Q50. - Algebra 1

Question

The table shows ticket sales during the first week. Write an equation of the regression line. Then estimate the daily ticket sales on the 15^th day after the movie opens.

Days Since Movie Opened	1	2	3	4	5	6	7
Daily Ticket Sales ($\$$)	85	92	89	78	65	68	55

Step-by-Step Solution

Answer

The required equation of the regression line is: $\mathit{y}\mathbf{=}\mathbf{-}\mathbf{5}\mathbf{.79}\mathit{x}\mathbf{+}\mathbf{99}\mathbf{.14}$. The ticket sales on 15th day after the movie opened is $\mathbf{\$}\mathbf{12}\mathbf{.}\mathbf{29}$.

Step 1. Rearrange the table.

Let 

$X$-axis = days since movie opened.

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

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

 

Days since sale began ($x$)	Daily Sales ($y$)
1	85
2	92
3	89
4	78
5	65
6	68
7	55

Step 2. Find the regression equation.

$\begin{array}{l}\text{Sum of }X=28\\ \text{Sum of }Y=532\\ \text{Mean }X=4\\ \text{Mean }Y=76\\ \text{Sum of squares }\left(S{S}_X\right)=28\\ \text{Sum of products }\left(SP\right)=-162\end{array}$

Regression Equation: $ŷ = bX + a$

$\begin{array}{l}b=SP/S{S}_X=-162/28=-5.78571\\ a=M_Y-b{M}_X=76-(-5.79\times 4)=99.14286\\ =-5.78571X+99.14286\end{array}$

Step 3. Plot the graph.

Graph the line $\widehat{y}=-5.79X+99.14$ with slope $a$ and intercept $b$.

Thus, the regression equation is $\widehat{y}=-5.79X+99.14$.

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 :

 $\begin{array}{c}\widehat{y}=-5.79X+99.14\\ \widehat{y}=-5.79\left(15\right)+99.14\\ \widehat{y}=-86.85+99.14\\ \widehat{y}=12.29\end{array}$

So, The ticket sales on 15th day after the movie opened = $\$12.29$.