Question

Which of the following pairs of numbers is possible to be the regression coefficients for a data of two variables

• A. (0.7, 3.2)
• B. (--0.6, 0.5)
• C. (0.85, 0.9)
• D. (0.4, 2.6)

Hint:

• Regression coefficients \(b_{yx}\) and \(b_{xy}\) must have the same sign (same as correlation coefficient \(r\)).
• The product of the regression coefficients is equal to the square of the correlation coefficient: \(b_{yx} \cdot b_{xy} = r^2\).
• Since \(|r| \le 1\), it must be that \(0 \le r^2 \le 1\). So, \(0 \le b_{yx} \cdot b_{xy} \le 1\).
• This also implies that if one regression coefficient is numerically greater than 1, the other must be numerically less than 1 (unless \(|r|=1\), in which case \(|b_{yx}| = 1/|b_{xy}|\)).

Answer 1

The Correct Option is C

We are given pairs of numbers and asked to determine which pair could represent the regression coefficients of two variables.

Let the two regression coefficients be \( b_{yx} \) and \( b_{xy} \). It is a known result in statistics that:

\[ b_{yx} \cdot b_{xy} = r^2 \]

where \( r \) is the Pearson correlation coefficient and \( -1 \leq r \leq 1 \), so \( 0 \leq r^2 \leq 1 \).

Hence, for any valid pair of regression coefficients, their product must satisfy:

\[ 0 \leq b_{yx} \cdot b_{xy} \leq 1 \]

Now we check each option:

1. \( 0.7 \cdot 3.2 = 2.24 \) (greater than 1)
2. \( -0.6 \cdot 0.5 = -0.3 \)  (negative, invalid)
3. \( 0.85 \cdot 0.9 = 0.765 \)  (valid, lies between 0 and 1)
4. \( 0.4 \cdot 2.6 = 1.04 \)  (greater than 1)

Final Answer: \( \boxed{(0.85,\ 0.9)} \)