Question

Question 3: Bernoulli distribution (23/100 points) Consider a random sample X1,...,Xn from a Bernoulli distribution with unknown parameter p that describes the probability that Xi is equal to 1. That is, Bernoulli(p), i = 1, ..., n. (10) The maximum likelihood (ML) estimator for p is given by ÔML = x (11) n It holds that NPML BIN(n,p). (12) 3.a) (1 point) Give the conservative 100(1 – a)% two-sided equal-tailed confidence interval for p based on ÔML for a given a and random sample X1,...,Xn. 3.b) (1 point) Give a 100(1 - a)% two-sided equal-tailed approximate confidence interval for p based on a limiting distribution of mL for a given a and random sample X1,...,Xn.

Answer 1

Solution 3.a)

Kindly check the attached picture for complete explanation of the above question subpart

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Solution 3.b)

The confidence level of choice is stated as 100(1 − α)%.
For a 95% confidence interval, α = 0.05.
For an 80% confidence interval, α = 0.20.
We can re-write the earlier Z probability as
P(−zα/2 ≤
X¯ − µ
σ/√
n
≤ zα/2
) = 1 − α
and this leads to the 100(1 − α)% confidence interval for µ
P(X¯ − zα/2
σ
√
n
≤ µ ≤ X¯ + zα/2
σ
√
n
) = 1 − α
In a two-sided confidence interval, the α amount is split between the
two tails, thus we see α/2 or specifically, zα/2
in the formula.