Chapter 2.5, Problem 14PPSE

Solution

a.

To find the line that best fits the data given in the table showing the relationshipbetween the number of Calories andgrams of fat.

The equation of the line that best fits the data given in the table relating the number of Calories and grams of fat:

  $y=0.07x-9.27$ .

Given:

Given the table:

Fast Food Calories
Restaurant	A	B	C	D	E	F	G	H	I
Number of Calories	$720$	$530$	$510$	$500$	$305$	$410$	$440$	$320$	$598$
Grams of fat	$46$	$30$	$27$	$26$	$13$	$20$	$25$	$13$	$26$

Concept Used:

Line of best fit:

Given some data points.

The line that best fits the set of data points is the one which minimises the distances between it and the data points. It is very reasonable to use a calculator to find such a line.

Calculation:

Plot the data points in the Cartesian coordinate systemso that the number of calories are represented along the $x$ -axis and the grams of fat are represented along the $y$-axis:

  

Find the line that best fits the data points with the help of a calculator:

  

Thus, the equation of the line that best fits the data given in the tablerelating the number of Calories with the gramsof fat:

  $y=0.07x-9.27$ .

Conclusion:

The equation of the line that best fits the data given in the tablerelating the number of Calories with the grams of fat:

  $y=0.07x-9.27$ .

b.

To determine the fat contained in a $330$ -Calorie hamburger.

As per the linear model that best fits the data, $13.83$ grams of fat is expected to be contained in a $330$ -Calorie hamburger.

Calculation:

The linear equation modelling the relationship between the number of Calories, $x$ , and grams of fat, $y$ , is found in part a:

  $y=0.07x-9.27$ .

Now, to determine the fat contained in a $330$ -Calorie hamburger, it is enough to solve the equation $y=0.07x-9.27$ for $y$ when $x=330$ .

Substitute $x=330$ in the equation $y=0.07x-9.27$ :

  $y=0.07\cdot 330-9.27$ .

Solve the equation $y=0.07\cdot 330-9.27$ for $y$ :

  $\begin{array}{l}y=0.07\cdot 330-9.27\\ y=13.83\end{array}$ .

Thus, it is found that $y=13.83$ when $x=330$ .

That is, as per the model, $13.83$ grams of fat is contained in a $330$ -Calorie hamburger.

Hence, $13.83$ grams of fat is expected to be contained in a $330$ -Calorie hamburger.

Conclusion:

As per the linear model that best fits the data, $13.83$ grams of fat is expected to be contained in a $330$ -Calorie hamburger.

c.

To determine which estimate is not reasonable from the estimates:

• $10\text{g}$ of fat for a $200$ -Calorie hamburger.
• $36\text{g}$ of fat for a $660$ -Calorie hamburger.

The first estimation that $10\text{g}$ of fat is contained in a $200$ -Calorie hamburger does not agree with the linear model representing the relationship between the fat content and number of Calorie contained in food items. Thus, the first estimation is not reasonable.

The second estimation that $36\text{g}$ of fat is contained in a $660$ -Calorie hamburger seems reasonable since it agrees with the mathematical model representing the relationship with a negligible error.

Given:

Given the estimates:

• $10\text{g}$ of fat for a $200$ -Calorie hamburger.
• $36\text{g}$ of fat for a $660$ -Calorie hamburger.

Calculation:

The linear equation modelling the relationship between the number of Calories, $x$ , and grams of fat, $y$ , is found in part a:

  $y=0.07x-9.27$ .

Consider the first estimate: $10\text{g}$ of fat for a $200$ -Calorie hamburger.

The first estimate corresponds to the values $y=10$ and $x=200$ .

Substitute $x=200$ in $y=0.07x-9.27$ and solve it for $y$ to determine whether it agrees with the model:

  $\begin{array}{l}y=0.07x-9.27\\ y=0.07\cdot 200-9.27\\ y=4.73\end{array}$ .

Thus, as per the model, $y$ is predicted to be $4.73$ when $x=200$ .

But, $y$ is estimated to be $10$ when $x=200$ .

That is, a huge error is happened while estimating the fat content of $200$ -Calorie hamburger.

Thus, the first estimation is not reasonable.

Consider the second estimate: $36\text{g}$ of fat for a $660$ -Calorie hamburger.

The second estimate corresponds to the values $y=36$ and $x=660$ .

Substitute $x=660$ in $y=0.07x-9.27$ and solve it for $y$ to determine whether it agrees with the model:

  $\begin{array}{l}y=0.07x-9.27\\ y=0.07\cdot 660-9.27\\ y=36.93\end{array}$ .

Thus, as per the model, $y$ is predicted to be $36.93$ when $x=660$ .

Now, $y$ is estimated to be $36$ when $x=660$ .

Observe that the error happened in the estimation of fat content is negligible.

Thus, the second estimation seems reasonable.

Conclusion:

The first estimation that $10\text{g}$ of fat is contained in a $200$ -Calorie hamburger does not agree with the linear model representing the relationship between the fat content and number of Calorie contained in food items. Thus, the first estimation is not reasonable.

The second estimation that $36\text{g}$ of fat is contained in a $660$ -Calorie hamburger seems reasonable since it agrees with the mathematical model representing the relationship with a negligible error.