The accompanying table illustrates the number of movie theaters showing a popular film and the film's weekly gross earnings, in millions of dollars.

Number of Theaters (x) Gross Earnings (y) (millions of dollars)
443 2.57
455 2.65
493 3.73
530 4.05
569 4.76
657 4.76
723 5.15
1064 9.35

Write the linear regression equation for this set of data, rounding values to three decimal places. Using this linear regression equation, find the approximate gross earnings, in millions of dollars, generated by 610 theaters.

Solution

Let $x$ denote the no. of theaters and $y$ denote the gross earnings in millions of dollars.

Let the simple linear regression model of $Y$ on $X$ is

$$y=\beta_0 + \beta_1x +e$$

By the method of least square, the estimates of $\beta_1$ and $\beta_0$ are respectively

$$ \begin{aligned} \hat{\beta}_1 & = \frac{n \sum xy - (\sum x)(\sum y)}{n(\sum x^2) -(\sum x)^2} \end{aligned} $$

and

$$ \begin{aligned} \hat{\beta}_0&=\overline{y}-\hat{\beta}_1\overline{x} \end{aligned} $$

scatter-plot
scatter-plot
$x$ $y$ $x^2$ $y^2$ $xy$
1 443 2.57 196249 6.6049 1138.51
2 455 2.65 207025 7.0225 1205.75
3 493 3.73 243049 13.9129 1838.89
4 530 4.05 280900 16.4025 2146.50
5 569 4.76 323761 22.6576 2708.44
6 657 4.76 431649 22.6576 3127.32
7 723 5.15 522729 26.5225 3723.45
8 1064 9.35 1132096 87.4225 9948.40
Total 4934 37.02 3337458 203.2030 25837.26

The sample mean of $x$ is

$$ \begin{aligned} \overline{x}&=\frac{1}{n} \sum_{i=1}^n x_i\\ &=\frac{4934}{8}\\ &=616.75 \end{aligned} $$

The sample mean of $y$ is

$$ \begin{aligned} \overline{y}&=\frac{1}{n} \sum_{i=1}^n y_i\\ &=\frac{37.02}{8}\\ &=4.6275 \end{aligned} $$

The estimate of $\beta_1$ is given by

$$ \begin{aligned} b_1 & = \frac{n \sum xy - (\sum x)(\sum y)}{n(\sum x^2) -(\sum x)^2}\\ & = \frac{8*25837.26-(4934)(37.02)}{8*(3337458)-(4934)^2}\\ &= \frac{24041.4}{2355308}\\ &= 0.0102. \end{aligned} $$

The estimate of intercept is

$$ \begin{aligned} b_0&=\overline{y}-b_1\overline{x}\\ &=4.6275-(0.01)*616.75\\ &=-1.6634. \end{aligned} $$

The best fitted simple linear regression model to predict gross earnings from no. of theaters is

$$ \begin{aligned} \hat{y} &= -1.6634+ (0.0102)*x \end{aligned} $$

The estimate of the amount gross earnings when the no. of theaters is $610$ is

$$ \begin{aligned} \hat{y}&=-1.6634 + (0.0102)\times 610\\ &= 4.5586\quad \text{ millions of dollars } \end{aligned} $$