How do I find these confidence intervals?

Asked Dec 04 '17 at 00:19

Active Dec 04 '17 at 00:19

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I am looking to calculate confidence intervals for the following:

If $X_1,...,X_n$ are iid following $N(\mu,\mu^2)$

How do I set about calculating confidence intervals for these pivots:

I have worked out that the distribution of $\bar{X}$ is $N(1,1/n)$ but I don't know how to use this to calculate $a_1$ and $a_2$ in the equation

$P(\frac{\bar{X}}{a_2}<\mu<\frac{\bar{X}}{a_1})=0.95$

Am I along the right lines?

asked Dec 04 '17 at 00:19

Thomas Smith

1

The distribution of $\bar{X}$ is $N(\mu,\mu^2/n)$ while the distribution of $\frac{\bar{X}}{\mu}$ is $N(1,1/n)$ – Henry Dec 04 '17 at 00:36
Meanwhile the distributions of $X_i-\bar{X}$ are $N(0,\mu^2(n-1)/n)$, though summing the absolute values is not straightforward – Henry Dec 04 '17 at 00:40
So how would I go about finding $a_1$ and $a_2$? – Thomas Smith Dec 04 '17 at 00:54
Can you find $L$ and $U$ such that $P(L < \bar X/\mu < U) = .95)$ and manipulate the inequality? – BruceET Dec 04 '17 at 06:21
But then L and U have infinitely many possibilities surely? – Thomas Smith Dec 04 '17 at 09:35
Can you please try answering a similar question [here] as I am stuck for days (https://math.stackexchange.com/questions/2909908/why-sampling-without-replacement-gives-better-ci-performance) – Parthiban Rajendran Sep 11 '18 at 07:02

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