Chapter 4, Problem 44SGR

To determine

To find:The equation of the regression line and estimating the daily ticket sale on day 15^th.

Answer to Problem 44SGR

The equation of regression line is $Y=199.143-5.786X$ and ticket sale estimated on day 15 is 112.353.

Explanation of Solution

Given: Daily ticket sale on different days since movie opened.

Concept Used: The general equation for a regression line is given as:

  $Y=a+bX$

Where, Y is dependent and X is an independent variable. Also, a is the y-intercept and b is the slope of the line.

The values of a and b can be obtained by the following formulae:

  $\begin{array}{l}a=\frac{({\displaystyle \sum y})({\displaystyle \sum x^2})-({\displaystyle \sum x})({\displaystyle \sum xy})}{n({\displaystyle \sum x^2})-{({\displaystyle \sum x})}^2}\\ b=\frac{n({\displaystyle \sum xy})-({\displaystyle \sum x})({\displaystyle \sum y})}{n({\displaystyle \sum x^2})-{({\displaystyle \sum x})}^2}\end{array}$

Here, n is the number of observations used to estimate the regression line.

Calculation:

The table showing all the values corresponding to each observation is shown below:

Observation	Day of Sale ( $x$ )	Daily Purchases ( $y$ )	$xy$	$x^2$	$y^2$
1	1	185	185	1	34225
2	2	192	384	4	36864
3	3	189	567	9	35721
4	4	178	712	16	31684
5	5	165	825	25	27225
6	6	168	1008	36	28224
7	7	155	1085	49	24025
${\displaystyle \sum }$	28	1232	4766	140	217968

From above table,

  $\begin{array}{l}a=\frac{({\displaystyle \sum y})({\displaystyle \sum x^2})-({\displaystyle \sum x})({\displaystyle \sum xy})}{n({\displaystyle \sum x^2})-{({\displaystyle \sum x})}^2}=\frac{(1232)(140)-(28)(4766)}{7(140)-{(28)}^2}\\ a=\frac{172480-133448}{980-784}=\frac{39032}{196}=199.143\\ b=\frac{n({\displaystyle \sum xy})-({\displaystyle \sum x})({\displaystyle \sum y})}{n({\displaystyle \sum x^2})-{({\displaystyle \sum x})}^2}=\frac{7(4766)-(28)(1232)}{7(140)-{(28)}^2}\\ a=\frac{33362-34496}{980-784}=\frac{-1134}{196}=-5.786\end{array}$

Hence, the equation of the regression line is obtained as:

  $\begin{array}{l}Y=a+bX\\ Y=199.143-5.786X\end{array}$

Now, Let us estimate the ticket sale on day 15 since the movie has opened. For this, $X=15$

Hence,

  $\begin{array}{l}Y=199.143-5.786(15)\\ \Rightarrow Y=199.143-86.79=112.353\end{array}$

Conclusion: Thus, the equation of regression line is $Y=199.143-5.786X$ and ticket sale estimated on day 15 is 112.353.