Understanding Correlation Coefficient and Linear Transformations

The question asks about the correlation coefficient between two variables, U and V, which are linear transformations of X and Y. We are given the correlation coefficient between X and Y.

Given Information

• Correlation coefficient between X and Y, denoted as $r(X, Y) = 0.7$.
• Variable U is defined as $U = X - 5$.
• Variable V is defined as $V = Y + 2$.

Impact of Linear Transformations on Correlation

A correlation coefficient measures the strength and direction of the linear relationship between two variables. Linear transformations of the variables affect the correlation coefficient in a specific way.

Consider two variables X and Y. If we transform them linearly to get U and V as follows:

$$U = aX + b$$

$$V = cY + d$$

where a, b, c, and d are constants, then the correlation coefficient between U and V is given by:

$$r(U, V) = r(aX + b, cY + d) = \frac{ac}{|ac|} r(X, Y)$$

This formula shows that the correlation coefficient remains the same in magnitude, but its sign might change depending on the signs of the coefficients 'a' and 'c'.

• If 'a' and 'c' have the same sign (both positive or both negative), $\frac{ac}{|ac|} = 1$, so $r(U, V) = r(X, Y)$.
• If 'a' and 'c' have different signs (one positive, one negative), $\frac{ac}{|ac|} = -1$, so $r(U, V) = -r(X, Y)$.

The addition or subtraction of a constant (b and d) does not affect the correlation coefficient at all.

Applying the Property to the Given Problem

In our case, the transformations are:

$$U = X - 5$$

$$V = Y + 2$$

Comparing these to the general forms $U = aX + b$ and $V = cY + d$, we can identify the constants:

• For U = X - 5, we have $a = 1$ and $b = -5$.
• For V = Y + 2, we have $c = 1$ and $d = 2$.

Now, let's look at the coefficients of X and Y, which are 'a' and 'c'. Here, $a = 1$ and $c = 1$. Both 'a' and 'c' are positive numbers. Since they have the same sign, the correlation coefficient between U and V will be the same as the correlation coefficient between X and Y.

Using the formula:

$$r(U, V) = \frac{(1)(1)}{|(1)(1)|} r(X, Y)$$

$$r(U, V) = \frac{1}{|1|} r(X, Y)$$

$$r(U, V) = 1 \times r(X, Y)$$

Given that $r(X, Y) = 0.7$, we substitute this value:

$$r(U, V) = 1 \times 0.7$$

$$r(U, V) = 0.7$$

Thus, the correlation coefficient between U and V is 0.7.

Conclusion

The correlation coefficient between U and V is 0.7, which is the same as the correlation coefficient between X and Y because the linear transformations involved adding/subtracting constants and multiplying by positive constants (which are implicitly 1 here).

Revision Table: Correlation Coefficient Transformations

Additional Information: Properties of Linear Transformations

Linear transformations affect different statistical measures in different ways. It's useful to understand how mean, variance, standard deviation, and covariance are affected compared to the correlation coefficient.

• Mean: If $U = aX + b$, then the mean of U is $E(U) = aE(X) + b$. The mean shifts and scales.
• Variance: If $U = aX + b$, then the variance of U is $Var(U) = a^2Var(X)$. Adding a constant does not change variance, but multiplying by 'a' scales variance by $a^2$.
• Standard Deviation: If $U = aX + b$, then the standard deviation of U is $SD(U) = |a|SD(X)$. It scales by the absolute value of 'a'.
• Covariance: If $U = aX + b$ and $V = cY + d$, then the covariance between U and V is $Cov(U, V) = ac \ Cov(X, Y)$. Covariance scales by the product of 'a' and 'c'.
• Correlation Coefficient: As shown above, $r(U, V) = \frac{ac}{|ac|} r(X, Y)$. Correlation is a standardized measure, so it is less affected by scaling and shifting than covariance, variance, and mean. It only changes sign if one variable is scaled by a negative factor relative to the other.

In this problem, $U = 1 \cdot X + (-5)$ and $V = 1 \cdot Y + 2$. Since the scaling factors (1 and 1) are both positive, the sign of the correlation coefficient is preserved, and its magnitude is always preserved under linear transformations.