Question

Which of the following statements are true?

I. The sampling distribution of ¯xx¯ has standard deviation σ√nσn even if the population is not normally distributed.
II. The sampling distribution of ¯xx¯ is normal if the population has a normal distribution.
III. When nn is large, the sampling distribution of ¯xx¯ is approximately normal even if the the population is not normally distributed.

I and II

I and III

II and III

I, II, and III

None of the above gives the complete set of true responses.

Answer 1

From The central limiting theorem

We know that the standard deviation of the distribution of is given by

So statement I is correct

If the population has normal distribution then is also normally distributed.

So statement II is also correct.

From central limiting theorem if population doesn't have normal distribution and the sample size is large enough then distribution of is normal

Hence statement III is also correct.

Therefore all the three statements are correct.

Fourth Option