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I'm trying to derive a test with significance level $\alpha$ based on a given statistic, but I just don't know what to do! Here's the setup:

  • $X$ is a symmetric random variable and $\text{var}(X) = 1$
  • Let $m$ be the median of $X$ (which we don't know)
  • Define the statistic $Z := \sqrt{n} \cdot \overline{X}$

Here's the question:

Determine a test based on $Z$ with significance level $\alpha$ of the following hypotheses:

\begin{align*} \text{null }H_0 : m &= 0 \\ \text{alternate }H_1 : m &< 0 \end{align*}

I've tried using Chebyshev's inequality to derive an upper bound for the significance, but I'm really lost ... can anyone help me?

Thanks! :)

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Assume the null is true. Given the variance is finite, the population mean must be finite and by symmetry equals the population median.

Consider the distribution of $\bar{X}$ - what's its population mean and sd? What would a standardized version of $\bar{X}$ look like (i.e. one with mean 0 and sd 1). How does it compare to the suggested statistic?

Now let's imagine you are using Chebyshev (it's possible to do better than that, though because this is an average). How many standard deviations ($k$) from the mean would you need your statistic to be before the probability of being outside that interval was no more than 0.05?

Now... notice that your alternative is actually one-tailed. So the two-tailed Chebyshev isn't quite what you want for that. However, recall you are told the original distribution is symmetric. This makes a difference.

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