Find:
\n$P(\\var{lower} \\lt X \\lt \\var{upper})=\\;\\;$[[0]]
\nCorrect to 4 decimal places.
\n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{m}", "minValue": "{m}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{sva+tol1}", "minValue": "{sva-tol1}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{m}", "minValue": "{m}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{svb+tol1}", "minValue": "{svb-tol1}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Let $\\overline{X}$ be the random variable given by the sample mean.
\nFind $\\operatorname{E}[ \\overline{X}]$ and $\\operatorname{Var}(\\overline{X})$ in the following cases:
\n1) A sample of size $\\var{sa}$
\n$\\operatorname{E}[ \\overline{X}]=\\;\\;$[[0]]$\\;\\;\\;\\operatorname{Var}(\\overline{X})=\\;\\;$[[1]]
\n2) A sample of size $\\var{sb}$
\n$\\operatorname{E}[ \\overline{X}]=\\;\\;$[[2]]$\\;\\;\\;\\operatorname{Var}(\\overline{X})=\\;\\;$[[3]]
\nEnter the variances to 2 decimal places.
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{probsam+tol}", "minValue": "{probsam-tol}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n \n \nAssuming that $\\overline{X}$ also follows a normal distribution.
\n \n \n \nFind $P(\\var{lower} \\lt \\overline{X} \\lt \\var{upper})$ in a sample of size $\\var{sa}$.
\n \n \n \n$P(\\var{lower} \\lt \\overline{X} \\lt \\var{upper})=\\;\\;$[[0]]
\n \n \n \nEnter the value correct to 4 decimal places.
\n \n \n ", "showCorrectAnswer": true, "marks": 0}], "statement": "The total {stuff} content of the blood plasma of {nationality} {animals} ($X$, in mg/100ml) is known to follow a $N(\\var{m},\\var{s^2})$ distribution.
", "tags": ["checked2015", "cr1", "distribution of sample mean", "distributions", "MAS1604", "MAS8380", "MAS8401", "normal distribution", "Normal distribution", "Probability", "probability", "random variables", "sample", "sample distribution", "sample mean", "sc", "statistics", "tested1", "z scores"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "13/07/2012:
\nAdded tags.
\nCannot check calculations as yet as cannot access stats extension.
\nSet new tolerance variable tol=0.0001 for numeric entries to 4 dps.
\nSet new tolerance variable tol=0 for numeric entries to 2 dps.
\n21/12/2012:
\nChecked calculations against standard tables, OK. Added tested1 tag.
\nCorrected a typos and improved display in Advice.
\nChecked rounding, OK. Added tag cr1.
\nThis has scenarios - could be extended. Added sc tag.
\n", "licence": "Creative Commons Attribution 4.0 International", "description": "
Normal distribution $X \\sim N(\\mu,\\sigma^2)$ given. Find $P(a \\lt X \\lt b)$. Find expectation, variance, $P(c \\lt \\overline{X} \\lt d)$ for sample mean $\\overline{X}$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a) Converting to $Z$ scores in $N(0,1)$ we have for $X \\sim N(\\var{m},\\var{s^2})$
\n\\[\\begin{eqnarray*} P(\\var{lower} \\lt X \\lt \\var{upper})&=&P(X \\lt \\var{upper})-P(X \\lt \\var{lower})\\\\ &=&P\\left(Z \\lt \\frac{\\var{upper}-\\var{m}}{\\var{s}}\\right)-P\\left(Z \\lt \\frac{\\var{lower}-\\var{m}}{\\var{s}}\\right)\\\\ &=&P(Z \\lt \\var{zup})-P( Z \\lt \\var{zlow})\\\\ &=&\\var{pup}-\\var{1-plow}\\\\ &=&\\var{prob} \\end{eqnarray*} \\] to 4 decimal places.
\nHere the probabilities could have been looked up from normal CDF tables. Alternatively we can simply do the whole calculation in
\nR by typing $\\operatorname{pnorm}(\\var{upper},\\var{m},\\var{s})-\\operatorname{pnorm}(\\var{lower},\\var{m},\\var{s})$.
\nb)
This part reminds you that for the sample mean for samples of size $n$ from a normal distribution $N(\\mu,\\sigma^2)$ has a normal distribution $\\displaystyle N\\left(\\mu,\\frac{\\sigma^2}{n}\\right)$.
Hence for a sample size $\\var{sa}$:
\n$\\displaystyle \\operatorname{E}[\\overline{X}] = \\mu=\\var{m}$ and $\\displaystyle \\operatorname{Var}(\\overline{X}) = \\frac{\\sigma^2}{n}=\\frac{\\var{s^2}}{\\var{sa}}=\\var{sva}$
\nHence for a sample size $\\var{sb}$:
\n$\\operatorname{E}[\\overline{X}] = \\mu=\\var{m}$ and $\\displaystyle \\operatorname{Var}(\\overline{X}) = \\frac{\\sigma^2}{n}=\\frac{\\var{s^2}}{\\var{sb}}=\\var{svb}$
\nc)
Since the sample size is $\\var{sa}$ we are dealing with the normal distribution $N(\\var{m},\\simplify[std]{{s^2}/{sa}})$.
Converting to $Z$ scores in $N(0,1)$ we have for $\\overline{X} \\sim N(\\var{m},\\simplify[std]{{s^2}/{sa}})$
\n\\[\\begin{eqnarray*} P(\\var{lower} \\lt \\overline{X} \\lt \\var{upper})&=&P(\\overline{X} \\lt \\var{upper})-P(\\overline{X}\\lt \\var{lower})\\\\ &=&P\\left(Z \\lt \\frac{\\sqrt{\\var{sa}}(\\var{upper}-\\var{m})}{\\var{s}}\\right)-P\\left(Z \\lt \\frac{\\sqrt{\\var{sa}}(\\var{lower}-\\var{m})}{\\var{s}}\\right)\\\\ &=&P(Z \\lt \\var{zupsam})-P( Z \\lt \\var{zlowsam})\\\\ &=&\\var{pupsam}-\\var{1-plowsam}\\\\ &=&\\var{probsam} \\end{eqnarray*} \\] to 4 decimal places.
\nHere the probabilities could have been looked up from normal CDF tables. Alternatively we can simply do the whole calculation in
\nR by typing $\\operatorname{pnorm}(\\var{upper},\\var{m},\\frac{\\var{s}}{\\sqrt{\\var{sa}}})-\\operatorname{pnorm}(\\var{lower},\\var{m},\\frac{\\var{s}}{\\sqrt{\\var{sa}}})$ i.e. $\\operatorname{pnorm}(\\var{upper},\\var{m},\\var{s/sqrt(sa)})-\\operatorname{pnorm}(\\var{lower},\\var{m},\\var{s/sqrt(sa)})$
", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}