Question

1. Use the following data to answer the questions below: 259 208 287 a. Fit a regression line to the data. b. Test the null hypothesis that the proportion of the variability in y explained by x is zero.

Use the data to fit the regression line.

Answer 1

The following data are passed:

X	Y
286	77
252	62
241	73
264	74
260	79
227	68
233	74
259	74
208	61
287	76

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

	X	Y	X*Y	X2	Y2
	286	77	22022	81796	5929
	252	62	15624	63504	3844
	241	73	17593	58081	5329
	264	74	19536	69696	5476
	260	79	20540	67600	6241
	227	68	15436	51529	4624
	233	74	17242	54289	5476
	259	74	19166	67081	5476
	208	61	12688	43264	3721
	287	76	21812	82369	5776
Sum =	2517	718	181659	639209	51892

Based on the above table, the following is calculated:

$\bar X = \frac{1}{n} \sum_{i=1}^{n} X_i = \frac{ 2517}{ 10} = 251.7$

$\bar Y = \frac{1}{n} \sum_{i=1}^{n} Y_i = \frac{ 718}{ 10} = 71.8$

$SS_{XX} = \sum_{i=1}^{n} X_i^2 - \displaystyle\frac{1}{n}\left(\sum_{i=1}^{n} X_i\right)^2 = 639209 - 2517^2/10 = 5680.1$

$SS_{YY} = \sum_{i=1}^{n} Y_i^2 - \displaystyle\frac{1}{n}\left(\sum_{i=1}^{n} Y_i\right)^2 = 51892 - 718^2/10 = 339.6$

$SS_{XY} = \sum_{i=1}^{n} X_i Y_i - \displaystyle\frac{1}{n}\left(\sum_{i=1}^{n} X_i\right) \left(\sum_{i=1}^{n} Y_i\right) = 181659 - 2517 \times 718/10 = 938.399$

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

$m = \frac{SS_{XY}}{SS_{XX}}$

$m = \frac{ 938.399}{ 5680.1}$

m = 0.1652

$n = \bar Y - \bar X \cdot m$

$n = 71.8 - 251.7 \times 0.1652$

n = 30.2171

Therefore, we find that the regression equation is:

Y^ = 30.2171 + 0.1652 X