Step 1: Understanding the Concept:
The t-distribution is utilized in hypothesis testing and confidence interval construction when the population variance is unknown and the sample size is small. This question seeks the fundamental assumptions for its valid application.
Step 2: Detailed Explanation:
Let's examine each statement:
(A) The variance of population is known.
This statement is false. The t-distribution is specifically applied when the population variance (\(\sigma^2\)) is unknown and must be estimated using the sample variance (\(s^2\)). If the population variance were known, the Z-distribution would be employed.
(B) The samples are drawn from a normally distributed population.
This statement is true. For the t-distribution to be applicable, particularly with small sample sizes, the underlying population from which the sample is drawn should be normal or approximately normal.
(C) Sample standard deviation is an unbiased estimate of the population variance.
This statement is false. The sample variance (\(s^2\)) serves as an unbiased estimator of the population variance (\(\sigma^2\)). However, the sample standard deviation (s) is a biased estimator of the population standard deviation (\(\sigma\)).
(D) It depends on a parameter known as degree of freedom.
This statement is true. The degrees of freedom (df) dictate the shape of the t-distribution. For a single sample, this is typically calculated as n-1, where n represents the sample size. As the degrees of freedom increase, the t-distribution converges towards the standard normal distribution.
Step 3: Final Answer:
The valid assumptions are that the samples originate from a normally distributed population (B) and that the distribution's form is contingent upon the degrees of freedom (D).