Need Critique on my solution. Find confidence interval

Question

I came across this problem in my self-study. I found a solution with the help from this forum (answer below) and posted my solution here.

Assume that $X_i$, $i\in\mathbb N$ is a sequence of I.I.D. random variables with a mean $\mu \neq 0$ and variance $\sigma^2$. We want to estimate $\dfrac{1}{\mu^2}$ using $\dfrac{1}{\overline X^2}$.

• Write the formula for confidence interval for $\mu$, say $(c_1, c_2)$.
• Write the formula for confidence interval for $\dfrac{1}{\mu^2}$, say $(d_1, d_2)$.
• Do we have $(\dfrac{1}{c_1^2},\dfrac{1}{ c_2^2})$=$(d_1, d_2)$

Would appreciate any constructive critique!

Answer 1

First, I think the statement that $X_i$ is some relatively large set of random variables, all i.i.d., tells you that you can use CLT on the sample mean to derive the confidence intervals for $\mu$ assuming Normality for $\bar{X}$.

That is, assuming $\frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \sim \mathcal{N}(0,1)$, we can construct confidence intervals for $\mu$.

Then, you can use the delta method to construct confidence intervals for the second part of the question involving $1/\mu^2$, given your answers from the first part.

Answer 2

So here is what I came up with after great advice form OrangeSleipnir