Find the expectation and variance of three estimators of the mean

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Bill Foster

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Has some problems

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Bill Foster commented 6 years, 5 months ago

Calculations are correct.

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Bill Foster 6 years, 5 months ago

Gave some feedback: Has some problems

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Bill Foster commented 6 years, 5 months ago

Two problems:

Suffices in part c) and in the solution are displayed as, for example, W_11.

Sometimes the multiple choice display is in two rows rather than 1 row.

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

Created this.

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Question statement

From previous studies it is known that, $W$ the weight of eggs laid by {birds}, has mean $\mu=\var{m}$g and standard deviation $\sigma=\var{sd}$g.

A random sample of $\var{sa}$ eggs was taken $\{W_1,\;W_2,\ldots,\;W_{\var{sa}}\}$ and their weights observed.

Find the expectation and variance of the following three estimators of $\mu$

Give any introductory information the student needs.

			Name	Type	Generated Value

sc5	integer	-1
vara	number	9.31
varc	number	3630
sc4	integer	0
birds	string	bluetits
s3	decimal	dec("8")
s2	decimal	dec("5")
s1	decimal	dec("3")
s5	integer	12
s4	decimal	dec("10")
sc3	integer	1
tol	integer	0
sc2	integer	-1
c3	integer	3
c2	integer	-4
c1	integer	1
c5	integer	-2
c4	integer	0
exc	integer	-104
mm	string	min
m	integer	52
su4	integer	0
t	integer	71
sa	integer	13
sd	integer	11

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-1

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$A = \overline{W}$

(where $\overline{W}$ denotes the random variable given by taking sample means of size $\var{sa}$)

$ \operatorname{E}[A]=\;\;$Gap 0.$\;\;\;\;\operatorname{Var}(A)=\;\;\;$Gap 1.

Input $\operatorname{Var}(A)$ to 2 decimal places.

Advice

We use the result throughout these solutions that for random variables $R$ and $S$ then
$\operatorname{E}[aR+bS]=a\operatorname{E}[R]+b\operatorname{E}[S]$ and

$\operatorname{Var}(aR+bS) = a^2\operatorname{Var}(R)+b^2\operatorname{Var}(S)$

a)
\[\begin{eqnarray*} \operatorname{E}[A] &=& \operatorname{E}(\overline{W})=\mu=\var{m}\\ \\ \operatorname{Var}(A)&=&\operatorname{Var}(\overline{W})=\frac{\sigma^2}{n}=\frac{\var{sd^2}}{\var{sa}}\\ =\var{vara} \end{eqnarray*} \] to 2 decimal places.

b)
\[\begin{eqnarray*} \operatorname{E}[B] &=& \operatorname{E}(W_{\var{s3}})=\mu=\var{m}\\ \\ \operatorname{Var}(B)&=&\operatorname{Var}(W_{\var{s3}})=\sigma^2=\var{sd^2} \end{eqnarray*} \]

c)
\[\begin{eqnarray*} \operatorname{E}[C] &=& \operatorname{E}(\simplify[std]{{c1} * W_{s1} + {c2} *W_{s2} + {c3} *W_{s3} + {c4} * W_{s4}+{c5} * W_{s5}})\\ &=&\simplify[std]{ ({c1} + {c2} + {c3} + {c4} + {c5}) * {m} }\\ &=& \var{exc}\\ \\ \operatorname{Var}(C)&=&\operatorname{Var}(\simplify[std]{{c1} * W_{s1} + {c2} *W_{s2} + {c3} *W_{s3} + {c4} * W_{s4}+{c5} * W_{s5}})\\ &=& \simplify[std]{({c1 ^ 2} + {c2 ^ 2} + {c3 ^ 2} + {c4 ^ 2} + {c5 ^ 2}) * {sd ^ 2} }\\ &=& \var{varc} \end{eqnarray*} \]

d)
$A$ and $B$ are unbiased as their expectations are both $\var{m}$.

e)
$A$ is the most efficient as it has the least variance.

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This question is used in the following exam:

Statistical inference by Newcastle University Mathematics and Statistics in Content created by Newcastle University.

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Find the expectation and variance of three estimators of the mean

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Bill Foster commented 6 years, 5 months ago

Bill Foster 6 years, 5 months ago

Bill Foster commented 6 years, 5 months ago

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

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Find the expectation and variance of three estimators of the mean

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Taxonomy: mathcentre

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Bill Foster commented 6 years, 5 months ago

Bill Foster 6 years, 5 months ago

Bill Foster commented 6 years, 5 months ago

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

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Generated value: integer

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Generated value: `integer`