ 10& 2.03\\ 12& 2.51\end{array}$

Explain why the number of hours spent playing video games is the independent variable and cumulative grade-point average is the dependent variable.

Calculation:

The number of hours spent playing video games is the independent variable is totally depends on students own wish, how long he plays a video game or there is no time restriction for students for playing video game as mentioned here.

Number of hours $h$ can hold any value without any restriction.

Further as cumulative grade point average of each student would directly depents on how long student plays video game as the more students plays game, the more game point, he would gain gain resulting a different point average.

Hence, variable $G$ is the dependent variable.

b

To determine

Use a graphing utility to draw a scatter diagram.

Answer to Problem 19AYU

  

Explanation of Solution

Given information:

Professor Grant Alexander wanted to find a linear model that relates the number of hours a student plays video games each week, $h$ , to the cumulative grade-point average, $G$ , of the student. He obtained a random sample of $10$ full-time students at his college and asked each student to disclose the number of hours spent playing video games and the student’s cumulative grade-point average.

  $\begin{array}{cc}\begin{array}{l}Hoursof\\ VideoGames\\ PerWeek,h\end{array}& \begin{array}{l}Grade-Po\mathrm{int}\\ Average,G\end{array}\\ 0& 3.49\\ 0& 3.05\\ 2& 3.24\\ 3& 2.82\\ 3& 3.19\\ 5& 2.78\\ 8& 2.31\\ 8& 2.54\\ 10& 2.03\\ 12& 2.51\end{array}$

Use a graphing utility to draw a scatter diagram.

Calculation:

Now consider the data given in table,

  $\left(0,3.49\right).\left(0,3.05\right).\left(2,3.24\right).\left(3,3.19\right).\left(5,2.31\right)$ , $\left(8,2.54\right).\left(10,2.03\right)and\left(12,2.51\right).$ to draw a scatter diagram.

  

The number of hours spent playing video games is the independent variable is totally depends on students own wish, how long he plays a video game or there is no time restriction for students for playing video game as mentioned here.

Number of hours $h$ can hold any value without any restriction.

Further as cumulative grade point average of each student would directly depents on how long student plays video game as the more students plays game, the more game point, he would gain gain resulting a different point average.

Hence, variable $G$ is the dependent variable.

c.

To determine

Use a graphing utility to draw a scatter diagram.

Answer to Problem 19AYU

  

Explanation of Solution

Given information:

Professor Grant Alexander wanted to find a linear model that relates the number of hours a student plays video games each week, $h$ , to the cumulative grade-point average, $G$ , of the student. He obtained a random sample of $10$ full-time students at his college and asked each student to disclose the number of hours spent playing video games and the student’s cumulative grade-point average.

  $\begin{array}{cc}\begin{array}{l}Hoursof\\ VideoGames\\ PerWeek,h\end{array}& \begin{array}{l}Grade-Po\mathrm{int}\\ Average,G\end{array}\\ 0& 3.49\\ 0& 3.05\\ 2& 3.24\\ 3& 2.82\\ 3& 3.19\\ 5& 2.78\\ 8& 2.31\\ 8& 2.54\\ 10& 2.03\\ 12& 2.51\end{array}$

Use a graphing utility to find the line of best fit that models the relation between numbers of hours of video game playing each week and grade-point average. Express the model using function notation.

Calculation:

Graphing utilities contain built-in-programs that find the line of best fit for a collection of points in the scatter diagram.

Upon executing the linear regression program, we obtain the results shown as,

The output that utility provides shows us the equation $y=mx+b$ where slope of the line is $m$ and $y$ intercept is $b$

The line of best fit relates the number of hours plying video games $\left(h\right)$ to cumulative grade point average $\left(G\right)$ may be expressed as the line $G\left(h\right)=-0.0942h+3.2763$

  $R^2=0.7539$ as shown in graph of line of best fit.

  

Hence, the graph of line of best fit.

d.

To determine

Interpret the slope.

Answer to Problem 19AYU

  $\boxed{m=-0.0942}$

Explanation of Solution

Given information:

Professor Grant Alexander wanted to find a linear model that relates the number of hours a student plays video games each week, $h$ , to the cumulative grade-point average, $G$ , of the student. He obtained a random sample of $10$ full-time students at his college and asked each student to disclose the number of hours spent playing video games and the student’s cumulative grade-point average.

  $\begin{array}{cc}\begin{array}{l}Hoursof\\ VideoGames\\ PerWeek,h\end{array}& \begin{array}{l}Grade-Po\mathrm{int}\\ Average,G\end{array}\\ 0& 3.49\\ 0& 3.05\\ 2& 3.24\\ 3& 2.82\\ 3& 3.19\\ 5& 2.78\\ 8& 2.31\\ 8& 2.54\\ 10& 2.03\\ 12& 2.51\end{array}$

Interpret the slope.

Calculation:

As we have that any equation in slope-intercept form is written as $y=mx+c$

Now on comparing it with equation of best line, we observe that $m=-0.0942$

Hence, slope of the line is $\boxed{m=-0.0942}$

d.

To determine

Interpret the slope.

Answer to Problem 19AYU

  $\boxed{m=-0.0942}$

Explanation of Solution

Given information:

Professor Grant Alexander wanted to find a linear model that relates the number of hours a student plays video games each week, $h$ , to the cumulative grade-point average, $G$ , of the student. He obtained a random sample of $10$ full-time students at his college and asked each student to disclose the number of hours spent playing video games and the student’s cumulative grade-point average.

  $\begin{array}{cc}\begin{array}{l}Hoursof\\ VideoGames\\ PerWeek,h\end{array}& \begin{array}{l}Grade-Po\mathrm{int}\\ Average,G\end{array}\\ 0& 3.49\\ 0& 3.05\\ 2& 3.24\\ 3& 2.82\\ 3& 3.19\\ 5& 2.78\\ 8& 2.31\\ 8& 2.54\\ 10& 2.03\\ 12& 2.51\end{array}$

Interpret the slope.

Calculation:

As we have that any equation in slope-intercept form is written as $y=mx+c$

Now on comparing it with equation of best line, we observe that $m=-0.0942$

Hence, slope of the line is $\boxed{m=-0.0942}$

e.

To determine

Find the average of grade points.

Answer to Problem 19AYU

  $\boxed{G\left(8\right)=2.5227}$

Explanation of Solution

Given information:

Professor Grant Alexander wanted to find a linear model that relates the number of hours a student plays video games each week, $h$ , to the cumulative grade-point average, $G$ , of the student. He obtained a random sample of $10$ full-time students at his college and asked each student to disclose the number of hours spent playing video games and the student’s cumulative grade-point average.

  $\begin{array}{cc}\begin{array}{l}Hoursof\\ VideoGames\\ PerWeek,h\end{array}& \begin{array}{l}Grade-Po\mathrm{int}\\ Average,G\end{array}\\ 0& 3.49\\ 0& 3.05\\ 2& 3.24\\ 3& 2.82\\ 3& 3.19\\ 5& 2.78\\ 8& 2.31\\ 8& 2.54\\ 10& 2.03\\ 12& 2.51\end{array}$

Predict the grade-point average of a student who plays video games for $8$ hours each week.

Calculation:

Consider $h=8hours$ .

  $G\left(h\right)=-0.0942h+3.2763$

  $\begin{array}{l}G\left(8\right)=-0.0942\times 8+3.2763\\ G\left(8\right)=-0.7536+3.2763\\ G\left(8\right)=2.5227\end{array}$

  $\boxed{G\left(8\right)=2.5227}$

Hence, the students who plays game for $8$ hours has grade points $\boxed{G\left(8\right)=2.5227}$ .

f.

To determine

Find the time to play game with average points.

Answer to Problem 19AYU

  $\boxed{G\left(8\right)=2.5227}$

Explanation of Solution

Given information:

Professor Grant Alexander wanted to find a linear model that relates the number of hours a student plays video games each week, $h$ , to the cumulative grade-point average, $G$ , of the student. He obtained a random sample of $10$ full-time students at his college and asked each student to disclose the number of hours spent playing video games and the student’s cumulative grade-point average.

  $\begin{array}{cc}\begin{array}{l}Hoursof\\ VideoGames\\ PerWeek,h\end{array}& \begin{array}{l}Grade-Po\mathrm{int}\\ Average,G\end{array}\\ 0& 3.49\\ 0& 3.05\\ 2& 3.24\\ 3& 2.82\\ 3& 3.19\\ 5& 2.78\\ 8& 2.31\\ 8& 2.54\\ 10& 2.03\\ 12& 2.51\end{array}$

How many hours of video game playing do you think a student plays whose grade- point average is $2.40$ ?

Calculation:

Consider $G=2.40$ .

  $G\left(h\right)=-0.0942h+3.2763$

  $\begin{array}{l}2.40=-0.0942h+3.2763\\ 2.40-3.2763=-0.0942h\\ -0.8763=-0.0942h\\ h=9.3\end{array}$

  $h=9.3hours$

Hence, the time to play game with average points $\boxed{9.3hours}$ .