Lines of Regression MCQ Quiz - Objective Question with Answer for Lines of Regression - Download Free PDF

Explanation:

Lines of regression of x on y

⇒x - 3y + 4 = 0

⇒x = 3y – 4

⇒ b_xy = 3

Line of regression of y on x

⇒ 2x – 7y+8 = 0

⇒ y =

⇒ b_yx = 2/7

Now

⇒ b _xy + 7b _yx  =  3 + 7× 2 /7 = 5

∴ Option (d) is correct

The correct answer is "Dependent variable".

Key Points

• Regression Analysis-
  • Regression analysis is a statistical technique for modeling and investigating the relationship between two or more variables, say a dependent variable, and one or more independent variables.
  • The variable researchers are trying to explain or predict is called the response variable or, "dependent variable" because it depends on another variable.
  • The earliest form of regression was the least squares method, published by Legendre in 1805, and by Gauss in 1809.
  • The term "regression" was coined by Francis Galton in the 19th century to describe a biological phenomenon.
• Uses of Regression Analysis-
  • Regression analysis is widely used for prediction and forecasting.
  • In some situations, regression analysis can be used to infer causal relationships between the independent and dependent variables.

Additional Information

• ​Dependent Variable-
  • ​Dependent variables depend, by some law or rule, on the values of other variables.
  • Depending on the context, a dependent variable is sometimes called a "response variable", "regressand", "criterion", "predicted variable", "measured variable", "explained variable", "experimental variable", "responding variable", "outcome variable", "output variable", "target" or "label".
• ​Independent Variable-
  • The independent variables are those that predict or forecast the values of the dependent variable in the model.
  • ​Independent variables are not seen as depending on any other variable in the scope of the experiment.
  • The term "predictor variable",  "explanatory variable",  'predictors', 'covariates' or 'features' is used by authors for the independent variable.

The correct answer is "Dependent variable".

Key Points

• Regression Analysis-
  • Regression analysis is a statistical technique for modeling and investigating the relationship between two or more variables, say a dependent variable, and one or more independent variables.
  • The variable researchers are trying to explain or predict is called the response variable or, "dependent variable" because it depends on another variable.
  • The earliest form of regression was the least squares method, published by Legendre in 1805, and by Gauss in 1809.
  • The term "regression" was coined by Francis Galton in the 19th century to describe a biological phenomenon.
• Uses of Regression Analysis-
  • Regression analysis is widely used for prediction and forecasting.
  • In some situations, regression analysis can be used to infer causal relationships between the independent and dependent variables.

Additional Information

• ​Dependent Variable-
  • ​Dependent variables depend, by some law or rule, on the values of other variables.
  • Depending on the context, a dependent variable is sometimes called a "response variable", "regressand", "criterion", "predicted variable", "measured variable", "explained variable", "experimental variable", "responding variable", "outcome variable", "output variable", "target" or "label".
• ​Independent Variable-
  • The independent variables are those that predict or forecast the values of the dependent variable in the model.
  • ​Independent variables are not seen as depending on any other variable in the scope of the experiment.
  • The term "predictor variable",  "explanatory variable",  'predictors', 'covariates' or 'features' is used by authors for the independent variable.

Given:

Let x + 2y + 1 = 0 and 2x + 3y + 4 = 0 be two lines of regression. If θ is the acute angle between them.

Concept Used:

The angle θ between two lines given by and is given by:

Where m₁ and m₂ are the slopes of the lines.

Explanation:

For line x + 2y + 1 = 0 :

⇒ 

For line 2x + 3y + 4 = 0 :

y = 



=   

Using the triple angle formula for tangent 

Substitute tan θ  = 1/8 

=  =  = 

Now we multiply this value by 488:

488 tan(3θ) = 488 x   = 191

Thus, the value of 488 tan(3θ) is 191.

Concept:

The regressing line of y on x is given by y - y̅ = byx(x - x̅), where

The regressing line of x on y is given by x - x̅ = bxy(y - y̅), where

Calculation:

Here, n = 4

∴ 

= 

= 

x̅ =  = 0

y̅ =  =  = 2.75

∴ Required line, y - y̅ = b_yx(x - x̅)

⇒ y - 2.75 = (x - 0)

⇒ y = 2.75 + 

Comparing with y = a + bx, a = 2.75 and b = 

⇒ 2a + 15b = 5.5 + 5.5 = 11

∴ The value of 2a + 15b is 11.

The correct answer is Option 2.

Concept:

Regression equation: Let x and y be two variables, then equation is given as:

Calculation:

Given: x̅ = 10, y̅ = 90, σ_x = 3, σ_y = 12 and r_xy = 0.8

So, regression equation is given as,

⇒ x - 10 = 0.2 × {y - 90}

⇒ x = 0.2y - 18 + 10

⇒ x = 0.2y - 8

⇒ X = 0.2Y - 8 

CONCEPT:

Where  = standard deviation of x;  = standard deviation of y

Regression Line y on x is given as 

CALCULATION:

Regression Line y on x is given as 

y – 10 = 0.25(x - 20)

y – 10 = 0.25x - 5

y = 0.25x – 5 + 10

y = 0.25x + 5

Concept:

Equation of regression of Y on X:

The equation of regression of Y on X with given values of  is given as follows:

Calculation:

It is given that X̅ = 65, Y̅ = 67, σX = 2.5, σY = 3.5 and r(X, Y) = 0.8 .

Using the equation of regression of Y on X we write:

Therefore, the required equation is y = 1.12x - 5.8

Concept:

Coefficient of correlation = 

Where, b_yx and b_xy are regression coefficients

or the slopes of the equation y on x and x on y are denoted as b_yx and b_xy

Standard deviation = 

,  and are the standard deviation of y and x series respectively.

Calculation:

Here, The two lines of regression are 8x - 10y = 66 and 40x - 18y = 214

⇒ 10y = 8x - 66

⇒ byx = 8/10 = 4/5

And, 

40x - 18y = 214

⇒ 40x =18y + 214

⇒ bxy = 18/40 = 9/20

Now, Coefficient of correlation = 

= 

bxy  > 0 and byx > 0.

So, r = 3/5

Here, variance of x series is 9

⇒ Standard deviation of x series is

 = √9 = 3

We know, 

So, 

= 4

Hence, option (2) is correct.

Concept:

The two lines of regression coincide and both pass through the common point (x̅, y̅). 

Where (x̅, y̅) is the point of intersection of the two regression lines.

Calculation:

Given two regression lines are

x - y + 1 = 0               .... (1)

2x - y + 4 = 0             .... (2)

Subtracting equation (2) from (1), we get

⇒ (2x - y + 4) - (x - y + 1) = 0 - 0

⇒ x + 3 = 0

∴ x = -3

Put the value of x in equation (1), we get

⇒ -3 - y + 1 = 0

∴ y = -2

Hence two regression lines pass through the point: (-3, -2)

Concept:

The correlation coefficient  is equal to the geometric mean of two regression coefficients.

The positive and negative sign of correlation coefficient depends on the sign of regression coefficients. 

Formula used:

where  are regression coefficients and r is correlation coefficient.

Calculation:

We have,

So, correlation coefficient is given as

But, sign of regression coefficient is negative so sign of correlation coefficient is also negative.

Hence, The correlation coefficient of two regression coefficient -0.1 and -0.9 is -0.3.

Concept:

Regression variable tells about the degree of dependence of one variable on the other 

Calculation:

When the regression line is linear: y = ax + b

This tells about dependency of variables and not the linear relationship

So, statement (1) is not correct but (2) is correct.

Hence, option (2) is correct.

CONCEPT:

byx = slope of the regression line of y on x , and calculated as –

Regression Line y on x is given as 

Calculation:

Given:

 , n = 5

Important Points Linear regression analysis:

• Linear regression analysis is used to predict the value of a variable based on the value of another variable.
• The variable you want to predict is called the dependent variable.
• The variable you are using to predict the other variable's value is called the independent variable.
• This form of analysis estimates the coefficients of the linear equation, involving one or more independent variables that best predict the value of the dependent variable.
• Linear regression fits a straight line or surface that minimizes the discrepancies between predicted and actual output values.
• There are simple linear regression calculators that use a “least squares” method to discover the best-fit line for a set of paired data.
• You then estimate the value of X (dependent variable) from Y (independent variable).

 , 

b = 1.50 

(Y = a + bX)

Putting the values in the equation, 

3.50 = a + (1.50*5.50)

or, 3.50 = a + 8.25

or, a = -4.75

Hence, the value of 'a' of the model is -4.75

The correct answer is: y = -0.8x + 5.5

 Key Points

• Regression line represents the best fit line for the given data points, which means that it describes the relationship between X and Y as accurately as possible.
• The slope of the regression line (b) represents the change in Y for a unit change in X, and the y-intercept (a) represents the value of Y when X is equal to 0.
• The regression line is used in various fields such as finance, economics, and engineering to model and make predictions based on the relationship between two variables.
• It is important to note that the regression line is not always the exact representation of the relationship between X and Y, but it is the closest possible approximation.
• There are different types of regression lines, including simple linear regression (one independent variable), multiple linear regression (more than one independent variable), and non-linear regression (a non-linear relationship between X and Y).

 Additional Information

• The regression line is used to describe the relationship between two variables X and Y.
• It can be found using the formula:
• where b is the slope and a is the y-intercept. Substituting the given values in the formula, we can find the equation of the regression line.