Math Problem Statement
In a statistical study relating to the prices of two shares, X and Y, the following two regression lines were found, where the share prices were expressed ni rupees: 8X- 10Y+70=0 20X - 9Y - 65 = 0 The standard deviation of the prices of share X is 3. You are required to calculate: (i)Average prices of shares Xand Y (11)Coefficient of correlation between the prices of shares Xand Y.
Solution
To solve this problem, we need to follow these steps for the two parts of the question:
(i) Average Prices of Shares X and Y
The regression equations are given as:
The general form of the regression equation is , where the averages of and (denoted as and ) satisfy both regression equations.
Step 1: Find the average prices by solving the two equations simultaneously.
From the first equation: From the second equation:
Now, we solve these two equations for and .
Step 2: Use substitution or elimination.
Multiply the first equation by 2.5 to align the coefficients of : Now, subtract the second equation from this result: Simplifying: Thus:
Step 3: Substitute into one of the original equations to find .
Substitute into :
Thus, the average prices of the shares are:
(ii) Coefficient of Correlation Between X and Y
We are given one standard deviation , but we need to determine the coefficient of correlation .
The formula for the coefficient of correlation from the regression coefficients is: Where:
- is the regression coefficient of on , and
- is the regression coefficient of on .
Step 1: Find the slopes (regression coefficients) from the given equations.
From the regression equation , rearranging gives:
From the regression equation , rearranging gives:
Step 2: Apply the formula for correlation.
Now, calculate the correlation:
However, the correlation coefficient must lie between -1 and 1, so we choose the negative value:
Hence, the correlation coefficient is: Note that this value of correlation may suggest an inconsistency, as correlation coefficients should always be between -1 and 1. Therefore, we might need to double-check the data or assumptions in practice.
Would you like more details or have any further questions?
5 Related Questions:
- What are the assumptions behind using linear regression in statistics?
- How do we interpret the sign of the correlation coefficient?
- What is the relationship between regression coefficients and correlation?
- How do changes in the slope affect the regression line interpretation?
- Can you explain how to find standard deviations from regression data?
Tip:
Always check that your correlation coefficient lies between -1 and 1, as values outside this range indicate calculation or model errors.
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Math Problem Analysis
Mathematical Concepts
Regression Analysis
Correlation
Simultaneous Equations
Formulas
Regression equation: aX + bY + c = 0
Coefficient of correlation: r = ±√(b_YX * b_XY)
Theorems
Linear regression equation properties
Correlation formula for regression coefficients
Suitable Grade Level
Grades 11-12 or Undergraduate
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