Let $X_1,\ldots,X_n$ be iid from $\chi^2_v$, where $v\in \mathbb{N}$. So $E(X_i) = v$ and $Var(X_i) = 2v$.
- Find a statistic $Y_n$ such that $$\frac{\sqrt{n}(\bar{X}_n-v)}{Y_n} \overset{D}{\to} N(0,1)$$
- Suppose $n = 100$ and $\bar{x_n} = 10$. Use the asymptotic result in part 1 to obtain an approximate 95% confidence interval of $v$.
Attempt: Since we have a random sample with common mean and variance we can use the central limit theorem. So we know that $$\frac{\sqrt{n}(\bar{X}_n-v)}{\sqrt{2v}} \overset{D}{\to} N(0,1) \text{ in distribution}$$
Since we need a statistic, let $Y_n$ be the sample standard deviation. Then we get the result using Slutsky's theorem because we know that the sample standard deviation converges in probability to the population standard deviation.
Since we are trying to construct a confidence interval for $v$, we know that the population variance is unknown. We could use the sample variance instead provided the sample size is large enough. But how can I do so without having the realized sample variance of the data? I can't deduce the sample variance using only the sample mean.
Any hints would be appreciated.