Question:

Which of the following statements are true ?
(A) Central limit theorem states that the sampling distribution of the mean (\(\bar x\)) approaches a normal distribution as the sample size increases.
(B) As per Central Limit Theorem, when the sample size increases, the mean (\(\bar x\)) for the data becomes closer to the mean of overall population.
(C) The shape of t-distribution does not depend on degree of freedom.
Choose the correct answer from the options given below :

Updated On: May 11, 2025

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The Correct Option is C

The correct answer is (A) only. Let's examine each statement to determine its validity:

(A) The Central Limit Theorem (CLT) states that when the sample size is sufficiently large, the sampling distribution of the sample mean \(\bar x\) will approximate a normal distribution, regardless of the shape of the population distribution. This is true.
(B) The CLT implies that as the sample size increases, the sample mean \(\bar x\) approaches a normal distribution but not necessarily closer to the actual population mean. Thus, this statement is false because it misrepresents the CLT.
(C) The shape of the t-distribution is affected by the degrees of freedom (df). As df increases, the t-distribution becomes similar to the normal distribution. Therefore, this statement is also false.

Based on the analysis, only statement (A) is correct.

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