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I'm working on an example in introductory statistics and I'm not sure if my answer is correct. I think 2,3,5 is correct and not sure about 1,4. Am I correct?

It's about choosing correct statements when central limit theorem is applied.

  1. sample $x_1, x_2, ..., x_n$ is approximately normal, as sample size gets bigger
  2. variance of sample mean $x$ bar gets smaller, as sample size gets bigger
  3. sum of sample $x_1$ to $x_n$ gets approximately normal, as sample size gets bigger
  4. sample variance gets approximately normal, as sample size gets bigger
  5. sample mean $x$ bar gets approximately normal, as sample size gets bigger

And one last question!

When the population follows normal distribution $N[\mu,\sigma^2]$, sample mean and sample variance are independent.

How do I prove this true or wrong?

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The question as phrased is unclear (especially what exactly is meant by "when central limit theorem is applied"? Does it mean "when the conditions for the CLT hold?" for example -- and if so, which CLT?), and you provide no reasoning for us to discuss, so I'll mostly confine myself to some general comments.

Lamentably the question seems to perpetuate all manner of misunderstandings, common in a host of texts (and sets of notes) of mostly very poor quality. It's little wonder misconceptions abound.

Firstly, the "distribution of the sample" is not at issue (that's discrete, for starters), its what is the population distribution that the sample was drawn from that's the concern. This means 1. and 3. as phrased don't really make sense. (5 skirts around saying the 'distribution of the sample' phrasing just enough for me to give it the benefit of the doubt.)

Let's also take - for simplicity of discussion - that the $X_i$ are assumed "independent, identically distributed". There are versions of the CLT that don't require those, but some of the options hint that it's a more classical CLT we're dealing with.

If we interpret them instead to be meaning to ask about the distribution from which the sample was drawn and the population distribution of sums of size $n$, then those options make more sense as statements to consider.

Even with that change, 1. is still obviously false. The population distribution doesn't change when you take a bigger sample from it.

Statement 2. isn't in any sense a consequence of CLT (indeed as stated, for a constant variance of the components, it's true whenever the random variables in the mean are not perfectly dependent). The classical CLT itself uses the fact (i.e. it relies on it) that with iid $X_i$, $\text{Var}(\bar{X})=\sigma/\sqrt{n}$. This follows from basic properties of variance.

While statement 5. is actually true under appropriate conditions, the CLT doesn't actually doesn't quite say that (to really say something like that you need something more than the CLT). It's important to know what the CLT does say -- at least a classic version. The Wikipedia page on the CLT isn't too bad; start with the statement of the Lindeberg–Lévy CLT; what it says is under certain conditions that

$\sqrt{n}\bigg(\bigg(\frac{1}{n}\sum_{i=1}^n X_i\bigg) - \mu\bigg)\ \xrightarrow{d}\ N(0,\;\sigma^2)$

(Statement 4. is very often true as well, but there's some additional subtleties with it.)

When the population follows normal distribution N[u,sigma^2], sample mean and sample variance are independent. How do I prove this true or wrong?

Well, there are a number of approaches. For example it follows from Cochran's theorem, or Basu's theorem.

However, this is quite a different sort of question and really deserves a question of its own.

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  • $\begingroup$ I am puzzled: If statement (3) is problematic, then so should statement (5) be, since the mean is a multiple of the sum. And what exactly is the problem with statement (5), assuming we understand the $x_i$ to be a sequence of realizations of iid variables having a finite variance? Doesn't the CLT include the assertion that the sample mean is approximately Normal in distribution for sufficiently large $n$? $\endgroup$
    – whuber ♦
    May 14, 2015 at 13:49
  • $\begingroup$ @whuber The difference I saw between 3 and was that 5 omitted some words that make a difference; I don't have to assume they're in 5, but I can't remove them from 3, and I think looking at the question overall the $x_i$ are observed sample values, so I can't quite reconcile 3. Option 5 also doesn't seem to say what you just said at the end there. I read it as effectively being about the way the approximation tends to improve as sample size goes from one finite value to a larger finite value. As I read it, then, option 5 says something much more like the Berry-Esseen theorem. $\endgroup$
    – Glen_b
    May 14, 2015 at 23:36
  • $\begingroup$ @whuber if there's some specific wording I have there that you think should be changed, please let me know. $\endgroup$
    – Glen_b
    May 14, 2015 at 23:41
  • $\begingroup$ In the context we ought to take the beginning of (5) to be a shorthand for "sum of sample $x_1$ through $x_n$ divided by $n$," making it directly parallel to (3). I suspect you might be making quite subtle distinctions which I doubt are apparent to most readers--least of all myself--which would render the usefulness of much of your answer doubtful without an explicit explanation of those distinctions. It just seems like the level and assumptions of this answer are not as well calibrated to the level of the question as most of your other answers have been. $\endgroup$
    – whuber ♦
    May 15, 2015 at 16:02
  • $\begingroup$ @whuber I can only agree with your assessment - while I believe I could explain the distinction I drew, you're right that it would not be a useful distinction to make in the first place. On that basis, 5 would then be in the same basket as 3 and your earlier comment would apply. I'll have to ponder how to reword, but I'll have to come back to it later. $\endgroup$
    – Glen_b
    May 15, 2015 at 16:28

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