Question

Which of the following is/are NOT TRUE?

• A. Arithmetic mean is always greater than or equal to the geometric mean of a set of positive values.
• B. Correlation coefficient between two variables varies between 0 and 1.
• C. Exact middle point of \( -\infty \) and \( \infty \) on the real line is 0.
• D. Domain of a random variable that follows \( \chi^2 \) distribution with degree of freedom one is [0, \( \infty \)].

Hint:

The correlation coefficient ranges from -1 to 1. A value of 1 indicates perfect positive correlation, and -1 indicates perfect negative correlation.

Answer 1

The Correct Option is B

Step 1: Understand the properties of each statement.
- Option (A) is true because the arithmetic mean is always greater than or equal to the geometric mean for any set of positive values (by the Arithmetic Mean-Geometric Mean inequality).
- Option (B) is incorrect. The correlation coefficient between two variables ranges from -1 to 1, not just between 0 and 1.
- Option (C) is true. The exact middle point of the real line between \( -\infty \) and \( \infty \) is 0, since it's the midpoint.
- Option (D) is true. The domain of the \( \chi^2 \) distribution with one degree of freedom is indeed [0, \( \infty \)], as \( \chi^2 \) distributions are defined for non-negative values.
Final Answer: \[ \boxed{\text{Correlation coefficient between two variables varies between 0 and 1 is NOT TRUE.}} \]