Which of the following statements are TRUE?
(A) The variable t of t-distribution ranges from \(-\infty\) to \(\infty\).
(B) The probability curve of the t-distribution is symmetric about the line \(t=0\)
(C) The variance of the t-distribution is greater than one.
(D) As the number of degrees of freedom decreases, the t-distribution curve moves closer to the standard normal probability curve.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (C) only
The t-distribution with \(\nu\) degrees of freedom is defined for every real t, is bell-shaped with mean 0, and is flatter than the standard normal curve, with heavier tails. Its variance (which exists for \(\nu \gt 2\)) is
\( \displaystyle \mathrm{Var}(t) = \frac{\nu}{\nu - 2} \)
As \(\nu\) increases, the curve gets closer and closer to the standard normal curve.
So t takes all real values, the curve is symmetric about t = 0, its variance \(\frac{\nu}{\nu - 2}\) is more than 1, and it approaches the normal curve as the degrees of freedom increase.
Checking the statements
(A) The variable t of t-distribution ranges from \(-\infty\) to \(\infty\). — Correct. t can take every real value.
(B) The probability curve of the t-distribution is symmetric about the line \(t=0\) — Correct. The curve is bell-shaped about its mean 0.
(C) The variance of the t-distribution is greater than one. — Correct. \(\nu \gt \nu - 2\), so \(\frac{\nu}{\nu - 2} \gt 1\).
(D) As the number of degrees of freedom decreases, the t-distribution curve moves closer to the standard normal probability curve. — Incorrect. It moves closer as the degrees of freedom increase.