Q13
1 markMCQSection A

The range of variable t of the t-distribution is :

Inferential Statistics
Range of the t-distribution

Options

(A)(0,1)(0, 1)
(B)(1,2)(1, 2)
(C)(−1,1)(-1, 1)
(D)(−∞,∞)(-\infty, \infty)
Official Answer

The correct option is (D) (-∞, ∞).


Reason

  • Like the normal curve, the t-distribution is symmetric about 0 and extends over the whole real line.
  • Hence the variable t can take any real value from -∞ to ∞.
t-distribution rangeminus infinity to infinitysymmetric distributionreal linet-variableunboundedbell-shaped curveinferential statistics

Marking Scheme

  • 11 mark: correct option (D) (−∞,∞)(-\infty, \infty).
  • 2No marks for the bounded intervals (A), (B) or (C).

Hint

The t-curve is bell-shaped like the normal curve, so t runs over all real numbers.

Quick Oral Answer

The t-distribution is symmetric and bell-shaped like the normal curve, so its variable t can take any real value from minus infinity to plus infinity.

Analysis & Explanation

The t-statistic is a standardised quantity that, like the z-score, can range over all real numbers.


Concept

  • The t-variable measures how many estimated standard errors a sample mean lies from the hypothesised mean.
  • This deviation can be arbitrarily large in either direction, so t is unbounded.

Why the key is right

  • (D) (−∞,∞)(-\infty, \infty) correctly captures that the density is defined and positive for every real t.

Why the distractors are wrong

  • (A) (0,1)(0, 1) is the range of a probability, not of the t-variable.
  • (B) (1,2)(1, 2) is an arbitrary finite interval with no basis.
  • (C) (−1,1)(-1, 1) wrongly bounds t; t-values well beyond ±1\pm 1 occur routinely (critical values are often around 2).

Common Mistakes

  1. 1Confusing the range of t with the range of a probability (0 to 1).
  2. 2Assuming t is bounded within (−1,1)(-1, 1) because standardised, when critical t-values often exceed 2.
  3. 3Thinking the t-variable is non-negative like a chi-square variable.

Interesting Facts

The t-distribution has heavier tails than the normal curve, so extreme t-values are more likely; this is why small-sample tests use t rather than z.

As degrees of freedom tend to infinity, the t-distribution converges to the standard normal, but both share the same full range from −∞-\infty to ∞\infty.

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Frequently Asked Questions

How is the t-distribution similar to the normal distribution?

Both are symmetric about zero, bell-shaped, and have a variable that ranges over the entire real line from −∞-\infty to ∞\infty. The t-distribution simply has heavier tails and depends on degrees of freedom.

Why is the range not (−1,1)(-1, 1)?

The t-variable is a standardised deviation, not a correlation or probability. Its values regularly exceed 1 in magnitude; typical critical values are near 2, so it cannot be confined to (−1,1)(-1, 1).