Question

Define and discuss "least-squares criterion" giving detailed computations for finding the equation of the least squares line and the coefficient of determination. What does coefficient of determination tell you about explained variation of y in a random sample of data pairs (x, y) ?

Answer 1

Solution: Here concept of simple linear regression is used.

Solution - Let x be a independent variable and y be a bondependent variable in the paired data (x,y). Let x,, 22, ---, an and Y., Z ., In be random sample of sizen from X and Y respec We want to establish a linear relationship between X and Y. Let y = atbx be the least square regression line between the care independent variable & and dependent variable . We shall obtain the slope of regression equation and y intercept with the help of principle of "bast square". aa Principle of least square consist of obtaining the slope of regression equation (6) and y-intercept (a) by minimizing the E= {(y; - a - bx;) E is known as som of squares due to error. Now, for minima of E DE = 0 = 2 2 (3;-a-62;) (-1) = 0 2 ( 4; - a - bx;) = 0 2x = na + bēx; - o 250 + 217;-a-bx;) (-xi) = 0 2 (x;y; - ax; - 62 ) = 0 2 (24) = a {x; + b3 x ® & Wi&WTEW 151

Equations and are known as normal equations. We shall solve ean and simultaneously in order to obtain the estimates of a and b. On solving equations and simultaneously we â = 7 - 6x 6= 2 (x; - x) (; }) I (2-)2 where, x = x; and j = 1 . Equation of least - square line is 2 = a + b = | i Coefficient of variation (² :- Coefficient of determination (R²) :- Coefficient of the determination tells us that what proportion of variation in y is explained by x. It is denoted by R² For example R² = 0.82 tells us that 82 t. of variations in y is explained by x. To compute R² we use following formula: ; where y, is actual value of y ŷ is predicted value of y R² = 1 - 1 (2 - 5 12 / 12 - 977) - J = 7 hy is sample mean of y.