Question

Question 2 (2 points) The data below gives the number of pages, x, and the cost, y, for a sample of six textbooks. Note that the data given in this problem is not necessarily the same as in the other problems. Find the equation of the regression line, y = a + bx, and correlation coefficient r for the data. Round all values to the nearest thousandth (to three decimal places). #pages, 1889416614 633 500 577 Cost, $189 S112 S152 S156 S123 S126 hp

Answer 1

The independent variable is Pages, and the dependent variable is Cost. In order to compute the regression coefficients, the following table needs to be used:

	Pages	Cost	Pages*Cost	Pages2	Cost2
	889	189	168021	790321	35721
	416	112	46592	173056	12544
	614	152	93328	376996	23104
	633	156	98748	400689	24336
	5005	123	615615	25050025	15129
	577	126	72702	332929	15876
Sum =	8134	858	1095006	27124016	126710

Based on the above table, the following is calculated:

$\bar X = \frac{1}{n} \sum_{i=1}^{n} X_i = \frac{ 8134}{ 6} = 1355.66$

$\bar Y = \frac{1}{n} \sum_{i=1}^{n} Y_i = \frac{ 858}{ 6} = 143$

$SS_{XX} = \sum_{i=1}^{n} X_i^2 - \displaystyle\frac{1}{n}\left(\sum_{i=1}^{n} X_i\right)^2 = 27124016 - 8134^2/6 = 16097023.33$

$SS_{YY} = \sum_{i=1}^{n} Y_i^2 - \displaystyle\frac{1}{n}\left(\sum_{i=1}^{n} Y_i\right)^2 = 126710 - 858^2/6 = 4016$

$SS_{XY} = \sum_{i=1}^{n} X_i Y_i - \displaystyle\frac{1}{n}\left(\sum_{i=1}^{n} X_i\right) \left(\sum_{i=1}^{n} Y_i\right) = 1095006 - 8134 \times 858/6 = -68156$

Therefore, based on the above calculations, the regression coefficients (the slope m, and the y-intercept n) are obtained as follows:

$m = \frac{SS_{XY}}{SS_{XX}} = \frac{ -68156}{ 16097023.333333} = -0.0042$

$n = \bar Y - \bar X \cdot m = 143 - 1355.6666666667 \times (-0.0042) = 148.74$

Therefore, we find that the regression equation is:

Cost = 148.74 - 0.0042 Pages

The sample correlation coefficient is computed as follows

$r = \frac{ SS_{XY}}{\sqrt{ SS_{XX} SS_{YY}}} = \frac{ -68156}{\sqrt{ 16097023.333 \times 4016}} = -0.268$

which completes the calculation.