Question

Which of the following estimators ((^) is typically considered unbiased for a population parameter θ ?

A. X is a binomial random variable with parameters n and p, the sample proportion p^​=X/n for estimating p.
B. X1​,…,Xn​ are iid from a distribution with mean μ and variance σ² then σ=∑√(X1​−Xˉ)²​​/n−1 for estimating σ.
C. X1​,…,Xn​ are iid from a distribution with mean μ and variance σ² then σ²=Σ(X1​−xˉ)²/n​ for estimating σ².
D. Both B and C.

Answer 1

The typically considered unbiased estimator for a population parameter θ is the sample proportion (p^) which is the quotient of the binomial random variable X and the number of trials n, used for estimating the probability of success p in option A.

Step-by-step explanation:

The estimator typically considered unbiased for a population parameter θ is A. X is a binomial random variable with parameters n and p, the sample proportion p^ = X/n for estimating p. This sample proportion is an unbiased estimator because its expected value is equal to the true population proportion p. When the population is large enough and the sample is randomly selected, the sample proportion p^ is a good approximation of the population proportion p, satisfying the condition of unbiasedness.

For B and C, while these estimators are commonly used for estimating the standard deviation (σ) and variance (σ²), respectively, they are slightly biased because they are based on a sample, not the entire population. Nevertheless, the correction factor of dividing by n-1 instead of n in option B does make the sample standard deviation, s, an unbiased estimator of the population standard deviation, σ. Meanwhile, option C is incorrect because dividing by n rather than n-1 when calculating variance from a sample results in a biased estimator for the population variance σ².