Chapter 7.2, Problem 54E

a.

To determine

To find:The least squares regression line $y=ax+b$ for the data.

Answer to Problem 54E

Theleast squares regression line $y=ax+b$ for the datais $y=-44.2105x+89.5263$ .

Explanation of Solution

Given information:

A store manager wants to know the demand for a product as a function of the price. The table shows the daily sales $y$ for different prices $x$ of the product.

  

System of equations to find the values of $a$ and $b$ is:

  $\{\begin{array}{c}3.00b+3.70a=105.00\\ 3.70b+4.69a=123.90\end{array}$

Concept Used:

To find the least squares regression line $y=ax+b$ for a set of points $(x_1,y_1),(x_2,y_2),\mathrm{....},(x_n,y_n)$ is by solving the system below for $a$ and $b.$

  $\{\begin{array}{c}ab+\left({\displaystyle \sum _{i=1}^n{x}_i}\right)a=\left({\displaystyle \sum _{i=1}^n{y}_i}\right)\\ \left({\displaystyle \sum _{i=1}^n{x}_i}\right)b+\left({\displaystyle \sum _{i=1}^n{x}_i{}^2}\right)a=\left({\displaystyle \sum _{i=1}^n{x}_i{y}_i}\right)\end{array}$

Calculations:

Solving the system of equations using substitution method, we have:

  $\begin{array}{l}3.00b+3.70a=105.00\\ \Rightarrow 3.00b=105.00-3.70a\\ \Rightarrow b=\frac{105.00-3.70a}{3.00}\end{array}$

Substituting the above finding in the second equation, we get

  $\begin{array}{l}3.70\left(\frac{105.00-3.70a}{3.00}\right)+4.69a=123.90\\ \Rightarrow 129.50-\frac{13.69a}{3.00}+4.69a=123.90\\ \Rightarrow \frac{19}{150}a=-\frac{28}{5}\\ \Rightarrow a=-\frac{28}{5}\times \frac{150}{19}=-\frac{840}{19}\\ \Rightarrow a=-44.21\end{array}$

Now, using the above finding value:

  $\begin{array}{l}b=\frac{105.00-3.70a}{3.00}\\ \Rightarrow b=\frac{105.00-3.70(-44.21)}{3.00}\\ \Rightarrow b=\frac{268.577}{3.00}\\ \Rightarrow b=89.52\end{array}$

Conclusion:

Using the graphing utility, we obtain the following results:

  

Therefore, the results we got are correct.

b.

To determine

To find: The prediction value of demand when the price is $1.75.

Answer to Problem 54E

The demand is 12 when the price is $1.75.

Explanation of Solution

Given information:

A store manager wants to know the demand for a product as a function of the price. The table shows the daily sales $y$ for different prices $x$ of the product.

  

System of equations to find the values of $a$ and $b$ is:

  $\{\begin{array}{c}3.00b+3.70a=105.00\\ 3.70b+4.69a=123.90\end{array}$

Calculations:

From the previous part, we got the linear model $y=-44.2105x+89.5263$ .

At $x=\$1.75$ ,

  $\begin{array}{l}y=-44.2105(1.75)+89.5263\\ \Rightarrow y=12.157925\end{array}$

Therefore, the demand is 12 when the price is $1.75.