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$\mathrm{E}\mathrm{\left[}\mathit{T}\mathrm{\right]}\mathrm{=}\mathit{E}\mathrm{\left[}{\mathit{a}}_{\mathrm{1}}{\mathit{X}}_{\mathrm{1}}\mathrm{+}{\mathit{a}}_{\mathrm{2}}{\mathit{X}}_{\mathrm{2}}\mathrm{\right]}\mathrm{=}{\mathit{a}}_{\mathrm{1}}\mathrm{E}\mathrm{\left[}{\mathit{X}}_{\mathrm{1}}\mathrm{\right]}\mathrm{+}{\mathit{a}}_{\mathrm{2}}\mathrm{E}\mathrm{\left[}{\mathit{X}}_{\mathrm{2}}\mathrm{\right]}\mathrm{=}{\mathit{a}}_{\mathrm{1}}\mathit{\mu }\mathrm{+}{\mathit{a}}_{\mathrm{2}}\mathit{\mu }\mathrm{=}\mathrm{\left(}{\mathit{a}}_{\mathrm{1}}\mathrm{+}{\mathit{a}}_{\mathrm{2}}\mathrm{\right)}\mathit{\mu }\mathrm{.}$

If we require $\mathit{E}\mathrm{\left[}\mathit{T}\mathrm{\right]}\mathrm{=}\mathit{\mu }$, then ${\mathit{a}}_{\mathrm{1}}\mathrm{+}{\mathit{a}}_{\mathrm{2}}\mathrm{=}\mathrm{1}$, so that ${\mathit{a}}_{\mathrm{2}}\mathrm{=}\mathrm{1}\mathrm{-}{\mathit{a}}_{\mathrm{1}}\mathrm{.}$

Since $\mathit{E}\mathrm{\left[}\mathit{T}\mathrm{\right]}\mathrm{=}\mathit{\mu }$, then $\mathit{T}$ is said to be an unbiased estimator of the mean $\mathit{\mu }\mathrm{.}$

Var $\mathrm{\left[}\mathit{T}\mathrm{\right]}\mathrm{=}$ Var $\mathrm{\left[}{\mathit{a}}_{\mathrm{1}}{\mathit{X}}_{\mathrm{1}}\mathrm{+}{\mathit{a}}_{\mathrm{2}}{\mathit{X}}_{\mathrm{2}}\mathrm{\right]}\mathrm{=}{\mathit{a}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{V}\mathrm{a}\mathrm{r}\mathrm{\left[}{\mathit{X}}_{\mathrm{1}}\mathrm{\right]}\mathrm{+}{\mathit{a}}_{\mathrm{2}}^{\mathrm{2}}\mathrm{V}\mathrm{a}\mathrm{r}\mathrm{\left[}{\mathit{X}}_{\mathrm{2}}\mathrm{\right]}\mathrm{=}{\mathit{a}}_{\mathrm{1}}^{\mathrm{2}}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{+}{\mathit{a}}_{\mathrm{2}}^{\mathrm{2}}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{=}\mathrm{\left(}{\mathit{a}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{+}{\mathit{a}}_{\mathrm{2}}^{\mathrm{2}}\mathrm{\right)}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{.}$

Since ${\mathit{a}}_{\mathrm{2}}\mathrm{=}\mathrm{1}\mathrm{-}{\mathit{a}}_{\mathrm{1}}\mathrm{,}$ $\mathrm{V}\mathrm{a}\mathrm{r}\mathrm{\left[}\mathit{T}\mathrm{\right]}\mathrm{=}\mathrm{\left\{}{\mathit{a}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{+}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathit{a}}_{\mathrm{1}}{\mathrm{\right)}}^{\mathrm{2}}\mathrm{\right\}}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{=}\mathrm{\left(}\mathrm{2}{\mathit{a}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{-}\mathrm{2}{\mathit{a}}_{\mathrm{1}}\mathrm{+}\mathrm{1}\mathrm{\right)}{\mathit{\sigma }}^{\mathrm{2}}$. Differentiate this with respect to ${\mathit{a}}_{\mathrm{1}}$ to find the minimum.

$\frac{\mathrm{d}}{\mathrm{d}{\mathit{a}}_{\mathrm{1}}}$ Var $\mathrm{\left[}\mathit{T}\mathrm{\right]}\mathrm{=}\mathrm{\left(}\mathrm{4}{\mathit{a}}_{\mathrm{1}}\mathrm{-}\mathrm{2}\mathrm{\right)}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{,}$

which is zero when ${\mathit{a}}_{\mathrm{1}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}$. Hence $\mathrm{V}\mathrm{a}\mathrm{r}\mathrm{\left[}\mathit{T}\mathrm{\right]}$ is a minimum when ${\mathit{a}}_{\mathrm{1}}\mathrm{=}{\mathit{a}}_{\mathrm{2}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}$ so $\mathit{T}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\left(}{\mathit{X}}_{\mathrm{1}}\mathrm{+}{\mathit{X}}_{\mathrm{2}}\mathrm{\right)}$ . Alternative derivation: write ${\mathit{a}}_{\mathrm{1}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{+}\mathit{\epsilon }\mathrm{,}$ ${\mathit{a}}_{\mathrm{2}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{-}\mathit{\epsilon }$. Then

Var $\mathrm{\left[}\mathit{T}\mathrm{\right]}\mathrm{=}\mathrm{\left(}{\mathit{a}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{+}{\mathit{a}}_{\mathrm{2}}^{\mathrm{2}}\mathrm{\right)}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{=}\mathrm{\left\{}\mathrm{\left(}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{+}\mathit{\epsilon }{\mathrm{\right)}}^{\mathrm{2}}\mathrm{+}\mathrm{\left(}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{-}\mathit{\epsilon }{\mathrm{\right)}}^{\mathrm{2}}\mathrm{\right\}}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{=}\mathrm{\left(}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{+}\mathrm{2}{\mathit{\epsilon }}^{\mathrm{2}}\mathrm{\right)}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{,}$

and is a minimum if $\mathit{\epsilon }\mathrm{=}\mathrm{0}\mathrm{.}$

What does this question show '? In part (a) you chose ${\mathit{a}}_{\mathrm{2}}$ to restrict attention to linear combinations of the ${\mathit{X}}_{\mathit{i}}$ which were unbiased estimators of the mean $\mathit{\mu }$, so $\mathrm{E}\mathrm{\left[}\mathit{T}\mathrm{\right]}\mathrm{=}\mathit{\mu }$. In part (b) you then showed that of all such unbiased estimators, the sample mean $\overline{\mathit{X}}$ is the one with smallest variance, so giving values closest to the true mean $\mathit{\mu }\mathrm{.}$

Worked Example: Lecture 15.

The following data give the noise level (in decibels) generated by fourteen different chain saws powered in one of two different ways.

 Petrol‐powered chain sawsElectric‐powered chain saws $\mathrm{1}\mathrm{0}\mathrm{3}\mathrm{ }\mathrm{1}\mathrm{0}\mathrm{3}\mathrm{ }\mathrm{1}\mathrm{0}\mathrm{5}\mathrm{ }\mathrm{1}\mathrm{0}\mathrm{6}\mathrm{ }\mathrm{1}\mathrm{0}\mathrm{8}\mathrm{ }\mathrm{1}\mathrm{0}\mathrm{5}\mathrm{ }\mathrm{1}\mathrm{0}\mathrm{6}$$\mathrm{9}\mathrm{7}\mathrm{ }\mathrm{9}\mathrm{5}\mathrm{ }\mathrm{9}\mathrm{4}\mathrm{ }\mathrm{9}\mathrm{3}\mathrm{ }\mathrm{9}\mathrm{1}\mathrm{ }\mathrm{9}\mathrm{5}\mathrm{ }\mathrm{9}\mathrm{4}$

At the 5% level of significance, test whether the average noise level of petrol‐powered chain saws is higher than for electric‐powered chain saws.

Answer: Testing ${\mathrm{H}}_{\mathrm{0}}$ : ${\mathit{\mu }}_{\mathrm{1}}\mathrm{=}{\mathit{\mu }}_{\mathrm{2}}$ vs. ${\mathrm{H}}_{\mathrm{1}}$ : ${\mathit{\mu }}_{\mathrm{1}}\mathrm{>}{\mathit{\mu }}_{\mathrm{2}}$, i.e. ${\mathrm{H}}_{\mathrm{0}}$ : ${\mathit{\mu }}_{\mathrm{1}}\mathrm{-}{\mathit{\mu }}_{\mathrm{2}}\mathrm{=}\mathrm{0}$ vs. ${\mathrm{H}}_{\mathrm{1}}$ : ${\mathit{\mu }}_{\mathrm{1}}\mathrm{-}{\mathit{\mu }}_{\mathrm{2}}\mathrm{>}\mathrm{0}\mathrm{.}$ Have two independent samples with unknown variance. Need to assume variances are equal.

Worked Example: Lecture 15.

The following data give the length (in mm.) of cuckoo (cuculus canorus) eggs found in nests belonging to wrens (A) and reed warblers (B).

$\overline{\mathrm{A}\mathrm{:}\mathrm{1}\mathrm{9}\mathrm{.}\mathrm{8}\mathrm{2}\mathrm{2}\mathrm{.}\mathrm{1}\mathrm{2}\mathrm{1}\mathrm{.}\mathrm{5}\mathrm{2}\mathrm{0}\mathrm{.}\mathrm{9}\mathrm{2}\mathrm{2}\mathrm{.}\mathrm{0}\mathrm{2}\mathrm{1}\mathrm{.}\mathrm{0}\mathrm{2}\mathrm{2}\mathrm{.}\mathrm{3}\mathrm{2}\mathrm{1}\mathrm{.}\mathrm{0}\mathrm{2}\mathrm{0}\mathrm{.}\mathrm{3}\mathrm{2}\mathrm{0}\mathrm{.}\mathrm{9}}$

$\mathrm{B}$: 23. $\mathrm{2}$ 22. $\mathrm{0}$ 22. $\mathrm{2}$ 21. $\mathrm{2}$ 21. $\mathrm{6}$ 21. $\mathrm{6}$ 21. $\mathrm{9}$ 22. $\mathrm{0}$ 22. $\mathrm{9}$ 22.8

Assuming the variances for each group are the same, is there any evidence at the 5% level to suggest that the egg size differs between the two host species?

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