Calculating Regression Coefficient, Correlation Coefficient, and Regression Lines from Given Data

Given data:
X: 1, 2, 3, 4, 5
Y: 2, 5, 3, 8, 7

Regression Coefficient:
Regression coefficient (b) can be calculated using the following formula:
b = (n∑(XY) - ∑X∑Y) / (n∑(X^2) - (∑X)^2)
where,
n = number of observations
X = values of independent variable
Y = values of dependent variable

Using the given data, we can calculate the regression coefficient as follows:
n = 5
∑X = 15
∑Y = 25
∑(XY) = 87
∑(X^2) = 55

Substituting these values in the formula, we get:
b = (5(87) - (15)(25)) / (5(55) - (15)^2)
b = 1.4

Therefore, the regression coefficient for the given data is 1.4.

Correlation Coefficient:
Correlation coefficient (r) can be calculated using the following formula:
r = (n∑(XY) - ∑X∑Y) / sqrt((n∑(X^2) - (∑X)^2)(n∑(Y^2) - (∑Y)^2))
where,
n = number of observations
X = values of independent variable
Y = values of dependent variable

Using the given data, we can calculate the correlation coefficient as follows:
n = 5
∑X = 15
∑Y = 25
∑(XY) = 87
∑(X^2) = 55
∑(Y^2) = 135

Substituting these values in the formula, we get:
r = (5(87) - (15)(25)) / sqrt((5(55) - (15)^2)(5(135) - (25)^2))
r = 0.529

Therefore, the correlation coefficient for the given data is 0.529.

Regression Lines:
We can calculate the regression lines using the following formulas:
Y = a + bX
where,
a = Y-intercept of the regression line
b = Regression coefficient
X = values of independent variable
Y = values of dependent variable

For the regression line of Y on X, we can calculate a as follows:
a = (∑Y - b∑X) / n
a = (25 - (1.4)(15)) / 5
a = 0.6

Therefore, the regression line of Y on X is:
Y = 0.6 + 1.4X

For the regression line of X on Y, we can calculate a as follows:
a = (∑X - b∑Y) / n
a = (15 - (1.4)(25)) / 5
a = -2

Therefore, the regression line of X on Y is:
X = -2 + 0.714Y

In summary, the regression coefficient for the given data is 1.4, the correlation coefficient is 0.529, the regression line of Y on X is