// Numbas version: finer_feedback_settings
{"name": "BES220: 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": [{"functions": {"regfun": {"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", "type": "html", "language": "javascript", "parameters": [["r1", "list"], ["r2", "list"], ["maxx", "number"], ["maxy", "number"], ["rsquared", "number"], ["sumr", "number"]]}, "regressline": {"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", "type": "html", "language": "javascript", "parameters": [["r1", "list"], ["r2", "list"], ["minx", "number"], ["maxx", "number"], ["miny", "number"], ["maxy", "number"]]}}, "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"], "name": "BES220: Find regression equation and correlation coefficient", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>{regfun(r1,r2,max(r1)+10,max(r2)+10,rsquared,sumr)}</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "<p>Calculate the sample correlation coefficient $r$ for these data:</p>\n<p>$r=\\;$[[0]] (enter to 2 decimal places).</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "corr+tol1", "minValue": "corr-tol1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "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>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"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>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "b+tol", "minValue": "b-tol", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry"}, {"allowFractions": false, "variableReplacements": [], "maxValue": "a+tol", "minValue": "a-tol", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "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>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "prediction+1", "minValue": "prediction-1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "extensions": ["stats", "jsxgraph"], "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} ($y$ thousand&nbsp;rands). 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, R1000s)</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>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"ch": {"definition": "random(0..11)", "templateType": "anything", "group": "Ungrouped variables", "name": "ch", "description": ""}, "prediction": {"definition": "round(a+b*thisval)", "templateType": "anything", "group": "Ungrouped variables", "name": "prediction", "description": ""}, "b1": {"definition": "random(0.25..0.45#0.05)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "owner": {"definition": "random(\"Kevin\",\"Mary\",\"Bill\",\"Doreen\",\"Peter\",\"Helen\",\"Michael\",\"Samantha\")", "templateType": "anything", "group": "Ungrouped variables", "name": "owner", "description": ""}, "sxy": {"definition": "sum(map(r1[x]*r2[x],x,0..n-1))", "templateType": "anything", "group": "Ungrouped variables", "name": "sxy", "description": ""}, "res": {"definition": "map(precround(r2[x]-(a+b*r1[x]),2),x,0..n-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "res", "description": ""}, "spxy": {"definition": "sxy-t[0]*t[1]/n", "templateType": "anything", "group": "Ungrouped variables", "name": "spxy", "description": ""}, "ls": {"definition": "precround(a+b*sc,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "ls", "description": ""}, "tol": {"definition": "0.001", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "tcorr": {"definition": "spxy/sqrt(ss[0]*ss[1])", "templateType": "anything", "group": "Ungrouped variables", "name": "tcorr", "description": ""}, "a": {"definition": "precround(1/n*(t[1]-spxy/ss[0]*t[0]),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "ssq": {"definition": "[sum(map(x^2,x,r1)),sum(map(x^2,x,r2))]", "templateType": "anything", "group": "Ungrouped variables", "name": "ssq", "description": ""}, "sumr": {"definition": "precround(sum(map(res[x]^2,x,0..n-1)),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "sumr", "description": ""}, "pub": {"definition": "random(\"Black Bull Inn\",\"County Inn\",\"Dog and Duck Pub\",\"Slug and Lettuce Pub\", \"Cross Keys Pub\",\"Newcastle Arms Pub\",\"Red Lion Pub\")", "templateType": "anything", "group": "Ungrouped variables", "name": "pub", "description": ""}, "a1": {"definition": "random(10..20)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "thisval": {"definition": "random(15..22)", "templateType": "anything", "group": "Ungrouped variables", "name": "thisval", "description": ""}, "corr": {"definition": "precround(tcorr,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "corr", "description": ""}, "tsqovern": {"definition": "[t[0]^2/n,t[1]^2/n]", "templateType": "anything", "group": "Ungrouped variables", "name": "tsqovern", "description": ""}, "b": {"definition": "precround(spxy/ss[0],3)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "obj": {"definition": "['Jan','Feb','March','April','May','June','July','August','Sept','Oct','Nov','Dec']", "templateType": "anything", "group": "Ungrouped variables", "name": "obj", "description": ""}, "r1": {"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)]", "templateType": "anything", "group": "Ungrouped variables", "name": "r1", "description": ""}, "r2": {"definition": "map(round(a1+b1*x+random(-9..9)),x,r1)", "templateType": "anything", "group": "Ungrouped variables", "name": "r2", "description": ""}, "ss": {"definition": "[ssq[0]-t[0]^2/n,ssq[1]-t[1]^2/n]", "templateType": "anything", "group": "Ungrouped variables", "name": "ss", "description": ""}, "tol1": {"definition": "0.01", "templateType": "anything", "group": "Ungrouped variables", "name": "tol1", "description": ""}, "n": {"definition": "12", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "beverage": {"definition": "random(\"home-brewed beer\",\"home-brewed lager\",\"specially-brewed beer\",\"super-strength lager\",\"cold-filtered lager\",\"ice-filtered cider\",\"cherry cider\")", "templateType": "anything", "group": "Ungrouped variables", "name": "beverage", "description": ""}, "t": {"definition": "[sum(r1),sum(r2)]", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}, "sc": {"definition": "r1[ch]", "templateType": "anything", "group": "Ungrouped variables", "name": "sc", "description": ""}, "rsquared": {"definition": "precround(spxy^2/(ss[0]*ss[1]),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "rsquared", "description": ""}}, "metadata": {"description": "<p>Find a regression equation given 12 months data on temperature and sales of a drink.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Elias Jakobus Willemse", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/846/"}]}]}], "contributors": [{"name": "Elias Jakobus Willemse", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/846/"}]}