### Reminder

Let $\hat{\theta}$ be an unbiased estimator of $\theta$, $S$ a sample on which we compute the estimation of $\theta$ and $\hat{\theta}_S$ the estimation of $\theta$ on $S$ with $\hat{\theta}$.
By definition, the confidence interval for a threshold of $\alpha$, noted \begin{align*} CI_{1-\alpha}(\theta;\hat{\theta},S) = CI(\theta;\hat{\theta},S,1-\alpha) &= \left[\hat{\theta}_S - z ; \hat{\theta}_S + z \right] \\ &= \left[\hat{\theta}_S \pm z \right] \end{align*} is defined by $\Pr(-z < \hat{\theta} < z) = 1 - \alpha$ Then, for computing a confidence interval for $\hat{\theta}_S$ we need to know the distribution of $\hat{\theta}$.

General case: $\hat{\theta}$ normally distributed

If $\hat{\theta}$ is distributed as a (non-standard) normal law, $\hat{\theta} \sim N(E[\hat{\theta}] = \theta,V[\hat{\theta}])$, we perform a variable change to come down to a standard distribution: $P\left(\mu - z_{1-\frac{\alpha}{2}}\sqrt{\frac{\sigma^2}{n}} < Z < \mu + z_{1-\frac{\alpha}{2}}\sqrt{\frac{\sigma^2}{n}}\right) = 1 - \alpha$ And then the confidence interval is given by $CI(\theta;\hat{\theta},S,1-\alpha) = \left[\hat{\theta}_S \pm z_{1 - \frac{\alpha}{2}}\sqrt{\frac{\sigma^2}{n}}\right]$ Furthermore $P(-z < Z < z) = 1 - \alpha$ Which is equivalent to $\displaystyle P(Z < z) = 1 - \frac{\alpha}{2}$.
For $\alpha = 0.05$, we have $z_{1 - \frac{\alpha}{2}} = 1.96$. (see: Z table)
For $\alpha = 0.10$, we have $z_{1 - \frac{\alpha}{2}} = 1.64$.

And the confidence interval is given by $CI(\theta;\hat{\theta},S,1-\alpha) = \left[\hat{\theta}_S \pm z_{1 - \frac{\alpha}{2}}\right]$

If $\hat{\theta}$ is distributed as a non-standard normal law, $\hat{\theta} \sim N(E[\hat{\theta}] = \theta,V[\hat{\theta}])$, we perform a variable change $P\left(\mu - z_{1-\frac{\alpha}{2}}\sqrt{\frac{\sigma^2}{n}} < Z < \mu + z_{1-\frac{\alpha}{2}}\sqrt{\frac{\sigma^2}{n}}\right) = 1 - \alpha$ And then the confidence interval is given by $CI(\theta;\hat{\theta},S,1-\alpha) = \left[\hat{\theta}_S \pm z_{1 - \frac{\alpha}{2}}\sqrt{\frac{\sigma^2}{n}}\right]$ Note: the standard deviation of an estimator is called the standard error. In the case of the estimator $\displaystyle \bar{X} = \sum_{i=1}^n X_i$, the
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