Question:
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:
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:
(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:
Show Hint
Remember the key difference between using a Z-test and a t-test: a Z-test is used when the population standard deviation (\(\sigma\)) is known, while a t-test is used when \(\sigma\) is unknown and estimated by the sample standard deviation (s).
Updated On: Mar 28, 2026
- (A), (B) and (D) only
- (A), (B) and (C) only
- (B) and (D) only
- (C) and (D) only
Show Solution
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept:
The t-distribution is used in hypothesis testing and for constructing confidence intervals when the population variance is unknown and the sample size is small. The question asks for the key assumptions required for its valid application.
Step 2: Detailed Explanation:
Let's analyze each statement:
(A) The variance of population is known.
This statement is incorrect. The t-distribution is specifically used 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 used.
(B) The samples are drawn from a normally distributed population.
This statement is correct. For the t-distribution to be applicable, especially 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 incorrect. The sample variance (\(s^2\)) is an unbiased estimator of the population variance (\(\sigma^2\)). 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 correct. The shape of the t-distribution is determined by its degrees of freedom (df), which for a single sample is typically calculated as n-1 (where n is the sample size). As the degrees of freedom increase, the t-distribution approaches the standard normal distribution.
Step 3: Final Answer:
The correct assumptions are that the samples are drawn from a normally distributed population (B) and that the distribution depends on degrees of freedom (D).
The t-distribution is used in hypothesis testing and for constructing confidence intervals when the population variance is unknown and the sample size is small. The question asks for the key assumptions required for its valid application.
Step 2: Detailed Explanation:
Let's analyze each statement:
(A) The variance of population is known.
This statement is incorrect. The t-distribution is specifically used 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 used.
(B) The samples are drawn from a normally distributed population.
This statement is correct. For the t-distribution to be applicable, especially 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 incorrect. The sample variance (\(s^2\)) is an unbiased estimator of the population variance (\(\sigma^2\)). 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 correct. The shape of the t-distribution is determined by its degrees of freedom (df), which for a single sample is typically calculated as n-1 (where n is the sample size). As the degrees of freedom increase, the t-distribution approaches the standard normal distribution.
Step 3: Final Answer:
The correct assumptions are that the samples are drawn from a normally distributed population (B) and that the distribution depends on degrees of freedom (D).
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