// Numbas version: finer_feedback_settings
{"name": "Conditional probability in normal distribution, ", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=0,heightm,heightw)", "description": "", "name": "m"}, "z": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-2.0..-0.2#0.1)", "description": "", "name": "z"}, "z2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.2..2.0#0.1 except [abs(z),z1])", "description": "", "name": "z2"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "name": "t"}, "age": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(20..40)", "description": "", "name": "age"}, "units1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'inches'", "description": "", "name": "units1"}, "w2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1-normalcdf(abs(z2),0,1)", "description": "", "name": "w2"}, "heightm": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(68..72#0.5)", "description": "", "name": "heightm"}, "lower1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "z1*s+m", "description": "", "name": "lower1"}, "upper1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "z2*s+m", "description": "", "name": "upper1"}, "prop": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(100*(1-normalcdf(abs(z),0,1)),1)", "description": "", "name": "prop"}, "s": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(5..7)", "description": "", "name": "s"}, "heightw": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(63..65#0.5)", "description": "", "name": "heightw"}, "mw": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=0,'men','women')", "description": "", "name": "mw"}, "z1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-2.0..-0.2#0.1 except z)", "description": "", "name": "z1"}, "lower": {"templateType": "anything", "group": "Ungrouped variables", "definition": "z*s+m", "description": "", "name": "lower"}, "prop1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(100*(w2)/(w1),1)", "description": "", "name": "prop1"}, "w1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "normalcdf(abs(z1),0,1)", "description": "", "name": "w1"}}, "ungrouped_variables": ["units1", "lower", "z", "age", "heightw", "m", "lower1", "prop", "heightm", "s", "t", "w1", "prop1", "mw", "upper1", "z1", "z2", "w2"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Conditional probability in normal distribution, ", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "prop+0.1", "minValue": "prop-0.1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>What proportion of such {mw} are less than $\\var{lower}$ {units1} tall?</p>\n      <p>Proportion =?[[0]]% (to 1 decimal place)..</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "prop1+0.1", "minValue": "prop1-0.1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>What proportion of such {mw} over $\\var{lower1}$ {units1} tall are more than $\\var{upper1}$ {units1} tall?</p>\n        <p>Proportion = ?[[0]]% (to one decimal place).</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Suppose that the height in {units1}, of $\\\\var{age}$ year old {mw} is a normal distribution with mean $\\var{m}$ and standard deviation $\\var{s}$.</p>", "tags": ["Baye's Theorem", "checked2015", "conditional probability", "converting to z scores", "MAS1604", "MAS2304", "mean ", "Normal distribution", "normal distribution", "normal tables", "Probability", "standard deviation", "statistics", "z scores"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>7/02/2013:</strong></p>\n                <p>Finished first draft. Tag sc included.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given a normal distribution $X \\sim N(m,\\sigma^2)$ find $P(X \\lt a),\\; a \\lt m$ and the conditional probability $P(X \\gt b | X \\gt c)$ where $b \\lt m$ and $c \\gt m$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>a)</p>\n    <p>Converting to $\\operatorname{N}(0,1)$</p>\n    <p>\\[\\begin{eqnarray*}P(X \\lt \\var{lower})&amp;=&amp;P\\left(Z \\lt \\frac{\\var{lower}-\\var{m}}{\\var{s}}\\right)\\\\&amp;=&amp;P(Z\\lt \\var{z})=1-P(Z \\lt \\var{-z})=1-\\var{normalcdf(-z,0,1)}\\\\&amp;=&amp;\\var{prop/100}\\end{eqnarray*}\\] to 3 decimal places.</p>\n    <p>Hence the proportion is $\\var{prop}$% to 1 decimal place.</p>\n    <p>b)</p>\n    <p>The proportion of such {mw} over $\\var{lower1}$ {units1} tall that are more than $\\var{upper1}$ {units1} tall is given by finding the conditional probability:</p>\n    <p>\\[\\begin{eqnarray*}P(X \\gt \\var{upper1}|X \\gt\\var{lower1})&amp;=&amp;\\frac{P(X \\gt \\var{upper1} \\text{ and } X \\gt \\var{lower1})}{P(X \\gt \\var{lower1})}\\\\&amp;=&amp;\\frac{P(X \\gt \\var{upper1})}{P(X \\gt \\var{lower1})}\\\\&amp;=&amp;\\frac{1-P(X \\le \\var{upper1})}{1-P(X \\le \\var{lower1})}\\\\&amp;=&amp;\\frac{1-\\Phi(\\var{z2})}{1-\\Phi(\\var{z1})}\\\\&amp;=&amp;\\frac{1-\\Phi(\\var{z2})}{\\Phi(\\var{-z1})}=\\frac{\\var{w2}}{\\var{w1}}\\\\&amp;=&amp;\\var{prop1/100}\\end{eqnarray*}\\]</p>\n    <p>to 3 decimal places.</p>\n    <p>hence the proportion is $\\var{prop1}$% to 1 decimal place.</p>", "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/"}]}