$A = \\overline{W}$
\n(where $\\overline{W}$ denotes the random variable given by taking sample means of size $\\var{sa}$)
\n$ \\operatorname{E}[A]=\\;\\;$[[0]]$\\;\\;\\;\\;\\operatorname{Var}(A)=\\;\\;\\;$[[1]]
\nInput $\\operatorname{Var}(A)$ to 2 decimal places.
\n ", "marks": 0}, {"scripts": {}, "gaps": [{"showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{m}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{m}"}, {"showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{sd^2}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{sd^2}"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\n$B = W_{\\var{s3}}$
\n$\\operatorname{E}[B]=\\;\\;$[[0]]$\\;\\;\\;\\;\\operatorname{Var}(B)=\\;\\;\\;$[[1]]
\n ", "marks": 0}, {"scripts": {}, "gaps": [{"showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{exc}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{exc}"}, {"showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{varc}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{varc}"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\n$\\simplify[std]{C = {c1} * W_{s1} + {c2} *W_{s2} + {c3} *W_{s3} + {c4} * W_{s4}+{c5} * W_{s5}}$
\n$ \\operatorname{E}[C]=\\;\\;$[[0]]$\\;\\;\\;\\;\\operatorname{Var}(C)=\\;\\;\\;$[[1]]
\nInput both as integers.
\n ", "marks": 0}, {"scripts": {}, "gaps": [{"displayType": "checkbox", "choices": ["$A$
", "$B$
", "$C$
"], "matrix": [1, 1, -1], "distractors": ["", "", ""], "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "warningType": "none", "scripts": {}, "maxMarks": 0, "type": "m_n_2", "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 3, "marks": 0}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\nWhich of the estimators $A,\\;B$ or $C$ above are unbiased for $\\mu$? Select the correct choices.
You will lose a mark for selecting a wrong choice.
[[0]]
\n ", "marks": 0}, {"scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["$A$
", "$B$
", "$C$
"], "matrix": [1, 0, 0], "distractors": ["", "", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "displayColumns": 3, "marks": 0}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\nWhich of the estimators $A,\\;B$ or $C$ above is the most efficient?
\n[[0]]
\n ", "marks": 0}], "variablesTest": {"condition": "exc<>m", "maxRuns": 100}, "statement": "\n \n \nFrom 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.
\n \n \n \nA random sample of $\\var{sa}$ eggs was taken $\\{W_1,\\;W_2,\\ldots,\\;W_{\\var{sa}}\\}$ and their weights observed.
\n \n \n \nFind the expectation and variance of the following three estimators of $\\mu$
\n \n \n ", "tags": ["biased", "checked2015", "cr1", "distributions", "efficient estimators", "estimators", "estimators of mean", "expectation", "expectations", "MAS2302", "mean", "mean ", "means", "random samples", "random variables", "sample means", "sc", "statistics", "tested1", "unbiased", "variance"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "13/07/2012:
\nAdded tags.
\nImproved and made consistent the display in various content areas.
\nSet new tolerance variable tol=0 for 2 dps numeric input question.
\nChecked calculation.
\nAdded description.
\n1/08/2012:
\nAdded tags.
\nQuestion appears to be working correctly.
\n21/12/2012:
\nChecked calculations, OK. Added tested1 tag.
\nChecked rounding, OK. Added cr1 tag.
\nCould have scenarios. Added sc tag.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Using a random sample from a population with given mean and variance, find the expectation and variance of three estimators of $\\mu$. Unbiased, efficient?
"}, "advice": "\nWe 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
$\\operatorname{Var}(aR+bS) = a^2\\operatorname{Var}(R)+b^2\\operatorname{Var}(S)$
\na)
\\[\\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.