Q16Multiple correct3 Marks22 Dec 2024

Suppose $X_1, X_2, \dots, X_{30}$ are i.i.d. samples from a binomial distribution with an unknown parameter $p$ and a known parameter $n$. Let $\hat{p}_1, \hat{p}_2$ and $\hat{p}_3$ be three estimators of $p$ given as: $\hat{p}_1 = \frac{\sum_{i=1}^{30} X_i}{30}$ $\hat{p}_2 = \frac{n \sum_{i=1}^{30} X_i}{30}$ $\hat{p}_3 = \frac{\sum_{i=1}^{30} X_i}{30n}$ Which of the following option(s) is(are) true?

A

$\hat{p}_1$ is an unbiased estimator of $p$.

B

$\hat{p}_2$ is a biased estimator of $p$.

C

$\hat{p}_3$ is an unbiased estimator of $p$.

D

All three are biased.