Edexcel IAL Mathematics S3.3.1 standard error and unbiased estimation

Practise estimating sample means, unbiased variances and standard errors, then comparing unbiased estimators by variance.

[Maximum number: 8]

A random sample of two observations $X_{1}$ and $X_{2}$ is taken from a population with unknown mean $\mu$ and unknown variance $\sigma^{2}$

(a)

Explain what you understand by an unbiased estimator for $\mu$

Two estimators for $\mu$ are $U_{1}$ and $U_{2}$ where

U_{1}=3 X_{1}-2 X_{2} \quad \text { and } \quad U_{2}=\frac{X_{1}+3 X_{2}}{4}

[ 1 ]

An estimator for $\mu$ is unbiased if its expected value is equal to $\mu$

B1

(1)

(b)

Show that both $U_{1}$ and $U_{2}$ are unbiased estimators for $\mu$

The most efficient estimator among a group of unbiased estimators is the one with the smallest variance.

[ 3 ]

$\mathrm{E}\left(U_{1}\right)=3 \mathrm{E}\left(X_{1}\right)-2 \mathrm{E}\left(X_{2}\right)$ or $\mathrm{E}\left(U_{2}\right)=\frac{1}{4}\left(\mathrm{E}\left(X_{1}\right)+3 \mathrm{E}\left(X_{2}\right)\right)$

M1

$\mathrm{E}\left(U_{1}\right)=3 \mu-2 \mu=\mu$ (therefore unbiased)

A1cso

$\mathrm{E}\left(U_{2}\right)=\frac{1}{4}(\mu+3 \mu)=\mu$ (therefore unbiased)

A1cso

(c)

By finding the variance of $U_{1}$ and the variance of $U_{2}$ state, giving a reason, the most efficient estimator for $\mu$ from these two estimators.

[ 4 ]

$\operatorname{Var}(U_1)=9\operatorname{Var}(X_1)+4\operatorname{Var}(X_2)=13\sigma^2$ $\operatorname{Var}(U_2)=\frac{1}{16}\operatorname{Var}(X_1)+\frac{9}{16}\operatorname{Var}(X_2)=\frac{5}{8}\sigma^2$
As $\operatorname{Var}(U_1)>\operatorname{Var}(U_2)$ , $U_2$ is the most efficient estimator for $\mu$ .
M1 for use of $a^2\operatorname{Var}(X_1)+b^2\operatorname{Var}(X_2)$ .
A1 for $13\sigma^2$ , A1 for $5\sigma^2/8$ , A1 for $U_2$ with a correct reason.

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S3.3.1 - Concepts of standard error, question 1

Question (a)

Question (b)

Question (c)