The coefficient of correlation for the variables x and y is 0.9926 from following data :
`{:(X,18,20, 25, 30, 31, 32),(Y, 21, 22, 28, 32, 35, 36):}`
Find the change in the correlation coefficient if each value of x in multiplied b 3 and subtracted by 4 and each value of y is multiplied by 2 and increased by 3.

Answer

To solve the problem, we need to determine the change in the correlation coefficient when the values of \( x \) and \( y \) are transformed. The original correlation coefficient is given as \( r = 0.9926 \).

### Step 1: Understand the transformations
The transformations applied to the variables are:
- For \( x \): Multiply by 3 and subtract 4, which can be expressed as \( x' = 3x - 4 \).
- For \( y \): Multiply by 2 and add 3, which can be expressed as \( y' = 2y + 3 \).

### Step 2: Analyze the effect of transformations on correlation
The correlation coefficient \( r \) is invariant under linear transformations of the form \( ax + b \) and \( cy + d \). Specifically:
- If \( x \) is transformed to \( x' = ax + b \) and \( y \) is transformed to \( y' = cy + d \), the new correlation coefficient \( r' \) will be given by:
\[ r' = \frac{r \cdot a \cdot c}{|a| \cdot |c|} = r \]
This means that the correlation coefficient remains unchanged if both transformations are linear.

### Step 3: Identify the constants in the transformations
In our case:
- For \( x \): \( a = 3 \) (the multiplier), and \( b = -4 \) (the constant added).
- For \( y \): \( c = 2 \) (the multiplier), and \( d = 3 \) (the constant added).

### Step 4: Calculate the new correlation coefficient
Since both transformations are linear, the correlation coefficient will not change:
\[ r' = r = 0.9926 \]

### Step 5: Conclusion
Thus, the change in the correlation coefficient is:
\[ \Delta r = r' - r = 0.9926 - 0.9926 = 0 \]

### Final Answer
The change in the correlation coefficient is \( 0 \).

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