There is a first question in training exercises for Test 3. I struggle to figure out what it necessary to perform in the exercise "Find a 95% confidence interval for θ in the following models (justify the formula)". For example, in "Pθ = Bernoulli(θ), θ ∈ [0, 1] ; θ_hat = X_hat ; observed value of X = 0.72." we have a mean given, so does it mean that we should calculate the intervals value or just show the final equation without calculation? and also, does it mean in this exercise that X_hat is the theta we are looking for, so that theta 97.5% CI can be computed like [X_hat - (t[n-1](97.5)*S/sqrt(n), X_hat + (t[n-1](2.5)*S/sqrt(n)] where n is sample size, S = (1-X_hat)X_hat? Could you, please, also elaborate on other examples?
It would be nice if we would have connection for some exercises.
In the Bernoulli case, you have to justify the formula (the law and the estimated s.e.) and compute the values of the CI bounds.
The parameter of the Bernoulli is its mean (theoretical mean, E(X_i)), estimated by the sample mean.
No, the CI is not based on the Student distribution as the sample is not Gaussian, and in this case you don't use the estimator S of the s.e. as you have a formula for the variance of a Bernoulli variable.
These examples are corrected in the reference books, it's your job to work these examples.
Remembering formulas is often misleading, it's better to understand them.
If you have other questions, i'll answer them.
Good work, M-A Poursat