Step 1: Understanding the Concept:
The t-distribution is employed in hypothesis testing and for constructing confidence intervals when the population variance is unknown and the sample size is small. The inquiry pertains to the essential assumptions for its valid application.
Step 2: Detailed Explanation:
Let's examine each assertion:
(A) The variance of the population is known.
This assertion is false. The t-distribution is specifically utilized when the population variance (\(\sigma^2\)) is unknown and must be estimated from the sample variance (\(s^2\)). If the population variance were known, the Z-distribution would be applicable.
(B) The samples are drawn from a normally distributed population.
This assertion is true. For the t-distribution to be applicable, particularly with small sample sizes, the underlying population from which the sample is drawn must be normal or approximately normal.
(C) Sample standard deviation is an unbiased estimate of the population variance.
This assertion is false. The sample variance (\(s^2\)) serves as an unbiased estimator of the population variance (\(\sigma^2\)). Conversely, 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 assertion is true. The characteristic shape of the t-distribution is dictated by its degrees of freedom (df). 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 is contingent upon the degrees of freedom (D).