// Numbas version: exam_results_page_options
{"name": "Find regression equation and correlation coefficient", "extensions": ["stats", "jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "r1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[random(0..6),random(1..7),random(4..10),random(7..13),random(12..18),random(4..20),random(16..22),random(19..25),random(17..23),random(13..19),random(8..14),random(3..9)]", "description": "", "name": "r1"}, "spxy": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sxy-t[0]*t[1]/n", "description": "", "name": "spxy"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": 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"anything", "group": "Ungrouped variables", "definition": "0.01", "description": "", "name": "tol1"}, "beverage": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(\"home-brewed beer\",\"home-brewed lager\",\"specially-brewed beer\",\"super-strength lager\",\"cold-filtered lager\",\"ice-filtered cider\",\"cherry cider\")", "description": "", "name": "beverage"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "12", "description": "", "name": "n"}, "ss": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[ssq[0]-t[0]^2/n,ssq[1]-t[1]^2/n]", "description": "", "name": "ss"}, "corr": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tcorr,2)", "description": "", "name": "corr"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(spxy/ss[0],3)", "description": "", "name": "b"}, "pub": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(\"Black Bull Inn\",\"County Inn\",\"Dog and Duck Pub\",\"Slug and Lettuce Pub\", \"Cross Keys Pub\",\"Newcastle Arms Pub\",\"Red Lion Pub\")", "description": "", "name": "pub"}, "res": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(precround(r2[x]-(a+b*r1[x]),2),x,0..n-1)", "description": "", "name": "res"}, "ch": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..11)", "description": "", "name": "ch"}, "sc": {"templateType": "anything", "group": "Ungrouped variables", "definition": "r1[ch]", "description": "", "name": "sc"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.25..0.45#0.05)", "description": "", "name": "b1"}, "sxy": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sum(map(r1[x]*r2[x],x,0..n-1))", "description": "", "name": "sxy"}, "thisval": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(15..22)", "description": "", "name": "thisval"}, "obj": {"templateType": "anything", "group": "Ungrouped variables", "definition": "['Jan','Feb','March','April','May','June','July','August','Sept','Oct','Nov','Dec']", "description": "", "name": "obj"}, "owner": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(\"Kevin\",\"Mary\",\"Bill\",\"Doreen\",\"Peter\",\"Helen\",\"Michael\",\"Samantha\")", "description": "", "name": "owner"}, "r2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(round(a1+b1*x+random(-9..9)),x,r1)", "description": "", "name": "r2"}, "ls": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(a+b*sc,2)", "description": "", "name": "ls"}, "tsqovern": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[t[0]^2/n,t[1]^2/n]", "description": "", "name": "tsqovern"}, "rsquared": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(spxy^2/(ss[0]*ss[1]),3)", "description": "", "name": "rsquared"}}, "ungrouped_variables": ["ch", "prediction", "b1", "owner", "sxy", "res", "spxy", "ls", "tol", "tcorr", "tsqovern", "ssq", "sumr", "thisval", "a1", "pub", "corr", "a", "b", "obj", "r1", "r2", "ss", "tol1", "n", "beverage", "t", "sc", "rsquared"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Find regression equation and correlation coefficient", "functions": {"regressline": {"type": "html", "language": "javascript", "definition": "\n var div = Numbas.extensions.jsxgraph.makeBoard('600px','600px',\n{boundingBox:[-5,maxy,maxx,-5],\n axis:true,\n showNavigation:false,\n grid:false});\n   var board = div.board;  \nvar l1=board.create('text',[maxx/2,-2,'Temperature']);\nvar l2=board.create('text',[-2,maxy/2,'Sales']);\nvar names = ['Jan','Feb','Mar','Apr','May','Jun','Jul','Aug','Sep','Oct','Nov','Dec'];\nfor (j=0;j<12;j++){ board.create('point',[r1[j],r2[j]],{fixed:true, style:3,name:'\\\\\\\\['+names[j]+'\\\\\\\\]'})};\nvar a1 = board.create('point',[minx+5,miny+5],{color:'blue'});\nvar b1 = board.create('point',[minx+7,miny+5],{color:'blue'});\nfunction updr(a,b){\n  var s=0;\n  for(var i=0;i<12;i++){\ns=s+Math.pow(r2[i]-a*r1[i]-b,2);}\ns=Numbas.math.niceNumber(Numbas.math.precround(s,2));\n$('#rsquared').text(s);}\n  var li=board.create('line',[a1,b1], {straightFirst:false, straightLast:false});\n  var a=0;\n  var b=0;\n  function dr(p){\n  p.on('drag',function(){\n  a = Numbas.math.niceNumber((b1.Y()-a1.Y())/(b1.X()-a1.X()));\n  b = Numbas.math.niceNumber((a1.Y()*b1.X()-a1.X()*b1.Y())/(b1.X()-a1.X()));\n  Numbas.exam.currentQuestion.parts[1].gaps[0].display.studentAnswer(a);\n  Numbas.exam.currentQuestion.parts[1].gaps[1].display.studentAnswer(b);\n  updr(a,b);\n  })};\n  dr(a1);\n  dr(b1);\n  \nreturn div;\n\n  \n", "parameters": [["r1", "list"], ["r2", "list"], ["minx", "number"], ["maxx", "number"], ["miny", "number"], ["maxy", "number"]]}, "regfun": {"type": "html", "language": "javascript", "definition": "\n  var div = Numbas.extensions.jsxgraph.makeBoard('600px','600px',\n{boundingBox:[-5,maxy,maxx,-5],\n axis:true,\n showNavigation:false,\n grid:true});\n   var board = div.board;  \nvar l1=board.create('text',[maxx/2,-2,'Temperature']);\nvar l2=board.create('text',[-2,maxy/2,'Sales']);\n  var names = ['Jan','Feb','Mar','Apr','May','Jun','Jul','Aug','Sep','Oct','Nov','Dec'];\n  for (j=0;j<12;j++){ board.create('point',[r1[j],r2[j]],{fixed:true, style:3, strokecolor:\"#0000a0\", name:'\\\\\\\\['+names[j]+'\\\\\\\\]'})};\nvar regressionPolynomial = JXG.Math.Numerics.regressionPolynomial(1, r1, r2);\nvar reg = board.create('functiongraph',[regressionPolynomial],{strokeColor:'blue',name:'Regression Line.',withLabel:true});        \n  //for(var i=0;i<12;i++){board.create(\"segment\",[[r1[i],r2[i]],[r1[i],regressionPolynomial(r1[i])]])};\nvar regExpression = regressionPolynomial.getTerm();\nvar regTeX = Numbas.jme.display.exprToLaTeX(regExpression,[],scope);\n\nvar t = board.create('text',[1,5,\nfunction(){ return \"\\\\[r(Y) = \" + regExpression +'\\\\]';}\n],\n{strokeColor:'black',fontSize:18});  \nvar t1 = board.create('text',[5,maxy,\nfunction(){ return \"\\\\[SSE = \" + sumr +'\\\\]';}\n],\n{strokeColor:'black',fontSize:18});  \nvar t2 = board.create('text',[20,maxy,\nfunction(){ return \"\\\\[R^2 = \" + rsquared +'\\\\]';}\n],\n{strokeColor:'black',fontSize:18});  \nreturn div;\n  \n", "parameters": [["r1", "list"], ["r2", "list"], ["maxx", "number"], ["maxy", "number"], ["rsquared", "number"], ["sumr", "number"]]}}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "corr+tol1", "minValue": "corr-tol1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>Calculate the sample correlation coefficient $r$ for these data:</p>\n<p>$r=\\;$[[0]] (enter to 2 decimal places).</p>", "showCorrectAnswer": true, "marks": 0}, {"stepsPenalty": 0, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "b+tol", "minValue": "b-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "a+tol", "minValue": "a-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>Calculate the equation of the best fitting regression line.</p>\n<p>\\[Y = \\beta_0 + \\beta_1X.\\] Find $\\beta_0$ and $\\beta_1$ to 5 decimal places, then input them below to 3 decimal places. You will use these approximate values in the rest of the question.&nbsp;</p>\n<p>$\\beta_1=\\;$[[0]], &nbsp; &nbsp; &nbsp;$\\beta_0=\\;$[[1]] (enter both to 3 decimal places).</p>\n<p>You can experiment by dragging the points A and B around to see if you can get close to the regression line.&nbsp;</p>\n<p>{regressline(r1,r2,min(r1)-10,max(r1)+10,min(r2)-10,max(r2)+10)}</p>\n<p></p>\n<p>Click on Show steps if you want more information on calculating $\\beta_0$ and $\\beta_1$. You will not lose any marks by doing so.</p>\n<p>&nbsp;</p>", "steps": [{"type": "information", "prompt": "<p>To find $\\beta_0$ and $\\beta_1$ you first find &nbsp;$\\displaystyle \\beta_1 = \\frac{SPXY}{SSX}$ where:</p>\n<p>$\\displaystyle SPXY=\\sum xy - \\frac{(\\sum x)\\times (\\sum y)}{\\var{n}}$</p>\n<p>$\\displaystyle SSX=\\sum x^2 - \\frac{(\\sum x)^2}{\\var{n}}$</p>\n<p>Then $\\displaystyle \\beta_0 = \\frac{1}{\\var{n}}\\left[\\sum y-\\beta_1 \\sum x\\right]$</p>\n<p>Now go back and fill in the values for $\\beta_0$ and $\\beta_1$.</p>", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "prediction+1", "minValue": "prediction-1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>Next month, the average temperature in {owner}'s town is forecast to be &nbsp;{thisval} Celsius. Use the regression equation in the second part to predict sales of the {beverage} in that month.</p>\n<p>What is the predicted value of sales (in hundreds of pounds) ?</p>\n<p>Use the values of $\\beta_0$ and $\\beta_1$ you input above to 3 decinal places.</p>\n<p>Enter the predicted sales here: &nbsp;[[0]] (hundreds of pounds to the nearest whole number).</p>\n<p></p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>{owner} owns the {pub}. {owner} believes that sales of {beverage} in the pub are linked to the average monthly temperature, with higher sales being recorded in months with higher temperatures. To investigate, {owner} records the average monthly temperature in the local town over a period of one year ($X$ degrees Celsius), along with total monthly sales of {beverage} ($Y$ hundred pounds). The results are shown in the table below:&nbsp;</p>\n<table border=\"1\">\n<tbody>\n<tr><th align=\"left\">Month</th><th>$\\var{obj[0]}$</th><th>$\\var{obj[1]}$</th><th>$\\var{obj[2]}$</th><th>$\\var{obj[3]}$</th><th>$\\var{obj[4]}$</th><th>$\\var{obj[5]}$</th><th>$\\var{obj[6]}$</th><th>$\\var{obj[7]}$</th><th>$\\var{obj[8]}$</th><th>$\\var{obj[9]}$</th><th>$\\var{obj[10]}$</th><th>$\\var{obj[11]}$</th></tr>\n<tr><th align=\"left\">$X$&nbsp;(temperature)</th>\n<td>$\\var{r1[0]}$</td>\n<td>$\\var{r1[1]}$</td>\n<td>$\\var{r1[2]}$</td>\n<td>$\\var{r1[3]}$</td>\n<td>$\\var{r1[4]}$</td>\n<td>$\\var{r1[5]}$</td>\n<td>$\\var{r1[6]}$</td>\n<td>$\\var{r1[7]}$</td>\n<td>$\\var{r1[8]}$</td>\n<td>$\\var{r1[9]}$</td>\n<td>$\\var{r1[10]}$</td>\n<td>$\\var{r1[11]}$</td>\n</tr>\n<tr><th align=\"left\">$Y$&nbsp;(sales, &pound;100s)</th>\n<td>$\\var{r2[0]}$</td>\n<td>$\\var{r2[1]}$</td>\n<td>$\\var{r2[2]}$</td>\n<td>$\\var{r2[3]}$</td>\n<td>$\\var{r2[4]}$</td>\n<td>$\\var{r2[5]}$</td>\n<td>$\\var{r2[6]}$</td>\n<td>$\\var{r2[7]}$</td>\n<td>$\\var{r2[8]}$</td>\n<td>$\\var{r2[9]}$</td>\n<td>$\\var{r2[10]}$</td>\n<td>$\\var{r2[11]}$</td>\n</tr>\n</tbody>\n</table>\n<p>You are given the following information:</p>\n<table border=\"1\">\n<tbody>\n<tr><th align=\"left\">$X$&nbsp;</th>\n<td>$\\sum x=\\;\\var{t[0]}$</td>\n<td>$\\sum x^2=\\;\\var{ssq[0]}$</td>\n</tr>\n<tr><th align=\"left\">$Y$</th>\n<td>$\\sum y=\\;\\var{t[1]}$</td>\n<td>$\\sum y^2=\\;\\var{ssq[1]}$</td>\n</tr>\n</tbody>\n</table>\n<p>Also you are given $\\sum xy = \\var{sxy}$.</p>", "tags": ["ACE2013", "checked2015", "correlation", "data analysis", "fitted value", "linear regression", "regression", "statistics"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>04/02/2014:</strong></p>\n<p>No advice as yet. Adapted from iassess question for ACE.</p>\n<p><strong>18/02/2014:</strong></p>\n<p>Slight changes in notation from Regression 3. No SSE</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find a regression equation given 12 months data on temperature and sales of a drink.&nbsp;</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>{regfun(r1,r2,max(r1)+10,max(r2)+10,rsquared,sumr)}</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/"}]}