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

You are watching: Which of the following statistics are unbiased estimators of population parameters?    An estimator T is called unbiased estimator for populace parameter 0 if Ε(T )-θ,νθ. The correct answers room as follows: B. Sample variance supplied to estimate populace variance. C. Sample proportion supplied to estimate populace proportion. D. Sample mean used to estimate populace mean. The proof for the exactly answers space as follows: allow X,,X, ,.., X, be an independent random sample from a population with mean u and variance o such the E(X,)= µ, i=1,2,..,n and also Var (X,) = o',i=1,2,..., n. To prove sample average is unbiased estimator of population mean, continue as follows: The sample can be composed as follows: Σχ. I=1 |X+X, +...+X, Now, we require to show that E(X)= µ. Therefore, take expectation because that the sample mean X together follows: x -x, E(X)= E |X, + X, +...+X, = (because X, s are independent) (E= µ,i =1,2,...,n) - пи Therefore, E(X)= µ. Thus, sample typical is unbiased estimator of populace mean.
Now, to display that sample variance is the unbiased estimator of population variance, that is, E(s³)= o². Start with L.H.S. And also 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, that is known that, Var (X,) = E(x} )- = E(x})=Væ (x.) - = E(x;)=o + ử Similarly, Var (X) = E(X²)- Var (X)+ = E(x³)=} - = E(X') = + µ²
Now substitute E(X²) and E(X;)in E(s²) together follows: E(-?+ p8) -n E(s*)= п-1 in i=1 n-1 п-1 n-1 = o? Therefore, E(s ) = o|. Thus, sample variance is the unbiased estimator that %3D populace variance.
Now to display that the sample ratio is the unbiased estimator of population proportion. Let sample proportion be denoted by p and is offered as p=÷, where x is the variety of successes in the sample and n is the sample size. Allow the population proportion it is in denoted by P. Then, E(p)= E %3D -E(00

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