Which of the following are correct about t-test statistics?
(A) The mean of t-test distribution is 0.
(B) It depends on the degrees of freedom.
(C) It depends on population standard deviation.
(D) With more degrees of freedom, it closely resembles standard normal distribution.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (B) and (D) only
The t statistic for a sample of size n is
\( \displaystyle t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \)
where s is the sample standard deviation. Its distribution, the t-distribution with \(n - 1\) degrees of freedom, is bell-shaped and symmetric about 0, flatter than the normal curve for small samples.
So the t-distribution has mean 0, its shape is fixed by the degrees of freedom alone (not by the population standard deviation), and it approaches the standard normal curve as the degrees of freedom grow.
Checking the statements
(A) The mean of t-test distribution is 0. — Correct. The curve is symmetric about 0.
(B) It depends on the degrees of freedom. — Correct. There is a different t-curve for each value of \(n - 1\).
(C) It depends on population standard deviation. — Incorrect. The t-test is used when \(\sigma\) is unknown; it uses the sample value s.
(D) With more degrees of freedom, it closely resembles standard normal distribution. — Correct. For large samples, t is almost the same as z.