Question

find the equation for the least squares regression line of the data described below. a reporter for a college magazine is writing a feature about student participation in on - campus organizations. he wants to see if there is a relationship between student population and the number of on - campus organizations. to begin, he recorded the number of students, x, and the number of officially recognized student organizations, y, for several colleges. students student organizations 2,079 343 3,597 260 3,749 243 4,374 218 5,518 408 6,478 487 7,725 494 round your answers to the nearest thousandth. y = x +

Explanation:

Step1: Calculate sums

Let \(n = 7\) (number of data - points).
Calculate \(\sum_{i = 1}^{n}x_{i}=2079 + 3597+3749+4374+5518+6478+7725=33520\)
\(\sum_{i = 1}^{n}y_{i}=343 + 260+243+218+408+487+494 = 2453\)
\(\sum_{i = 1}^{n}x_{i}^{2}=2079^{2}+3597^{2}+3749^{2}+4374^{2}+5518^{2}+6478^{2}+7725^{2}\)
\(=4322241+12938409+14055001+19131876+30448324+41964484+59675625 = 182536960\)
\(\sum_{i = 1}^{n}x_{i}y_{i}=2079\times343+3597\times260+3749\times243+4374\times218+5518\times408+6478\times487+7725\times494\)
\(=713097+935220+909007+953532+2251344+3154786+3816150=12743136\)

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the least - squares regression line is \(m=\frac{n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}}{n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}\)
Substitute \(n = 7\), \(\sum_{i = 1}^{n}x_{i}=33520\), \(\sum_{i = 1}^{n}y_{i}=2453\), \(\sum_{i = 1}^{n}x_{i}^{2}=182536960\) and \(\sum_{i = 1}^{n}x_{i}y_{i}=12743136\) into the formula:
\[

$$\begin{align*} m&=\frac{7\times12743136-33520\times2453}{7\times182536960 - 33520^{2}}\\ &=\frac{89201952-82224560}{1277758720 - 1123590400}\\ &=\frac{6977392}{154168320}\\ &\approx0.045 \end{align*}$$

\]

Step3: Calculate intercept \(b\)

The formula for the intercept \(b\) is \(b=\bar{y}-m\bar{x}\), where \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}=\frac{33520}{7}\approx4788.571\) and \(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}=\frac{2453}{7}\approx350.429\)
\(b = 350.429-0.045\times4788.571\)
\(=350.429 - 215.486\)
\(=134.943\)

Answer:

\(y = 0.045x+134.943\)