Chapter 3.2, Problem 17AYU

a

To determine

Draw a scatter diagram of the data treating weight as the independent variable.

Answer to Problem 17AYU

  

Explanation of Solution

Given information:

The following data represents the weight (in grams) of various candy bars and the corresponding number of calories.

  $\begin{array}{ccc}CandyBar& Weight,X& CaloriesY\\ Hershey\text{'}sMilkChocolate& 44.28& 230\\ Nestle\text{'}sCrunch& 44.84& 230\\ Butterfinger& 61.30& 270\\ BabyRuth& 66.45& 280\\ AlmondJoy& 47.33& 220\\ Twix& 58.00& 280\\ Snic\ker s& 61.12& 280\\ Heath& 39.52& 210\end{array}$

Draw a scatter diagram of the data treating weight as the independent variable.

Calculation:

Draw a scatter diagram that relates the weight (in grams) of various candy bars and the corresponding number of calories.

  

Hence, the result shown in scatter plot.

b.

To determine

What type of relation appears to exist between the weight of a candy bar and the number of calories?

Answer to Problem 17AYU

A linear relation exists.

Explanation of Solution

Given information:

The following data represents the weight (in grams) of various candy bars and the corresponding number of calories.

  $\begin{array}{ccc}CandyBar& Weight,X& CaloriesY\\ Hershey\text{'}sMilkChocolate& 44.28& 230\\ Nestle\text{'}sCrunch& 44.84& 230\\ Butterfinger& 61.30& 270\\ BabyRuth& 66.45& 280\\ AlmondJoy& 47.33& 220\\ Twix& 58.00& 280\\ Snic\ker s& 61.12& 280\\ Heath& 39.52& 210\end{array}$

What type of relation appears to exist between the weight of a candy bar and the number of calories?

Calculation:

In the scatter diagram, the plotted points lie somewhat along a straight line that slants downward from left to right.

Hence, a linear relation exists.

c.

To determine

Find a linear model that contains the points.

Answer to Problem 17AYU

  $\boxed{y=2.5993x+107.2757}$

Explanation of Solution

Given information:

The following data represents the weight (in grams) of various candy bars and the corresponding number of calories.

  $\begin{array}{ccc}CandyBar& Weight,X& CaloriesY\\ Hershey\text{'}sMilkChocolate& 44.28& 230\\ Nestle\text{'}sCrunch& 44.84& 230\\ Butterfinger& 61.30& 270\\ BabyRuth& 66.45& 280\\ AlmondJoy& 47.33& 220\\ Twix& 58.00& 280\\ Snic\ker s& 61.12& 280\\ Heath& 39.52& 210\end{array}$

Select two points and find a linear model that contains the points.

Calculation:

Find a line that fits the data given, select two points from scatter plot and find the slope of line that contains the points.

Consider points, $\left(39,52,210\right)$ and $\left(66,45,280\right)$ now find the slope of line that contains the same points.

Slope $m$ of line containing points $P\left(x_1,y_1\right)$ and $Q\left(x_2,y_2\right)$ is obtained by the formula,

  $m=\frac{y_2-y_1}{x_2-x_1}=\frac{\Delta y}{\Delta x}$

So,

  $m=\frac{280-210}{66.45-39.52}=\frac{70}{26.93}=2.5993$

Now find the equation of line that contains the points $\left(39,52,210\right)$ using point-slope form.

  $y-y_1=m\left(x-x_1\right)$

Now substitute $m=2.5990,x_1=39.52,y_1=210$

  $\begin{array}{l}y-210=2.5993\left(x-39.52\right)\\ y-210=2.5993x-102.7243\\ y=2.5993x-102.7243+210\\ y=2.5993x+107.2757\end{array}$

Hence, the equation of line is, $\boxed{y=2.5993x+107.2757}$

d.

To determine

Graph the line on scatter plot.

Answer to Problem 17AYU

  

Explanation of Solution

Given information:

The following data represents the weight (in grams) of various candy bars and the corresponding number of calories.

  $\begin{array}{ccc}CandyBar& Weight,X& CaloriesY\\ Hershey\text{'}sMilkChocolate& 44.28& 230\\ Nestle\text{'}sCrunch& 44.84& 230\\ Butterfinger& 61.30& 270\\ BabyRuth& 66.45& 280\\ AlmondJoy& 47.33& 220\\ Twix& 58.00& 280\\ Snic\ker s& 61.12& 280\\ Heath& 39.52& 210\end{array}$

Graph the line on scatter plot drawn in part $\left(a\right)$ .

Calculation:

Let Graph the line on scatter plot $y=2.5993x+107.2757$ .

  

Hence, the graph of line is plotted.

e.

To determine

Use the linear model to predict the number of calories in a candy bar.

Answer to Problem 17AYU

  $\boxed{269calories.}$

Explanation of Solution

Given information:

The following data represents the weight (in grams) of various candy bars and the corresponding number of calories.

  $\begin{array}{ccc}CandyBar& Weight,X& CaloriesY\\ Hershey\text{'}sMilkChocolate& 44.28& 230\\ Nestle\text{'}sCrunch& 44.84& 230\\ Butterfinger& 61.30& 270\\ BabyRuth& 66.45& 280\\ AlmondJoy& 47.33& 220\\ Twix& 58.00& 280\\ Snic\ker s& 61.12& 280\\ Heath& 39.52& 210\end{array}$

Use the linear model to predict the number of calories in a candy bar that weight $62.3grams.$Calculation:

Use the linear model to predict the number of calories in a candy bar that weight $62.3grams.$

Put $x=62.3$ in linear equation, $y=2.5993x+107.2757$

  $\begin{array}{l}y=2.5993x+107.2757\\ y=2.5993\left(62.3\right)+107.2757\\ y=269.21\\ y=269calories.\end{array}$

Hence, the candy bar that weight $62.3grams.$ has $\boxed{269calories.}$

f.

To determine

Interpret the slope of the line.

Answer to Problem 17AYU

  $\boxed{2.599}$

Explanation of Solution

Given information:

The following data represents the weight (in grams) of various candy bars and the corresponding number of calories.

  $\begin{array}{ccc}CandyBar& Weight,X& CaloriesY\\ Hershey\text{'}sMilkChocolate& 44.28& 230\\ Nestle\text{'}sCrunch& 44.84& 230\\ Butterfinger& 61.30& 270\\ BabyRuth& 66.45& 280\\ AlmondJoy& 47.33& 220\\ Twix& 58.00& 280\\ Snic\ker s& 61.12& 280\\ Heath& 39.52& 210\end{array}$

Use the linear model to predict the number of calories in a candy bar that weight $62.3grams.$

Interpret the slope of the line found in part $\left(c\right)$ .

Calculation:

Interpret the slope of the line.

  $m=\frac{y_2-y_1}{x_2-x_1}=\frac{\Delta y}{\Delta x}$ is average rate of change of $y$ with respect $x$ .which is equal to $2.599$ .

If the weight of candy bar is increased by $1gm$ then number of calories increased by

  $\frac{\Delta y}{\Delta x}\times 1=2.599\times 1=2.599$

Hence, as the candy bar is increased by $1gm$ then number of calories increased by $\boxed{2.599}$