1. Converting to $\\operatorname{N}(0,1)$
\n$\\simplify[all,!collectNumbers]{P(X < {lower}) = P(Z < ({lower} -{m}) / {s}) = P(Z < {lower-m}/{s}) = 1 -P(Z < {m-lower}/{s})} = 1 -\\var{p} = \\var{precround(1 -p,4)}$ to 4 decimal places.
\n2. Converting to $\\operatorname{N}(0,1)$
\n$\\simplify[all,!collectNumbers]{P(X > {upper}) = P(Z > ({upper} -{m}) / {s}) = P(Z > {upper-m}/{s}) = 1 -P(Z < {upper-m}/{s})} = 1-\\var{p1} = \\var{precround(1 -p1,4)}$ to 4 decimal places.
", "rulesets": {}, "parts": [{"prompt": "Find the probability that in a particular week the {amount} is less than {lower} {units1}:
\nProbability = [[0]] (to 4 decimal places)
\nFind the probability that in a particular week the {amount} is greater than {upper} {units1}:
\nProbability = [[1]] (to 4 decimal places)
", "marks": 0, "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "scripts": {}, "precision": "4", "maxValue": "prob1+tol", "minValue": "prob1-tol", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "scripts": {}, "precision": "4", "maxValue": "prob2+tol", "minValue": "prob2-tol", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "The {amount}, $X$, of {stuff} is normally distributed with mean {m}{units1} and standard deviation {s}{units1}.
\ni.e. \\[X \\sim \\operatorname{N}(\\var{m},\\var{s}^2)\\]
\n", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"units1": {"definition": "\"kWh\"", "templateType": "anything", "group": "Ungrouped variables", "name": "units1", "description": ""}, "upper": {"definition": "random(m+0.5s..m+1.5*s#5)", "templateType": "anything", "group": "Ungrouped variables", "name": "upper", "description": ""}, "lower": {"definition": "random(m-1.5*s..m-0.5s#5)", "templateType": "anything", "group": "Ungrouped variables", "name": "lower", "description": ""}, "p1": {"definition": "normalcdf((upper-m)/s,0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "p1", "description": ""}, "m": {"definition": "random(750..1250#50)", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "s": {"definition": "random(60..100#10)", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "p": {"definition": "normalcdf((m-lower)/s,0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "amount": {"definition": "\"electricity consumption\"", "templateType": "anything", "group": "Ungrouped variables", "name": "amount", "description": ""}, "stuff": {"definition": "\"a frozen foods warehouse each week in the summer months \"", "templateType": "anything", "group": "Ungrouped variables", "name": "stuff", "description": ""}, "tol": {"definition": "0.0001", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "prob2": {"definition": "precround(1-normalcdf(upper,m,s),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "prob2", "description": ""}, "prob1": {"definition": "precround(normalcdf(lower,m,s),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "prob1", "description": ""}}, "metadata": {"notes": "\n \t\t \t\t
1/1/2012:
\n \t\t \t\tCan be configured to other applications using the string variables suppplied. Included tag sc.
\n \t\t \n \t\t", "description": "Given a random variable $X$ normally distributed as $\\operatorname{N}(m,\\sigma^2)$ find probabilities $P(X \\gt a),\\; a \\gt m;\\;\\;P(X \\lt b),\\;b \\lt m$.
", "licence": "None specified"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Naomi's copy of business: steph and naomi 3 NEW", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Naomi Hannaford", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/342/"}], "functions": {}, "ungrouped_variables": ["units1", "lower", "upper1", "age", "heightw", "m", "s", "prop", "heightm", "mw", "t", "prop1", "z", "z1", "z2", "w2"], "tags": ["Baye's Theorem", "Normal distribution", "Probability", "conditional probability", "converting to z scores", "mean", "normal distribution", "normal tables", "probabilities", "probability", "standard deviation", "statistics", "z scores"], "advice": "Converting to $\\operatorname{N}(0,1)$
\n\\[\\begin{eqnarray*}P(X \\lt \\var{lower})&=&P\\left(Z \\lt \\frac{\\var{lower}-\\var{m}}{\\var{s}}\\right)\\\\&=&P(Z\\lt \\var{z})=1-P(Z \\lt \\var{-z})=1-\\var{normalcdf(-z,0,1)}\\\\&\\approx&\\var{prop/100}\\end{eqnarray*}\\]
\nHence the proportion is $\\var{prop}$% to 1 decimal place.
\nThe proportion of {mw} more than $\\var{upper1}$ {units1} tall is given by finding the probability:
\n\\begin{align}
\\operatorname{P}(X>\\var{upper1} &= 1 - \\operatorname{P}(X<\\var{upper1})\\\\
&= 1 - \\operatorname{P}(Z<\\dfrac{\\var{upper1}-\\var{m}}{\\var{s}})\\\\
&= 1 - \\var{w2}\\\\
&\\approx \\var{prop1/100}
\\end{align}
Hence the proportion is $\\var{prop1}$% to 1 decimal place.
", "rulesets": {}, "parts": [{"prompt": "What proportion of such {mw} are less than $\\var{lower}$ {units1} tall?
\nProportion = [[0]]% (to 1 decimal place).
", "marks": 0, "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "scripts": {}, "precision": "1", "maxValue": "prop+0.1", "minValue": "prop-0.1", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "What proportion of such {mw} are more than $\\var{upper1}$ {units1} tall?
\nProportion = [[0]]% (to 1 decimal place).
", "marks": 0, "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "scripts": {}, "precision": "1", "maxValue": "prop1+0.1", "minValue": "prop1-0.1", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "Suppose that the heights, in {units1}, of $\\var{age}$-year-old {mw} are normally distributed with mean $\\var{m}$ {units1} and standard deviation $\\var{s}$ {units1}.
", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"units1": {"definition": "'inches'", "templateType": "anything", "group": "Ungrouped variables", "name": "units1", "description": ""}, "lower": {"definition": "z*s+m", "templateType": "anything", "group": "Ungrouped variables", "name": "lower", "description": ""}, "upper1": {"definition": "z2*s+m", "templateType": "anything", "group": "Ungrouped variables", "name": "upper1", "description": ""}, "age": {"definition": "random(20..40)", "templateType": "anything", "group": "Ungrouped variables", "name": "age", "description": ""}, "heightw": {"definition": "random(63..65#0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "heightw", "description": ""}, "m": {"definition": "if(t=0,heightm,heightw)", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "s": {"definition": "random(5..7)", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "prop": {"definition": "precround(100*(1-normalcdf(abs(z),0,1)),1)", "templateType": "anything", "group": "Ungrouped variables", "name": "prop", "description": ""}, "heightm": {"definition": "random(68..72#0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "heightm", "description": ""}, "mw": {"definition": "switch(t=0,'men','women')", "templateType": "anything", "group": "Ungrouped variables", "name": "mw", "description": ""}, "t": {"definition": "random(0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}, "prop1": {"definition": "precround(100*(1-normalcdf(abs(z2),0,1)),1)", "templateType": "anything", "group": "Ungrouped variables", "name": "prop1", "description": ""}, "z": {"definition": "random(-2.0..-0.2#0.1)", "templateType": "anything", "group": "Ungrouped variables", "name": "z", "description": ""}, "z1": {"definition": "random(-2.0..-0.2#0.1 except z)", "templateType": "anything", "group": "Ungrouped variables", "name": "z1", "description": ""}, "z2": {"definition": "random(0.2..2.0#0.1 except [abs(z),z1])", "templateType": "anything", "group": "Ungrouped variables", "name": "z2", "description": ""}, "w2": {"definition": "normalcdf(abs(z2),0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "w2", "description": ""}}, "metadata": {"notes": "7/02/2013:
\nFinished first draft. Tag sc included.
", "description": "Given a normal distribution $X \\sim N(m,\\sigma^2)$ find $P(X \\lt a),\\; a \\lt m$ and find $P(X\\gt b), \\; b\\gt m$.
", "licence": "None specified"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Stephanie's copy of business: steph and naomi ready for exam", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Stephanie Greaves", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/340/"}], "functions": {}, "ungrouped_variables": ["zl", "lower1", "perx", "this", "m", "athing", "l", "p", "s", "percentile", "u", "something", "tol", "zu", "upper1"], "tags": ["Normal distribution", "Probability", "continuous distributions", "distributions", "mean", "normal distribution", "normal tables", "percentiles", "probability", "sc", "standard deviation", "statistics", "z scores"], "preamble": {"css": "", "js": ""}, "advice": "Converting to $\\operatorname{N}(0,1)$:
\n\\[\\begin{eqnarray*}P(\\var{lower1} \\lt X \\lt \\var{upper1})&=&P(X \\lt \\var{upper1})- P(X \\lt \\var{lower1})\\\\&=&P\\left(Z \\lt \\frac{\\var{upper1}-\\var{m}}{\\var{s}}\\right)-P\\left(Z \\lt \\frac{\\var{lower1}-\\var{m}}{\\var{s}}\\right)\\\\&=&P(Z \\lt \\var{zu})-P(Z \\lt \\var{zl})=\\var{precround(u,4)}-\\var{precround(l,4)},\\\\&=&\\var{p}\\end{eqnarray*}\\] (to 3 decimal places.)
\nWe need to find $x$ such that:
\n\\[\\begin{eqnarray*}P(X \\lt x) &\\le& \\var{percentile/100}\\\\ \\Rightarrow P\\left(Z \\le \\frac{x-\\var{m}}{\\var{s}}\\right) &\\le& \\var{percentile/100}\\\\ \\Rightarrow \\frac{x-\\var{m}}{\\var{s}} &\\le& \\Phi^{-1}(\\var{percentile/100})=\\var{precround(normalinv(percentile/100,0,1),4)}\\\\ \\Rightarrow x &\\le&\\var{precround(normalinv(percentile/100,0,1),4)}\\times\\var{s}+\\var{m}=\\var{perX}\\end{eqnarray*}\\] to the nearest whole number.
\nHence $X=\\var{perX}$ gives the $\\var{percentile}$% percentile.
\n", "rulesets": {}, "parts": [{"prompt": "
What is the probability that {athing} has {this} between $\\var{lower1}$ and $\\var{upper1}$?
\nProbability = [[0]] (to 3 decimal places).
", "marks": 0, "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "marks": 1, "maxValue": "p+tol", "strictPrecision": false, "minValue": "p-tol", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "What value of $X$ is equivalent to the $\\var{percentile}$% percentile? (i.e. $\\var{percentile}$% of medium sized factories in Europe that have a workcount below this value)
\n$\\var{percentile}$% percentile = [[0]] (round to the nearest whole number).
", "marks": 0, "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "marks": 1, "maxValue": "perX+0.01", "strictPrecision": false, "minValue": "perX-0.01", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": 0, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "Let $X$ be the {something}. It is believed that $X$ has a normal distribution, with mean $\\var{m}$ and standard deviation $\\var{s}$. That is, $X \\sim \\operatorname{N}(\\var{m},\\var{s^2})$.
", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"zl": {"definition": "random(-2.0..-0.2#0.1)", "templateType": "anything", "group": "Ungrouped variables", "name": "zl", "description": ""}, "lower1": {"definition": "precround(zl*s+m,0)", "templateType": "anything", "group": "Ungrouped variables", "name": "lower1", "description": ""}, "upper1": {"definition": "precround(zu*s+m,0)", "templateType": "anything", "group": "Ungrouped variables", "name": "upper1", "description": ""}, "this": {"definition": "'a workforce count'", "templateType": "anything", "group": "Ungrouped variables", "name": "this", "description": ""}, "m": {"definition": "random(270..290#1)", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "athing": {"definition": "'a particular medium sized factory'", "templateType": "anything", "group": "Ungrouped variables", "name": "athing", "description": ""}, "l": {"definition": "1-normalcdf(abs(zl),0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "l", "description": ""}, "p": {"definition": "precround(u-l,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "s": {"definition": "random(5..8#1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "percentile": {"definition": "//perhaps should stick to standard values 90,95,99,99.9\n random(40..99 except 50)", "templateType": "anything", "group": "Ungrouped variables", "name": "percentile", "description": ""}, "u": {"definition": "normalcdf(zu,0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "u", "description": ""}, "something": {"definition": "'number of employees in a medium sized factory in Europe'", "templateType": "anything", "group": "Ungrouped variables", "name": "something", "description": ""}, "tol": {"definition": "0.001", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "zu": {"definition": "random(0.2..2.0 except abs(zl))", "templateType": "anything", "group": "Ungrouped variables", "name": "zu", "description": ""}, "perx": {"definition": "precround(normalinv(percentile/100,m,s),0)", "templateType": "anything", "group": "Ungrouped variables", "name": "perx", "description": ""}}, "metadata": {"notes": "7/02/2013:
\nAdded scenario tag sc. Have to use $\\operatorname{normalinv}(p/100,m,\\sigma)$ for a given percentile $p$% where $p$ ranges from 40 to 99. Might be better to choose standard values for p.
", "description": "Given normal distribution $\\operatorname{N}(m,\\sigma^2)$ find $P(a \\lt X \\lt b),\\; a \\lt m,\\;b \\gt m$ and also find the value of $X$ corresponding to a given percentile $p$%.
", "licence": "None specified"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}], "pickingStrategy": "all-ordered"}], "allQuestions": true, "pickQuestions": 0, "navigation": {"onleave": {"action": "none", "message": ""}, "browse": true, "reverse": true, "preventleave": true, "showfrontpage": true, "allowregen": true, "showresultspage": "never"}, "timing": {"timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}, "allowPause": true}, "metadata": {"notes": "", "description": "A resource for Business students to practise calculating Normal probabilities and finding $Z$-values.
", "licence": "Creative Commons Attribution 4.0 International"}, "name": "mathcentre: calculating probabilities from a normal distribution (Business)", "percentPass": 0, "shuffleQuestions": false, "type": "exam", "showQuestionGroupNames": false, "questions": [], "feedback": {"allowrevealanswer": true, "showtotalmark": true, "advicethreshold": 0, "showanswerstate": true, "showactualmark": true}, "duration": 0, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "extensions": ["stats"], "custom_part_types": [], "resources": []}