Q49. - Algebra 1

Question

The table shows the number of sales made at an outerwear store during a sale. Write an equation of the regression line. Then estimate the daily sales on day 10 of the sale

Days Since Sale Began	1	2	3	4	5	6	7
Daily Sales ($\$$)	15	21	32	30	40	38	51

Step-by-Step Solution

Answer

The required equation of the regression line is: $\mathit{y}\mathbf{=}\mathbf{5}\mathbf{.36}\mathit{x}\mathbf{+}\mathbf{11}$. The daily sales on day 10 of the sale is $\$$65.

Step 1. Rearrange the table.

Let 

$X$-axis = days since sale begin.

$Y$-axis = daily sales ($\$$).

 

Using the regression calculator to find the equation of the regression line:

Days since sale began ($x$)	Daily Sales ($y$)
1	15
2	21
3	32
4	30
5	40
6	38
7	51

Step 2. Find the regression equation.

$\begin{array}{l}\text{Sum of }X=28\\ \text{Sum of }Y=227\\ \text{Mean }X=4\\ \text{Mean }Y=32.4286\\ \text{Sum of squares }\left(S{S}_X\right)=28\\ \text{Sum of products }\left(SP\right)=150\end{array}$

Regression Equation: $ŷ = bX + a$

$\begin{array}{l}b=SP/S{S}_X=150/28=5.35714\\ a=M_Y-b{M}_X=32.43-(5.36\times 4)=11\\ y=5.35714X+11\end{array}$ 

Step 3. Plot the graph.

Graph the line $\widehat{y}=5.35714X+11$ with slope $a$ and intercept $b$.

Thus, the regression equation is $\widehat{y}=5.36X+11$.

Step 4. Put the value of X to find the estimated sales.

Substituting the value of $x=10$ in regression equation to find the daily sales on day 10 of  the sale :

$\begin{array}{c}\widehat{y}=5.36X+11\\ \widehat{y}=5.36\left(10\right)+11\\ \widehat{y}=53.6+11\\ \widehat{y}=64.6\approx 65\end{array}$

So, the daily sales on day 10 of the sale = $\$$65.