// Numbas version: finer_feedback_settings
{"name": "Pearson2", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {"darr": {"definition": "if(n=1,a,darr(n-1,m,[random(1..m except a)]+a))", "type": "list", "language": "jme", "parameters": [["n", "number"], ["m", "number"], ["a", "list"]]}, "rk": {"definition": "\n        /*This gives the ranking of the entries in a, c counts the ties */\n        var out = [];\n      for(var j=0;j<a.length;j++){\n        var s=0;\n        var c=0;\n        for(var i=0;i<a.length;i++){\n      if(a[j]>a[i]){s+=1;}\n      else\n        if(a[j]==a[i]){c+=1;}\n        }\n      out[j]=(2*s+c+1)/2;\n      }\n      return out;\n        \n      ", "type": "list", "language": "javascript", "parameters": [["a", "list"]]}, "pstdev": {"definition": "sqrt(abs(l)/(abs(l)-1))*stdev(l)", "type": "number", "language": "jme", "parameters": [["l", "list"]]}, "marr": {"definition": "if(length(a)=2,max(a[0],a[1]),max(a[0],marr(a[1..length(a)])))", "type": "number", "language": "jme", "parameters": [["a", "list"]]}, "tesarr": {"definition": "if(marr(map(abs(x),x,list(vector(a)-vector(b))))<d,b,tesarr(a,darr(length(a),m,[random(1..m)]),d,m))", "type": "list", "language": "jme", "parameters": [["a", "list"], ["b", "list"], ["d", "number"], ["m", "number"]]}}, "name": "Pearson2", "tags": ["Pearson correlation coefficient", "correlation coefficient", "data analysis", "hypothesis testing", "statistics"], "advice": "<p>The answers to all parts are given on revealing.</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n        <table border=\"1\">\n        <tbody>\n        <tr><th align=\"left\">Wife $(X)$</th>\n        <td>$\\sum x=\\;$[[0]]</td>\n        <td>$\\sum x^2=\\;$[[1]]</td>\n        </tr>\n        <tr><th align=\"left\">Husband $(Y)$</th>\n        <td>$\\sum y=\\;$[[2]]</td>\n        <td>$\\sum y^2=\\;$[[3]]</td>\n        </tr>\n        </tbody>\n        </table>\n        <p>Also find $\\sum xy=\\;$[[4]] and then:</p>\n        <p>$\\displaystyle SSX = \\;$[[5]]</p>\n        <p>$\\displaystyle SSY = \\;$[[6]]</p>\n        <p>$\\displaystyle SPXY = \\;$[[7]]</p>\n        <p>Hence calculate the correlation coefficient $r$:</p>\n        <p>$r=\\;$[[8]]</p>\n        <p>&nbsp;</p>\n        \n      ", "gaps": [{"minvalue": "t[0]", "type": "numberentry", "maxvalue": "t[0]", "marks": 0.5, "showPrecisionHint": false}, {"minvalue": "ssq[0]", "type": "numberentry", "maxvalue": "ssq[0]", "marks": 0.5, "showPrecisionHint": false}, {"minvalue": "t[1]", "type": "numberentry", "maxvalue": "t[1]", "marks": 0.5, "showPrecisionHint": false}, {"minvalue": "ssq[1]", "type": "numberentry", "maxvalue": "ssq[1]", "marks": 0.5, "showPrecisionHint": false}, {"minvalue": "sxy", "type": "numberentry", "maxvalue": "sxy", "marks": 0.5, "showPrecisionHint": false}, {"minvalue": "ss[0]", "type": "numberentry", "maxvalue": "ss[0]", "marks": 0.5, "showPrecisionHint": false}, {"minvalue": "ss[1]", "type": "numberentry", "maxvalue": "ss[1]", "marks": 0.5, "showPrecisionHint": false}, {"minvalue": "spxy", "type": "numberentry", "maxvalue": "spxy", "marks": 0.5, "showPrecisionHint": false}, {"minvalue": "corrcoef-tol", "type": "numberentry", "maxvalue": "corrcoef+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n        <p>Give the value of the correlation coefficient you have found, choose the range for the $p$ value by looking up the relevant table. Input the required values from the table here:</p>\n        <table border=\"1\">\n        <tbody>\n        <tr>\n        <td>$10\\%$</td>\n        <td>$5\\%$</td>\n        <td>$1\\%$</td>\n        <td>$0.2\\%$</td>\n        </tr>\n        <tr>\n        <td>[[0]]</td>\n        <td>[[1]]</td>\n        <td>[[2]]</td>\n        <td>[[3]]</td>\n        </tr>\n        </tbody>\n        </table>\n        <p>Then make a decision based on the $p$-value you have found by choosing one of these options:</p>\n        <p>[[4]]</p>\n        \n      ", "gaps": [{"minvalue": 0.549, "type": "numberentry", "maxvalue": 0.549, "marks": 0.25, "showPrecisionHint": false}, {"minvalue": 0.632, "type": "numberentry", "maxvalue": 0.632, "marks": 0.25, "showPrecisionHint": false}, {"minvalue": 0.765, "type": "numberentry", "maxvalue": 0.765, "marks": 0.25, "showPrecisionHint": false}, {"minvalue": 0.847, "type": "numberentry", "maxvalue": 0.847, "marks": 0.25, "showPrecisionHint": false}, {"maxanswers": 0.0, "distractors": ["", "", "", "", ""], "matrix": "v", "shufflechoices": false, "minanswers": 0.0, "choices": ["$p \\leq 0.002$, very strong evidence to reject the null hypothesis that there is no association.", "$0.002 \\lt p \\leq 0.01$, strong evidence to reject the null  hypothesis that there is no association.", "$0.01 \\lt p \\leq 0.05$, there is evidence to reject the null hypothesis that there is no association.", "$0.05 \\lt p \\leq 0.1$, weak evidence  against, do not reject the null hypothesis but consider further investigation.", "$p \\gt 0.1$, no evidence to reject the null hypothesis that there is no association."], "displaytype": "radiogroup", "maxmarks": 0.0, "marks": 0.0, "displaycolumns": 1.0, "type": "1_n_2", "minmarks": 0.0}], "type": "gapfill", "marks": 0.0}], "extensions": ["stats"], "statement": "\n  <p>It is well known that similarity in attitudes, beliefs and interests plays an important role in interpersonal attraction. A researcher developed a questionnaire which was completed by 10 married couples. One question sought to place each individual on a 20 point scale in which low scores represent liberal attitudes and high scores represent conservative attitudes. The data were:</p>\n    <table border=\"1\">\n    <tbody>\n    <tr><th align=\"left\">Couple</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></tr>\n    <tr><th align=\"left\">Wife $(X)$</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    </tr>\n    <tr><th align=\"left\">Husband $(Y)$</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    </tr>\n    </tbody>\n    </table>\n    <p>In this exercise you will find the Pearson correlation coefficent for the above paired data and&nbsp;comment on the significance of the calculated correlation.</p>\n    <p>The null hypothesis you are testing is:</p>\n    <p>$H_0$: There is no association between the attitudes of wives and husbands.</p>\n  ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"aspcoef": {"definition": "abs(spcoef)", "name": "aspcoef"}, "spcoef": {"definition": "precround(1-6*ssd/(n*(n^2-1)),3)", "name": "spcoef"}, "vs": {"definition": "switch(aspcoef >=0.952,[1,0,0,0,0],aspcoef>=0.881,[0,1,0,0,0],aspcoef>=0.738,[0,0,1,0,0],aspcoef>=0.643,[0,0,0,1,0],[0,0,0,0,1])", "name": "vs"}, "sxy": {"definition": "sum(map(r1[x]*r2[x],x,0..n-1))", "name": "sxy"}, "spxy": {"definition": "sxy-t[0]*t[1]/n", "name": "spxy"}, "tol": {"definition": 0.001, "name": "tol"}, "ssq": {"definition": "[sum(map(x^2,x,r1)),sum(map(x^2,x,r2))]", "name": "ssq"}, "corrcoef": {"definition": "precround(spxy/sqrt(ss[0]*ss[1]),3)", "name": "corrcoef"}, "ssd": {"definition": "sum(map(x^2,x,d))", "name": "ssd"}, "rr2": {"definition": "rk(r2)", "name": "rr2"}, "rr1": {"definition": "rk(r1)", "name": "rr1"}, "d": {"definition": "list(vector(rr1)-vector(rr2))", "name": "d"}, "tsqovern": {"definition": "[t[0]^2/n,t[1]^2/n]", "name": "tsqovern"}, "obj": {"definition": "['A','B','C','D','E','F','G','H','I','J']", "name": "obj"}, "r1": {"definition": "darr(n,m,[random(1..20)])", "name": "r1"}, "r2": {"definition": "tesarr(r1,darr(n,m,[random(1..m)]),11,m)", "name": "r2"}, "ss": {"definition": "[ssq[0]-t[0]^2/n,ssq[1]-t[1]^2/n]", "name": "ss"}, "k": {"definition": 3.0, "name": "k"}, "m": {"definition": 20.0, "name": "m"}, "n": {"definition": 10.0, "name": "n"}, "t": {"definition": "[sum(r1),sum(r2)]", "name": "t"}, "v": {"definition": "switch(corrcoef >=0.847,[1,0,0,0,0],corrcoef>=0.765,[0,1,0,0,0],corrcoef>=0.632,[0,0,1,0,0],corrcoef>=0.549,[0,0,0,1,0],[0,0,0,0,1])", "name": "v"}}, "metadata": {"notes": "\n    \t\t                                    \t\t<p><strong>30/09/2102:</strong></p>\n    \t\t                                    \t\t                                                        <p>Introduced three functions:</p>\n    \t\t                                    \t\t                                                        <p>1. To produce the ranking of a list of 8 numbers.</p>\n    \t\t                                    \t\t                                                        <p>2. To produce a list of 8 numbers from a scale of 1..20 which are all distinct.</p>\n    \t\t                                    \t\t                                                        <p>3. To produce the maximum of the numbers in a list.</p>\n    \t\t                                    \t\t                                                        <p>4. Given an array such as in 2. to find another such array which has max diff between any two corresponding entries less than a given number. This is to ensure that the two array produced do not differ too much, as the point of the exercise is to show that there is a positive high correlation.</p>\n    \t\t                                    \t\t                                                        <p>&nbsp;</p>\n    \t\t                                    \n    \t\t", "description": "<p>Calculate the Pearson correlation coefficient on paired data and comment on the significance.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}