Test: Correlation Analysis - Civil Engineering (CE) MCQ

Given

Multiple correlation coefficient = R_1.23

Explanation

For multiple correlation coefficient,

⇒ R_1.23 ≥ r₁₂ × r₁₃ × r₂₃

From given expression we can say that

∴ The multiple correlation coefficient R_1,23 as compared to any simple correlation coefficients between the distinct variable X₁ ,X₂, and X₃ is not less than any r₁₂, r₁₃ and r₂₃

Given

r₁₂ = r₁₃ = r₂₃ = r,

Formula

r_12.3 = (r₁₂ - r₁₃.r₂₃)/√[(1 - r₁₂²)(1 - r₂₃²)

Calculation

According to question

⇒ (r - r × r)/√(1 - r²)(1 - r²)

⇒ (r - r²)/(1 - r²)

⇒ r(1 - r)/(1 - r)(1 + r)

⇒ The value of r_12.3 is r/(1 + r)

The simplest device for ascertaining whether two variables are related is to prepare a dot chart called scatter diagram. When this method is used the given data are plotted on a graph paper in the form of dots.

∴ If a data set contains n paired values on two variables x(independent) and y(dependent), then their plot is called Scatter diagram.

The regression line of Y on X is Y = 30 - 0.9x

⇒ Y - 30 = - 0.9x

The regression equation line of Y on X is = y - y₁ = r(s_y/s_x)(x - x₁)

Comparing both equations, we get

⇒ r(s_y/s_x) = -0.9

⇒ r(9/2) = -0.9

⇒  r = (-0/9 × 2)/9 = - 0.2

∴ The value of the correlation coefficient rxy is -0.2

Given

The correlation coefficient between two variables X and Y = 0.4

Concept used

The correlation coefficient (r) is independent of origin and scale and depend on the sign of variables

Calculation

The correlation coefficient between the two variables is the measure of the slope between the variables in the regression graph. It is given that the correlation coefficient between X and Y is 0.4 and the correlation coefficient is independent of change of origin and scale but it depends on variables 

∴ The correlation coefficient between 2X and (-Y) is - 0.4

Important Points: 

The value of simple correlation coefficient in the interval of [-1, 1]

The regression coefficient is independent of the change of origin. But, they are not independent of the change of the scale. It means there will be no effect on the regression coefficient if any constant is subtracted from the values of x and y

Key Points

• The correlation coefficient only measures linear relationships, so it does not necessarily indicate causality.
• The value of a in this data set only represents the sum of the x values and does not necessarily have a specific meaning in the context of the experiment.
•  The value of a in the data set is 7.
• The correlation coefficient of -0.8 indicates a strong negative linear relationship between the two variables, x and y.
• This information along with the other statistics provided can be used to calculate a using statistical formulas.

Additional Information 

• The correlation coefficient (r) measures the strength and direction of a linear relationship between two variables.
• A correlation of -1 indicates a perfect negative linear relationship, while a correlation of 1 indicates a perfect positive linear relationship.
• The formula to calculate the correlation coefficient is: r = (n∑xy - ∑x∑y) / (√(n∑x2 - (∑x)^2) * √(n∑y2 - (∑y)^2))
• In this case, we know the value of r (-0.8), so we can rearrange the formula and solve for a.

Concept:

Coin flipping, coin tossing, or heads or tails is the practice of throwing a coin in the air and checking which side is showing when it lands, in order to choose between two alternatives, heads or tails, sometimes used to resolve a dispute between two parties. It is a form of sortition which inherently has two possible outcomes. The party that calls the side that is facing up when the coin lands win.

Calculation:

Let X be the no. of heads and Y be the no. of tails.

Given:

X + Y = x

E[X + Y] = E[x]

∴ E(X) + E(Y) = E(x)

or, X - E(X) + Y - E(Y) = 0

∴ X - E(X) = - (Y - E(Y))

Cov (X,Y) = E[(X - E(X))(Y - E(Y))]

= - E [X - E(X)]²

= - var(X)

Also, var(X) = var(Y)

The correct answer is option (a).

Key Points 

A - Interpolation (II - Newton):
Interpolation is a statistical topic that involves estimating values within a known set of data points. Newton's method is a commonly used method for interpolation. It uses polynomial interpolation to approximate the values between data points based on the differences or changes in the data.

B - Regression coefficients (I - Probabilistic):
Regression analysis is a statistical topic that examines the relationship between a dependent variable and one or more independent variables. Regression coefficients are the estimated parameters that quantify the relationship between the variables in the regression model. The estimation of these coefficients is done using probabilistic methods, such as ordinary least squares (OLS) regression, maximum likelihood estimation (MLE), or Bayesian regression.

C - Sampling (IV - Deviations):
Sampling is a statistical topic that involves selecting a subset of individuals or units from a larger population to gather information and make inferences about the entire population. Deviations, in this context, refer to the differences or variations observed between the sample and the population. By studying and analyzing these deviations, statisticians can draw conclusions about the characteristics of the population based on the sample.

D - Correlation Analysis (III - Scatter Diagram):
Correlation analysis is a statistical topic that examines the relationship between two variables. It measures the strength and direction of the association between variables. One common method to visualize and analyze the correlation between two variables is by using a scatter diagram. Scatter diagrams plot the values of one variable against the values of another variable, allowing for visual inspection of their relationship and the calculation of correlation coefficients.

Given

σ_x = √16 = 4

r = 0.48

Covariance = ∑xy/N = 36

Fomula

Covariance = ∑xy/N

r = ∑xy/N.σ_x × σ_y

Calculation

According to question

⇒ 0.48 = 36/4 × σ_y

⇒ σ_y = 9/0.48

∴ The standard deviation of y(σ_y) is 18.75

The general formula for multiple correlation coefficient

A multiple correlation coefficient is a non-negative coefficient.

It is the value ranges between 0 and 1. It cannot assume a minus value.

If the multiple correlation coefficient of X₁ on X₂ and X₃ is zero, then:

If R_1.23 = 0 then r₁₂ = 0 and r₁₃ = 0

R_1.23 is the same as R_1.32