Chapter 4, Problem 43SGR

To determine

To find: The equation of the regression line and estimating the daily purchase on day 10 of the sale.

Answer to Problem 43SGR

The equation of regression line is $Y=11+5.357X$ and sale estimated on day 10 is 64.57.

Explanation of Solution

Given: Daily purchases on different days since sale begun is given.

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	15	15	1	225
2	2	21	42	4	441
3	3	32	96	9	1024
4	4	30	120	16	900
5	5	40	200	25	1600
6	6	38	228	36	1444
7	7	51	357	49	2601
${\displaystyle \sum }$	28	227	1058	140	8235

From above table,

  $\begin{array}{l}a=\frac{({\displaystyle \sum y})({\displaystyle \sum x^2})-({\displaystyle \sum x})({\displaystyle \sum xy})}{n({\displaystyle \sum x^2})-{({\displaystyle \sum x})}^2}=\frac{(227)(140)-(28)(1058)}{7(140)-{(28)}^2}\\ a=\frac{31780-29624}{980-784}=\frac{2156}{196}=11\\ b=\frac{n({\displaystyle \sum xy})-({\displaystyle \sum x})({\displaystyle \sum y})}{n({\displaystyle \sum x^2})-{({\displaystyle \sum x})}^2}=\frac{7(1058)-(28)(227)}{7(140)-{(28)}^2}\\ a=\frac{7406-6356}{980-784}=\frac{1050}{196}=5.357\end{array}$

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

  $\begin{array}{l}Y=a+bX\\ Y=11+5.357X\end{array}$

Now, Let us estimate the sale on day 10 since the sale has begun. For this, $X=10$

Hence,

  $\begin{array}{l}Y=11+5.357(10)\\ \Rightarrow Y=11+53.57=64.57\end{array}$

Conclusion: Thus, the equation of regression line is $Y=11+5.357X$ and sale estimated on day 10 is 64.57.