Question

The coefficient of correlation between x and y is 0.85. If $u = \frac{x+1.5}{12}$ and $v = \frac{y-2.4}{30}$ Then the correlation coefficient between u and v is

• A. -0.85
• B. 0.85
• C. $\frac{0.85-0.9}{6}$
• D. $\frac{0.85+0.9}{15}$

Hint:

• The correlation coefficient $r$ is a measure of linear association between two variables.
• Property of correlation coefficient under linear transformation: If $u = ax+b$ and $v = cy+d$, then $r_{uv} = r_{xy}$ if $ac>0$ (i.e., $a$ and $c$ have the same sign).
• $r_{uv} = -r_{xy}$ if $ac<0$ (i.e., $a$ and $c$ have opposite signs).
• The magnitude $|r_{uv}| = |r_{xy}|$.
• "Change of origin" (adding/subtracting constants $b,d$) does not affect $r$. "Change of scale" (multiplying by $a,c$) affects the sign of $r$ only if the signs of $a$ and $c$ are different.
• Here, $a=1/12>0$ and $c=1/30>0$. So $ac>0$, thus $r_{uv} = r_{xy}$.

Answer 1

The Correct Option is B

Understanding the Problem:

• We are given the correlation coefficient between variables \( x \) and \( y \) as \( \rho_{xy} = 0.85 \).
• New variables \( u \) and \( v \) are defined as linear transformations of \( x \) and \( y \): \[ u = \frac{x + 1.5}{12}, \quad v = \frac{y - 2.4}{30} \]
• We need to find the correlation coefficient between \( u \) and \( v \), denoted as \( \rho_{uv} \).

Key Concepts:

• The correlation coefficient is invariant under linear transformations of the form: \[ u = a x + b, \quad v = c y + d \] where \( a \) and \( c \) are positive constants.
• If either \( a \) or \( c \) is negative, the sign of the correlation coefficient flips, but its magnitude remains the same.

Analyzing the Transformations:

• For \( u \): \[ u = \frac{x}{12} + \frac{1.5}{12} \] This is a linear transformation with \( a = \frac{1}{12} \) (positive) and \( b = \frac{1.5}{12} \).
• For \( v \): \[ v = \frac{y}{30} - \frac{2.4}{30} \] This is a linear transformation with \( c = \frac{1}{30} \) (positive) and \( d = -\frac{2.4}{30} \).

Effect on Correlation Coefficient:

• Since both \( a \) and \( c \) are positive, the correlation coefficient remains unchanged in sign and magnitude.
• Thus: \[ \rho_{uv} = \rho_{xy} = 0.85 \]

Verification:

• The transformations involve only scaling (division by positive constants) and shifting (addition/subtraction of constants), which do not affect the correlation coefficient.
• If either transformation had a negative scaling factor (e.g., \( u = -\frac{x}{12} \)), the sign of \( \rho_{uv} \) would flip.

Final Answer:

Option 2: \( 0.85 \).