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Correct option is (c) Regression line of sales on advertisement cost

The linear regression model is of the form Y = α + βX + u; where α and β are constants, u is error variable.

A mathematical or functional relationship between two correlated variables which helps in estimating the value of dependent variable for some given value of independent variable is called Linear Regression.

A regression model expressing the dependent variable Y as a linear function of independent variable X is called a

Linear regression model. It is of the following form :

Y = α + βx + u
Where, Y = Dependent variable
X = Independent variable α,
β = Constants
u = Disturbance variable of model

Variable u shows the incompletness of Linear correlation between two variables X and Y.

• If there is perfect linear correlation between two variables X and Y, then the linear regression model is of the form
  Y = α + β X. In such case the value of disturbance variable u will be 0 (zero).
• If there is partial linear correlation between two variables X and Y, the form of linear regression- model is
  Y = α +βx + u.

\(b = \frac{nΣxy−(Σx)(Σy)}{nΣx^2−(Σx)^2}\)

Correct option is (b) Regression

In the regression line ŷ = a + bx, a is called the intercept of the regression line of Y on X and b is called the regression coefficient of the regression line of Y on X. It is also called the slope of the regression line of Y on X.

If the values of both the variable are doubled then the value of the regression coefficient will not change, because C_x = \(\frac{1}{2}\) and C_y = \(\frac{1}{2}\).

ŷ = 11 + 3x

∴ b = 3

S_x : S_y = 3 : 10,

∴ S_x = 3, S_y = 10.

\(Now, b = r . \frac{S_y} {S_x}\)

∴ 3 = r × \(\frac{10}{3}\)

∴ r = 3 × \(\frac{3}{10}=\frac{9}{10}\)= 0.9

Now, coefficient of determination

R² = (r)²

∴ R² = (0.9)²

= 0.81

ŷ = 31.5 + 1.85x

Putting x = 10 in ŷ = 31.5 + 1.85 x, we get

ŷ = 31.5 + 1.85 (10)

= 31.5 + 18.5 = 50

Hence, the estimate of Y for X = 10 obtained is 50.

X̄ = 30, ȳ = 20, b = 0.6

Intercept a = ȳ – bx̄

∴ a = 20 – 0.6(30) = 20 – 18 = 2

Putting a = 2 and b = 0.6 in ŷ = a + bx,

the equation of the regression line obtained is ŷ = 2 + 0.6x.

r = 0.5, S_x = 2, S_y = 4

Now, b_yx = r ∙ \(\frac{S_y}{S_x}\);

putting the values given.

∴ b_yx = 0.5 × \(\frac{4}{2}\) = 0.5 × 2 = 1

b_yx = 5 means a unit increase in the value of variable X results in estimated increase of 5 units in the value of variable Y.

Here, n = 8;

x̄ = 100; ȳ = 100;

Σ(x – x̄)² = 130;

Σ(y – ȳ)² = 145 and

Σ(x – x̄) (y – ȳ) =115 are given.

\(Now, b = \frac{Σ(x−\bar x)(y−\bar y)}{Σ(x−\bar x)^2}\)

\(∴ b = \frac{115}{130}\)

= 0.88

a = ȳ – bx̄
∴ a = 100 – 0.88 (100)
= 100 – 88
= 12

Regression line of Y on X is ŷ = a + bx

Putting a = 12 and b = 0.88 in ŷ = a + bx,

the regression line of Y on X obtained is ŷ = 12 + 0.88x.

Here, X = Rainfall and Y = Yield of crop
∴ x̄ = 18 cm;

ȳ = 970 kg,

S_x = 2,

S_y = 38,

r = 0.6.

\(Now, b = r .\frac{S_y}{S_x}\)

∴ b = 0.6 × \(\frac{38}{2}\)

= 11.4

a = ȳ – bx̄
∴ a = 970 – 11.4 (18)
= 970 – 205.2
= 764.8

The regression line of yield of crop (Y) on rainfall (X) :

Putting a = 764.8 and b = 11.4 in ŷ = a + bx, we get
ŷ = 764.8 + 11.4x

Estimate of yield of crop (Y) if rainfall X = 20 cms :

Putting x = 20 in ŷ = 764.8 + 11.4x, we get
ŷ = 764.8 + 11.4 (20)
= 764.8 + 228
= 992.8

Hence, the estimate of yield of crop obtained is 992.8 kg.

The square of correlation coefficient between the observed value y of the dependent variable Y and
the corresponding estimated value y of Y by the regression line y = a + bx is called the coefficient of determination. It is denoted by R².

Thus, R² = [Cor (y, ŷ)]²
= [Cor (y, a + bx)]²
= [Cor (y, x)]²

• If R² = 1, estimates obtained on the basis of regression line are 100% reliable. There is perfect linear correlation between the variable Y and X.
• If R² = 0 estimates obtained on the basis of regression line are not reliable. There is lack of linear correlation between the variables Y and X.
• If R² is near to 1 (i.e., 0.5 ≤ R² < 1), then the assumption of linear regression is said to be proper and estimates obtained by regression line are reliable,
• If R² is near to ‘0’ (i.e., 0 ≤ R² < 0.5), then the assumption of linear regression is said to be improper and the estimates obtained by regression line are not reliable.

The following precautions are necessary while using the regression:

• We should make the use of the estimates obtained only after examining the assumption of linear regression by coefficient of determination R².
• The regression relation obtained by the scatter diagram or by the least squares should not be used for the values which are very far from the given values of independent variable.
• If the correlation between Y and X is found near to perfect linear correlation then only the conclusions drawn from the regression line should be used.
• If the linear regression model is not found proper, then in such a case only after obtaining the proper linear regression model using statistical methods, we should make use of estimates and conclusions from the regression model.

The square of correlation coefficient between the observed value y of the dependent variable Y and the corresponding estimated value ŷ of y by the regression line ŷ = a + bx is called the coefficient of determination. It Is denoted by R².

Thus, R² = [Coy (y, ŷ)]²

= [Cov(y, a + bx)]²

= [Cov(y, x)]²

= r²

Uses:
(i) To determine the reliability of estimates obtained from the regression lines:

• If R² = 1, estimates obtained on the basis of regression line are 100 % reliable. There is perfect linear correlation between the variable Y and X.
• If R² = 0 estimates obtained on the basis of regression line are not reliable. There is lack of linear correlation between the variables Y and X.

(ii) For the interpretation regarding the assumption of linear regression between two random variables X and Y:

• If the value of R² obtained is close to 1 (i.e., 0.5 ≤ R² < 1), then it can be said that correlation between variables Y and X is close to the perfect linear correlation and hence it is said that the assumption of linear regression between variables X and Y is proper.
• If the value of R² obtained is close to zero (0) (i.e., 0 ≤ R² < 0.5) then it can be said that the correlation between variables Y and X is far from the perfect linear correlation and hence it is said that the assumption of linear regression between variables X and Y is not proper.

The properties of regression coefficient are as follows :

• The signs of correlation coefficient r and regression coefficient b are same.
• Value of regression coefficient b is independent of change of origin but not of scale.
• From regression coefficient the estimated change in the value of Y, per unit change in the value of X can be known.
• The value of b can be less than 1 or greater than 1.
• The regression coefficient is a relative measure.
• Sign of ‘b’ depends on the sign of Cov(x, y).
• If b > 0, a unit increase in X, implies an estimated increase of b units in Y.
• If b < 0, a unit increase in X, implies an estimated decrease of b units in Y.

Suppose n observations of a data are (x₁, y₁), (x₂, y₂), …….. (x_n, y_n) based on a sample drawn from a bivariate population. From this data we have to fit the line of regression of Y on X. First of all, we will draw scatter diagram by plotting the n pairs of observations (x₁, y₁), (x₂, y₂), …….. (x_n, y_n). We will draw a line in such a way that it is close to almost all the points of the scatter diagram and gives the best estimate of relationship between Y and X.

Such line of best fit is called the regression line of, Y on X. We can find the equation of the regression line by choosing any two points on the line. This equation is called the regression line. This method to fit the line of regression is easy and quick and does not require any computation.

Limitations:

• The regression line obtained by this method does not give the best estimate of relation between Y and X.
• In this method, from the same data, different persons can select different points and get the different regression lines. Hence, using them the conclusions derived are different.
• This method is not objective but it is rather subjective in sense.

The utility of regression can be described as follows :

• We can know the functional relationship between two correlated variables.
• We can estimate the unknown value of Y, for a given value of independent value of X.
• We can find the estimated change in Y, per unit change in X with the help of regression.
• We can determine the error occured in finding the estimated value of dependent variable by using the regression line.
•  It is useful for economists, planners, businessmen, administrators, researchers, etc.

The method to obtain the best fitted regression line is ‘Least square method’.