Question

Linear Regression and Correlation.

x	y
2	4.64
3	6.76
4	3.08
5	5.5
6	-5.88
7	1.04
8	0.56
9	-2.42

Compute the equation of the linear regression line in the form y = mx + b, where m is the slope and b is the intercept.

Use at least 3 decimal places. (Round if necessary)
y =_____ x + _____

Compute the correlation coeficient for this data set. Use at least 3 decimal places. (Round if necessary)
r= ____

Compute the P-value (Use H_A: slope ≠ 0 for the alternative hypothesis.)

Use at least 3 decimal places. (Round if necessary)
P-value = ____

At the alpha = 0.05 significance level, is the correlation significant?

• No
• Yes, significant correlation

Answer 1

The following data are passed:

X	Y
2	4.64
3	6.76
4	3.08
5	5.5
6	-5.88
7	1.04
8	0.56
9	-2.42

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
	2	4.64	9.28	4	21.5296
	3	6.76	20.28	9	45.6976
	4	3.08	12.32	16	9.4864
	5	5.5	27.5	25	30.25
	6	-5.88	-35.28	36	34.5744
	7	1.04	7.28	49	1.0816
	8	0.56	4.48	64	0.3136
	9	-2.42	-21.78	81	5.8564
Sum =	44	13.28	24.08	284	148.7896

Based on the above table, the following is calculated:

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

Therefore, we find that the regression equation is:

Y = 8.0714 - 1.1657 X

Therefore, based on this information, the sample correlation coefficient is computed as follows

which completes the calculation.

The following needs to be tested:

The sample size is n = 8 , so then the number of degrees of freedom is df = n-2 = 8 - 2 = 6

The corresponding t-statistic to test for the significance of the correlation is:

The p-value is computed as follows:

Since we have that p = 0.0685 ≥0.05, it is concluded that the null hypothesis H0​ is not rejected.

Therefore, based on the sample correlation provided, it is concluded that there is not enough evidence to claim that the population correlation ρ is different than 0, at the 0.05 significance level.