Question

Which of the following statistics are unbiased estimators of population parameters? Choose the correct answer below. Select all that apply. □ A. Sample range used to estimate a population range. □ B. Sample variance used to estimate a population variance □ C. Sample proportion used to estimate a population proportion. D. Sample mean used to estimate a population mean □ E. Sample standard deviation used to estimate a population standard deviation. □ F. Sample median used to estimate a population median

Answer 1

An estimator T is called unbiased estimator for population parameter 0 if Ε(T )-θ,νθ. The correct answers are as follows: B. Sample variance used to estimate population variance. C. Sample proportion used to estimate population proportion. D. Sample mean used to estimate population mean. The proof for the correct answers are as follows: Let X,,X, ,.., X, be an independent random sample from a population with mean u and variance o such that E(X,)= µ, i=1,2,..,n and Var (X,) = o',i=1,2,..., n. To prove sample mean is unbiased estimator of population mean, proceed as follows: The sample can be written as follows: Σχ. i=1 |X+X, +...+X, Now, we need to show that E(X)= µ. Therefore, take expectation for the sample mean X as follows: x -x, E(X)= E |X, + X, +...+X, =[E(X,)+E(X,)+.+ E(X,)] (because X, s are independent) (E[X,]= µ,i =1,2,...,n) -[u+µ+...+ µ] пи Therefore, E(X)= µ. Thus, sample mean is unbiased estimator of population mean.

Now, to show that sample variance is the unbiased estimator of population variance, that is, E(s³)= o². Begin with L.H.S. and proceed as follows: E(3)--Σ-1 - X)² E(s?)= E n-12(X,- F)² п-1 i=1 Σ(χ+X _2 Σ) i=1 Ε ΣΧ+nX-2XΣ. -1 i=1 i=1 ·EE(x;)+nX² – 2nx? п-1 i=1 Σ(x)-n п-1 i=1 ΣΕ(x)-Ε ( I') i=1 Now, it is known that, Var (X,) = E(x} )-[E(X)] = E(x})=Væ (x.) -[E(X)] = E(x;)=o + ử Similarly, Var (X) = E(X²)-[E(X)] Var (X)+[E(X)] = E(x³)=} - = E(X') = + µ²

Now substitute E(X²) and E(X;)in E(s²) as follows: E(-?+ p8) -n E(s*)= п-1 in i=1 [no? +nự – ơ - nự - 1 no² -o²] n-1 п-1 n-1 = o? Therefore, E(s ) = o|. Thus, sample variance is the unbiased estimator of %3D population variance.

Now to show that the sample proportion is the unbiased estimator of population proportion. Let sample proportion be denoted by p and is given as p=÷, where x is the number of successes in the sample and n is the sample size. Let the population proportion be denoted by P. Then, E(p)= E %3D -E(<) -(x) E(x) Therefore, E(p)= P. Thus, the sample proportion is the unbiased estimator of population proportion.