For certain x and y series which are correlated, the two lines of regression are 5x-6y+9=0 , 15x-8y+130=0. The correlation coefficient is :
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No idea
To find the correlation coefficient, we can use the formula:
r = ± √[(nΣxy - ΣxΣy) / (√(nΣx^2 - (Σx)^2) * √(nΣy^2 - (Σy)^2))]
where:
n = number of data points
Σxy = sum of the product of x and y
Σx = sum of x
Σy = sum of y
Σx^2 = sum of squared x
Σy^2 = sum of squared y
In this case, the equations of the regression lines are:
1) 5x - 6y + 9 = 0
2) 15x - 8y + 130 = 0
Comparing these equations to the standard form of a linear equation (Ax + By + C = 0), we can determine the values of A, B, and C for the two lines:
1) Line 1: A = 5, B = -6, C = 9
2) Line 2: A = 15, B = -8, C = 130
Now we need to calculate the necessary sums and calculate the correlation coefficient. However, we would need the specific x and y values to perform these calculations. Please provide them so that I can assist you further.
To find the correlation coefficient for the given x and y series, we can use the formulas for the slopes of the regression lines, which are related to the correlation coefficient.
The equation of the first regression line is: 5x - 6y + 9 = 0.
Let's rearrange this equation in slope-intercept form (y = mx + b):
-6y = -5x - 9 => y = (5/6)x + (9/6) => y = (5/6)x + 3/2.
Comparing this with the general form of a linear equation (y = mx + b), we can see that the slope of the first regression line is 5/6.
Similarly, the equation of the second regression line is: 15x - 8y + 130 = 0.
Rearranging it in slope-intercept form gives us:
-8y = -15x - 130 => y = (15/8)x + (130/8) => y = (15/8)x + 65/4.
Again comparing this with the general form, we can see that the slope of the second regression line is 15/8.
The correlation coefficient (r) is related to the slopes of the regression lines by the formula:
r = sqrt[(m2^2) / (m1^2 + m2^2)], where m1 and m2 are the slopes of the regression lines.
Substituting the values of m1 = 5/6 and m2 = 15/8 into the formula, we have:
r = sqrt[((15/8)^2) / ((5/6)^2 + (15/8)^2)].
Calculating this further, we get:
r = sqrt[(225/64) / (25/36 + 225/64)].
Combining the fractions in the denominator, we get:
r = sqrt[(225/64) / (1600/2304 + 225/64)].
Further, simplifying the denominator, we have:
r = sqrt[(225/64) / (3850/2304)].
Dividing the numerator by the denominator, we get:
r = sqrt[(225/64) * (2304/3850)].
Multiplying these fractions, we have:
r = sqrt[(225 * 2304) / (64 * 3850)].
Calculating this further, we obtain:
r = sqrt[518400 / 246400].
Taking the square root, we get:
r = sqrt(2.1).
Therefore, the correlation coefficient is approximately 1.449.
Note: The correlation coefficient ranges from -1 to 1, where values closer to -1 indicate a strong negative correlation, values closer to 1 indicate a strong positive correlation, and a value of 0 indicates no correlation.