Correlation And Regression

Step 1: Understanding the Concept:
This question deals with the effect of change of origin and scale on the regression coefficient. The variables X and Y are transformed into U and V. We need to find the relationship between the regression coefficient of U on V ( ) and the regression coefficient of X on Y ( ).

Step 2: Key Formula or Approach:
The regression coefficient of X on Y is given by . Properties to use: 1. Correlation coefficient is independent of change of origin and scale: . 2. Effect of transformation on standard deviation: .

Step 3: Detailed Explanation:
The transformations are and . We need to find , the regression coefficient of U on V. The formula for is: Let's find the components in terms of X and Y. 1. The correlation coefficient is invariant, so . 2. The standard deviation of U is: 3. The standard deviation of V is: Now, substitute these into the formula for : Since , we can substitute this into the equation:
Step 4: Final Answer:
The regression coefficient is .

Step 1: Understanding the Concept:
The two lines of regression (Y on X and X on Y) intersect at the point . The angle ( ) between these two lines depends on the correlation coefficient and the standard deviations of the variables.

Step 2: Key Formula or Approach:
The acute angle between the two regression lines is given by the formula: where is the correlation coefficient, and are the standard deviations.

Step 3: Detailed Explanation:
We are given the following values: - Correlation coefficient, - Standard deviation of X, - Standard deviation of Y, First, calculate the components of the formula: - - - - Now, substitute these values into the formula for : Converting the fraction to a decimal: Therefore, the angle is:
Step 4: Final Answer:
The angle between the two lines of regression is .

Step 1: Understanding the Concept:
The correlation coefficient, , can be determined from the two regression lines. One line represents the regression of Y on X, and the other represents the regression of X on Y. The slopes of these lines are the regression coefficients, and . The square of the correlation coefficient is the product of these two regression coefficients.

Step 2: Key Formula or Approach:
The correlation coefficient is given by the geometric mean of the regression coefficients: The sign of is the same as the sign of the regression coefficients.

Step 3: Detailed Explanation:
Let's assume the first equation is the regression of Y on X and the second is the regression of X on Y. Line 1 (Y on X): Rearrange to the form : So, the regression coefficient of Y on X is . Line 2 (X on Y): Rearrange to the form : So, the regression coefficient of X on Y is . Now, calculate the square of the correlation coefficient: The correlation coefficient is the square root of this value. Since both and are positive, must also be positive. Note: Our assumption was correct because . If we had assumed the opposite, we would get and , whose product is greater than 1, which is impossible.
Step 4: Final Answer:
The correlation coefficient between X and Y is 0.6.

Step 1: Understanding the Concept:
The question asks for the "regression coefficient of Mumbai prices over Kolkata prices". This phrasing means we are predicting Mumbai prices based on Kolkata prices. Therefore, Mumbai price is the dependent variable (Y) and Kolkata price is the independent variable (X). We need to find the regression coefficient .

Step 2: Key Formula or Approach:
The formula for the regression coefficient of Y on X is: where is the correlation coefficient, is the standard deviation of Y, and is the standard deviation of X.

Step 3: Detailed Explanation:
From the table, we identify the necessary values: - Dependent variable Y: Mumbai prices. So, . - Independent variable X: Kolkata prices. So, . - Correlation coefficient, . Now, substitute these values into the formula: The average prices and the sample size are extra information not needed for this specific calculation.
Step 4: Final Answer:
The regression coefficient of Mumbai prices over Kolkata prices is 0.48.