Which of the following are the assumptions underlying the use of t-distribution?

(A) The variance of population is known.

(B) The samples are drawn from a normally distributed population.

(C) Sample standard deviation is an unbiased estimate of the population variance.

(D) It depends on a parameter known as degree of freedom.

Choose the correct answer from the options given below:

The t-distribution is a statistical tool used under specific conditions. Each option needs to be evaluated to determine which are true assumptions.

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Option (A) states the variance of population is known.

This is incorrect. The t-distribution is used specifically when the population variance is unknown. When the population variance is known, the Z-distribution (normal distribution) is used instead. The purpose of the t-distribution is to handle situations where the population variance must be estimated from the sample.

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Option (B) states the samples are drawn from a normally distributed population.

This is a key assumption. The t-distribution requires that data comes from a population following a normal distribution. For larger samples (n > 30), this assumption becomes less critical due to the Central Limit Theorem.

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Option (C) states sample standard deviation is an unbiased estimate of population variance.

This is not an underlying assumption for using the t-distribution. While sample standard deviation is used in t-distribution calculations, this statement describes a technical property rather than a prerequisite condition for applying the t-distribution.

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Option (D) states it depends on a parameter known as degree of freedom.

This is correct. The t-distribution requires degrees of freedom, calculated as $df = n - 1$ where $n$ is the sample size. Different degrees of freedom produce different t-distribution shapes, with smaller degrees of freedom resulting in distributions with heavier tails.

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The valid assumptions are (B) and (D).

The t-distribution is applied when the population variance is unknown, the sample comes from a normal population, and degrees of freedom can be determined.

The correct answer is Option 3: (B) and (D) only.