Query regarding finding CI

Student

Hi.
https://www.imo.universite-paris-saclay.fr/~poursat/STAT-AI/Qtest3.pdf
For Question 1,
1. For the non-normal distributions does (\overline{X}\ - \mu)/\widehat{\sigma} ~ N(0, 1) hold for n = 16 (isn't the sample size too low)?
2. To find the CI, for various standard distributions, the variance is known (for example, in exponential, poisson it equals the mean, for normal and binomial too we know). So when finding the CI, is it expected in these questions to find the estimators for variance via the either of the two methods we've gone through and then arrive at the CI?
Thank You.

Student

Hi,
I also have some questions about how to calculate the variance. For example in the first question, for the Bernoulli distributrion, we know that the variance can be calculated as Var[X_i] = \theta(1-\theta), so in this case should we
A) just substitute \sigma = \sqrt{Var[X_i]} = \sqrt{0.72·(1-0.72)} (and for the CI then divide by \sqrt{n}
So CI(\theta)= \overline{X} \pm q_{97.5\%} · \frac{ \sqrt{0.72·(1-0.72)}}{\sqrt{n}}
or should we
B)do \sigma = \sqrt{Var[X]} =... = \sqrt{\frac{Var[X_i]}{n}} =\sqrt{\frac{ 0.72·(1-0.72)}{n}} (and then divide again by \sqrt{n} for the CI)?
So CI(\theta)= \overline{X} \pm q_{97.5\%} · \frac{ \sqrt{\frac{0.72·(1-0.72)}{n}}}{\sqrt{n}}

Teacher

Hello,
To compute a CI you have to estimate the s.e. : when it is a function of the parameter of interest, you "plug" the estimator in the variance formula.
For the Bernoulli(p) distribution: the s.e. of the sample mean is \sqrt{p(1-p)}/\sqrt{n} so you estimate it by \sqrt(.72(1-.72)/\sqrt{n}. There is no reason to add another [imath]\sqrt{n}[/imath].
It would not be wrong but aukward (and less accurate) to estimate the s.e. using the sample variance in these cases.

When you use the CLT to find an approximate normal distribution of the normalized sample mean, i agree that the approximation is better if n is large. In practice, you still use it with n=16, unless you have another method for small n, like the bootstrap for example. If the bootstrap CI and the CLT CI are similar, you conclude that you can use the CLT approximation for your problem ; if not, prefer the bootsrap CI.
M-A Poursat

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