// Numbas version: exam_results_page_options
{"name": "Adaptive marking", "allQuestions": true, "questions": [{"variables": {"roots": {"name": "roots", "description": "", "templateType": "anything", "group": "Quadratic equation", "definition": "sort(shuffle(list(-5..5))[0..2])"}, "b": {"name": "b", "description": "<p>Greatest root of the quadratic</p>", "templateType": "anything", "group": "Quadratic equation", "definition": "roots[1]"}, "ys": {"name": "ys", "description": "<p>y coordinates corresponding to the given x coordinates</p>", "templateType": "anything", "group": "Quadratic equation", "definition": "map((x-a)*(x-b),x,xs)"}, "a": {"name": "a", "description": "<p>Least root of the quadratic equation</p>", "templateType": "anything", "group": "Quadratic equation", "definition": "roots[0]"}, "xs": {"name": "xs", "description": "<p>Three given x coordinates</p>", "templateType": "anything", "group": "Quadratic equation", "definition": "shuffle(-5..5 except [a,b])[0..3]"}}, "name": "Adaptive marking: quadratic equation", "parts": [{"gaps": [{"maxValue": "a", "marks": "1", "variableReplacements": [], "type": "numberentry", "showPrecisionHint": false, "allowFractions": false, "correctAnswerFraction": false, "minValue": "a", "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst"}, {"maxValue": "b", "marks": "1", "variableReplacements": [], "type": "numberentry", "showPrecisionHint": false, "allowFractions": false, "correctAnswerFraction": false, "minValue": "b", "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "scripts": {}, "marks": 0, "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>A quadratic function&nbsp;$f(x)$ goes through the points $(\\var{xs[0]},\\var{ys[0]})$,&nbsp;$(\\var{xs[1]},\\var{ys[1]})$ and&nbsp;$(\\var{xs[2]},\\var{ys[2]})$.</p>\n<p>Find the two roots of $f$. Write the&nbsp;least root first.</p>\n<p>Least root:&nbsp;[[0]]</p>\n<p>Greatest root:&nbsp;[[1]]</p>", "variableReplacementStrategy": "originalfirst"}, {"gaps": [{"showpreview": true, "marks": 1, "expectedvariablenames": [], "answer": "(x-{a})(x-{b})", "vsetrangepoints": 5, "variableReplacements": [{"part": "p0g0", "variable": "a", "must_go_first": false}, {"part": "p0g1", "variable": "b", "must_go_first": false}], "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "scripts": {}, "marks": 0, "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>Write&nbsp;a quadratic function $g(x)$ whose roots are the values you entered above, with $x^2$ coefficient $1$.</p>\n<p>$g(x) =&nbsp;$&nbsp;[[0]]</p>", "variableReplacementStrategy": "originalfirst"}, {"maxValue": "mean(roots)", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false, "variableReplacements": [{"part": "p0", "variable": "roots", "must_go_first": false}], "showCorrectAnswer": true, "prompt": "<p>What is the midpoint of the two roots of $f$?</p>", "type": "numberentry", "scripts": {}, "allowFractions": false, "minValue": "mean(roots)", "variableReplacementStrategy": "originalfirst"}], "variable_groups": [{"variables": ["roots", "a", "b", "xs", "ys"], "name": "Quadratic equation"}], "functions": {}, "tags": [], "ungrouped_variables": [], "statement": "<p>(This question doesn't make a lot of pedagogic sense, it just shows off how adaptive marking works)</p>", "preamble": {"js": "", "css": ""}, "metadata": {"description": "<p data-bind=\"html: 'hi'\">Give the student three points lying on a quadratic, and ask them to find the roots.</p>\n<p data-bind=\"html: 'hi'\">Then ask them to find the equation of the quadratic, using their roots. Error in calculating the roots is carried forward.</p>\n<p data-bind=\"html: 'hi'\">Finally, ask them to find the midpoint of the roots (just for fun). Error is carried forward again.</p>", "notes": "", "licence": "None specified"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "advice": "", "rulesets": {}}, {"variables": {"rad_rotation": {"name": "rad_rotation", "description": "<p>Angle of rotation in radians</p>", "templateType": "anything", "group": "Rotation matrix", "definition": "radians(rotation)"}, "det_m": {"name": "det_m", "description": "", "templateType": "anything", "group": "Rotation matrix", "definition": "det(mat)"}, "mat": {"name": "mat", "description": "<p>Rotation matrix for the given rotation</p>", "templateType": "anything", "group": "Rotation matrix", "definition": "matrix([\n  [cos(rad_rotation),-sin(rad_rotation)],\n  [sin(rad_rotation),cos(rad_rotation)]\n])"}, "rotation": {"name": "rotation", "description": "<p>Angle of rotation in degrees</p>", "templateType": "anything", "group": "Rotation matrix", "definition": "random(10..90)"}}, "name": "Adaptive marking: rotation matrix", "parts": [{"gaps": [{"correctAnswerFractions": false, "precision": "2", "strictPrecision": false, "marks": "4", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacements": [], "showCorrectAnswer": true, "markPerCell": true, "allowResize": false, "type": "matrix", "scripts": {}, "allowFractions": false, "correctAnswer": "mat", "numRows": "2", "numColumns": "2", "tolerance": 0, "precisionPartialCredit": 0, "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "scripts": {}, "marks": 0, "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>Write a matrix $\\mathrm{M}$ corresponding to a rotation of $\\var{rotation}^{\\circ}$ anti-clockwise about the origin.</p>\n<p>Round numbers to 2 decimal places.</p>\n<p>$\\mathrm{M} = $&nbsp;[[0]]</p>", "variableReplacementStrategy": "originalfirst"}, {"gaps": [{"maxValue": "det(mat)", "variableReplacements": [{"part": "p0g0", "variable": "mat"}], "precision": "2", "showPrecisionHint": false, "marks": 1, "precisionType": "dp", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "type": "numberentry", "scripts": {}, "allowFractions": false, "strictPrecision": false, "minValue": "det(mat)", "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "scripts": {}, "marks": 0, "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>$\\operatorname{det}\\mathrm{M} = $&nbsp;[[0]] (Round your answer to 2 decimal places)</p>", "variableReplacementStrategy": "originalfirst"}, {"gaps": [{"correctAnswerFractions": false, "precision": "2", "strictPrecision": false, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacements": [{"part": "p0g0", "variable": "mat"}, {"part": "p1g0", "variable": "det_m"}], "showCorrectAnswer": true, "markPerCell": false, "allowResize": false, "type": "matrix", "scripts": {}, "allowFractions": false, "correctAnswer": "matrix([[mat[1][1],-mat[0][1]],[-mat[1][0],mat[0][0]]])/det_m", "numRows": "2", "numColumns": "2", "tolerance": 0, "precisionPartialCredit": 0, "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "scripts": {}, "marks": 0, "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>$\\mathrm{M}^{-1} = $&nbsp;[[0]]</p>\n<p></p>", "variableReplacementStrategy": "originalfirst"}], "variable_groups": [{"variables": ["rotation", "rad_rotation", "mat", "det_m"], "name": "Rotation matrix"}], "functions": {}, "tags": [], "ungrouped_variables": [], "statement": "", "preamble": {"js": "", "css": ""}, "metadata": {"description": "<p data-bind=\"html: 'hi'\">Ask the student to find a&nbsp;matrix corresponding to a given rotation about the origin.</p>\n<p data-bind=\"html: 'hi'\">Then ask them to find the determinant. Their answer is marked against the matrix they gave, not just the correct one.</p>\n<p data-bind=\"html: 'hi'\">Finally, ask them to find the inverse of their matrix. Marking is&nbsp;against the matrix and determinant they gave.</p>", "notes": "", "licence": "None specified"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "advice": "", "rulesets": {}}, {"variables": {"p_value_range": {"name": "p_value_range", "description": "<p>Describe where the p-value lies in relation to the critical values</p>", "templateType": "anything", "group": "Advice messages", "definition": "['is less than $0.001$','lies between $0.001$ and $0.01$','lies between $0.01$ and $0.05$','lies between $0.05$ and $0.10$','is greater than $0.10$'][scenario]"}, "sd2": {"name": "sd2", "description": "<p>Sample standard deviation of sample 2</p>", "templateType": "anything", "group": "Stats", "definition": "precround(pstdev(r2),3)"}, "mu1": {"name": "mu1", "description": "<p>Population mean of sample 1 (we'll generate samples from different distributions to produce different outcomes)</p>", "templateType": "anything", "group": "Setup", "definition": "random(55..75#0.5)"}, "sd1": {"name": "sd1", "description": "<p>Sample standard deviation of sample 1</p>", "templateType": "anything", "group": "Stats", "definition": "precround(pstdev(r1),3)"}, "sigma2": {"name": "sigma2", "description": "<p>Population standard deviation of sample 2</p>", "templateType": "anything", "group": "Setup", "definition": "random(8..10#0.2)"}, "decision_marking_matrix": {"name": "decision_marking_matrix", "description": "<p>Marking matrix for the multiple choice questions</p>", "templateType": "anything", "group": "Advice messages", "definition": "[\n  [1,0,0,0,0],\n  [0,1,0,0,0],\n  [0,0,1,0,0],\n  [0,0,0,1,0],\n  [0,0,0,0,1]\n][scenario]"}, "r2": {"name": "r2", "description": "<p>Sample 2</p>", "templateType": "anything", "group": "Samples", "definition": "repeat(round(normalsample(mu2,sigma2)),n2)"}, "evidence_strength": {"name": "evidence_strength", "description": "<p>How much evidence is there against the null hypothesis?</p>", "templateType": "anything", "group": "Advice messages", "definition": "['very strong','strong','slight','no','no'][scenario]"}, "n2": {"name": "n2", "description": "<p>Size of sample 2</p>", "templateType": "anything", "group": "Setup", "definition": "10"}, "s": {"name": "s", "description": "<p>Used in the&nbsp;formula for the t statistic</p>", "templateType": "anything", "group": "Stats", "definition": "precround(sqrt(((n1-1)*sd1^2+(n2-1)*sd2^2)/(n1+n2-2)),3)"}, "t99": {"name": "t99", "description": "", "templateType": "anything", "group": "Critical t-values", "definition": "2.878"}, "mean1": {"name": "mean1", "description": "<p>Sample mean of sample 1</p>", "templateType": "anything", "group": "Stats", "definition": "mean(r1)"}, "t95": {"name": "t95", "description": "", "templateType": "anything", "group": "Critical t-values", "definition": "2.101"}, "reject": {"name": "reject", "description": "<p>Do we reject the null hypothesis?</p>", "templateType": "anything", "group": "Advice messages", "definition": "if(scenario<2,'do reject','do not reject')"}, "t999": {"name": "t999", "description": "", "templateType": "anything", "group": "Critical t-values", "definition": "3.922"}, "sigma1": {"name": "sigma1", "description": "<p>Population standard deviation of sample 1</p>", "templateType": "anything", "group": "Setup", "definition": "random(8..10#0.2)"}, "t_statistic": {"name": "t_statistic", "description": "", "templateType": "anything", "group": "Stats", "definition": "(mean1-mean2)*sqrt(n1*n2)/(s*sqrt(n1+n2))"}, "r1": {"name": "r1", "description": "<p>Sample 1</p>", "templateType": "anything", "group": "Samples", "definition": "repeat(round(normalsample(mu1,sigma1)),n1)"}, "t_statistic_range": {"name": "t_statistic_range", "description": "<p>Describe where the t-statistic lies in relation to the critical values</p>", "templateType": "anything", "group": "Advice messages", "definition": "['is greater than $\\\\var{t999}$','lies between $\\\\var{t99}$ and $\\\\var{t999}$','lies between $\\\\var{t95}$ and $\\\\var{t99}$','lies between $\\\\var{t90}$ and $\\\\var{t95}$','is less than $\\\\var{t90}$'][scenario]"}, "n1": {"name": "n1", "description": "<p>Size of sample 1</p>", "templateType": "anything", "group": "Setup", "definition": "10"}, "mean2": {"name": "mean2", "description": "<p>Sample&nbsp;mean of sample 1</p>", "templateType": "anything", "group": "Stats", "definition": "mean(r2)"}, "p_value": {"name": "p_value", "description": "<p>p-value&nbsp;corresponding to the t-statistic</p>", "templateType": "anything", "group": "Stats", "definition": "ttest(abs(t_statistic),19,2)"}, "t90": {"name": "t90", "description": "", "templateType": "anything", "group": "Critical t-values", "definition": "1.734"}, "mu2": {"name": "mu2", "description": "<p>Population mean of sample 2</p>", "templateType": "anything", "group": "Setup", "definition": "random(65..75#0.5)"}, "scenario": {"name": "scenario", "description": "<p>Which scenario are we in - how many critical values of the t distribution does t_statistic exceed?</p>", "templateType": "anything", "group": "Advice messages", "definition": "sum(map(abs(t_statistic)<x,x,[t90,t95,t99,t999]))"}}, "name": "Adaptive marking: Independent two sample t-test", "parts": [{"stepsPenalty": 0, "gaps": [{"maxValue": "mean1", "variableReplacements": [], "precision": "3", "showPrecisionHint": false, "marks": 0.5, "precisionType": "dp", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "type": "numberentry", "scripts": {}, "allowFractions": false, "strictPrecision": false, "minValue": "mean1", "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst"}, {"maxValue": "sd1", "variableReplacements": [], "precision": "3", "showPrecisionHint": false, "marks": 0.5, "precisionType": "dp", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "type": "numberentry", "scripts": {}, "allowFractions": false, "strictPrecision": false, "minValue": "sd1", "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst"}, {"maxValue": "mean2", "variableReplacements": [], "precision": "3", "showPrecisionHint": false, "marks": 0.5, "precisionType": "dp", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "type": "numberentry", "scripts": {}, "allowFractions": false, "strictPrecision": false, "minValue": "mean2", "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst"}, {"maxValue": "sd2", "variableReplacements": [], "precision": "3", "showPrecisionHint": false, "marks": 0.5, "precisionType": "dp", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "type": "numberentry", "scripts": {}, "allowFractions": false, "strictPrecision": false, "minValue": "sd2", "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst"}, {"maxValue": "t_statistic", "variableReplacements": [{"part": "p0g0", "variable": "mean1", "must_go_first": false}, {"part": "p0g1", "variable": "sd1", "must_go_first": false}, {"part": "p0g2", "variable": "mean2", "must_go_first": false}, {"part": "p0g3", "variable": "sd2", "must_go_first": false}], "precision": "3", "showPrecisionHint": false, "marks": 1, "precisionType": "dp", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "type": "numberentry", "scripts": {}, "allowFractions": false, "strictPrecision": false, "minValue": "t_statistic", "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "scripts": {}, "marks": 0, "steps": [{"type": "information", "scripts": {}, "marks": 0, "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>The two-sample <em>t</em>-statistic for two independent sets of data where one set has $n_1$ data points and the other set $n_2$ data points is calculated as follows:</p>\n<p>\\[T = \\frac{(\\overline{x}_1-\\overline{x}_2)-(\\mu_1-\\mu_2)}{s\\times\\sqrt{\\frac{1}{n_1}+\\frac{1}{n_2}}}\\;\\;\\;\\]</p>\n<p>where $\\overline{x}_1,\\;\\overline{x}_2$ are the sample means and&nbsp;</p>\n<p>\\[s^2=\\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}\\]</p>\n<p>where $s_1,\\;s_2$ are the sample standard deviations.</p>\n<p>Use the values you calculated to 3 decimal places in order to find $T$.</p>", "variableReplacementStrategy": "originalfirst"}], "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>Find the mean and standard deviations of the scores of the two groups.&nbsp;Round your answers to 3 decimal places.</p>\n<table>\n<thead>\n<tr><th></th><th>Mean</th><th>Standard deviation</th></tr>\n</thead>\n<tbody>\n<tr><th>Group 1</th>\n<td>[[0]]</td>\n<td>[[1]]</td>\n</tr>\n<tr><th>Group 2</th>\n<td>[[2]]</td>\n<td>[[3]]</td>\n</tr>\n</tbody>\n</table>\n<p>Now find the two sample <em>t</em>-test statistic $T$ using the values you have just calculated and enter it here: [[4]]</p>", "variableReplacementStrategy": "originalfirst"}, {"displayColumns": "1", "minMarks": 0, "scripts": {}, "marks": 0, "matrix": "decision_marking_matrix", "variableReplacements": [{"part": "p0g4", "variable": "t_statistic", "must_go_first": false}], "showCorrectAnswer": true, "displayType": "radiogroup", "choices": ["<p>$p$ is less than $0.1\\%$</p>", "<p>$p$ lies between $0.1\\%$ and $1\\%$</p>", "<p>$p$ lies between  $1 \\%$ and $5\\%$</p>", "<p>$p$ lies between  $5 \\%$ and $10\\%$</p>", "<p>$p$ is greater than $10\\%$</p>"], "prompt": "<p>Give the value $|T|$ of the <em>t</em>-statistic you have found, choose the range for the $p$ value by looking up the <em>t</em>&nbsp;tables:</p>", "shuffleChoices": false, "distractors": ["", "", "", "", ""], "maxMarks": 0, "type": "1_n_2", "variableReplacementStrategy": "originalfirst"}, {"displayColumns": 0, "minMarks": 0, "scripts": {}, "marks": 0, "matrix": "decision_marking_matrix", "variableReplacements": [{"part": "p0g4", "variable": "t_statistic", "must_go_first": false}], "showCorrectAnswer": true, "displayType": "radiogroup", "choices": ["<p>Very Strong Evidence</p>", "<p>Strong Evidence</p>", "<p>Evidence</p>", "<p>Weak Evidence</p>", "<p>No Evidence</p>"], "prompt": "<p>Given the $p$-value and the range you have found, what is the strength of evidence against the null hypothesis that there is no difference in the average times for the left and right hands?</p>", "shuffleChoices": false, "distractors": ["", "", "", "", ""], "maxMarks": 0, "type": "1_n_2", "variableReplacementStrategy": "originalfirst"}, {"displayColumns": "1", "minMarks": 0, "scripts": {}, "marks": 0, "matrix": "decision_marking_matrix", "variableReplacements": [{"part": "p0g4", "variable": "t_statistic", "must_go_first": false}], "showCorrectAnswer": true, "displayType": "radiogroup", "choices": ["<p>We reject the null hypothesis at the $0.1\\%$ level</p>", "<p>We reject the null hypothesis at the $1\\%$ level.</p>", "<p>We reject the null hypothesis at the $5\\%$ level.</p>", "<p>We do not reject the null hypothesis but consider further investigation.</p>", "<p>We do not reject the null hypothesis.</p>"], "prompt": "<p>What do you&nbsp;decide&nbsp;based on the above analysis?</p>", "shuffleChoices": false, "distractors": ["", "", "", "", ""], "maxMarks": 0, "type": "1_n_2", "variableReplacementStrategy": "originalfirst"}], "variable_groups": [{"variables": ["n1", "n2", "mu1", "sigma1", "mu2", "sigma2"], "name": "Setup"}, {"variables": ["r1", "r2"], "name": "Samples"}, {"variables": ["mean1", "sd1", "mean2", "sd2", "s", "t_statistic", "p_value"], "name": "Stats"}, {"variables": ["scenario", "decision_marking_matrix", "reject", "evidence_strength", "t_statistic_range", "p_value_range"], "name": "Advice messages"}, {"variables": ["t90", "t95", "t99", "t999"], "name": "Critical t-values"}], "functions": {"pstdev": {"type": "number", "parameters": [["l", "list"]], "definition": "sqrt(len(l)/(len(l)-1))*stdev(l)", "language": "jme"}}, "tags": ["average", "data analysis", "differences", "elementary statistics", "hypothesis testing", "mean", "standard deviation", "statistics", "stats", "t-test", "two sample t-test", "variance"], "ungrouped_variables": [], "statement": "<p>An educational psychologist claimed that the order in which questions were asked affected the&nbsp;student&rsquo;s ability to answer them correctly and hence their total score. In order to test this, $20$ students were randomly divided into two groups of $10$. The first group were given questions&nbsp;in increasing order of difficulty and the second group in decreasing order of difficulty. The&nbsp;ordered test scores obtained were:</p>\n<table border=\"1\">\n<tbody>\n<tr><th align=\"left\">Group 1</th>\n<td>{r1[0]}</td>\n<td>{r1[1]}</td>\n<td>{r1[2]}</td>\n<td>{r1[3]}</td>\n<td>{r1[4]}</td>\n<td>{r1[5]}</td>\n<td>{r1[6]}</td>\n<td>{r1[7]}</td>\n<td>{r1[8]}</td>\n<td>{r1[9]}</td>\n</tr>\n<tr><th align=\"left\">Group 2</th>\n<td>{r2[0]}</td>\n<td>{r2[1]}</td>\n<td>{r2[2]}</td>\n<td>{r2[3]}</td>\n<td>{r2[4]}</td>\n<td>{r2[5]}</td>\n<td>{r2[6]}</td>\n<td>{r2[7]}</td>\n<td>{r2[8]}</td>\n<td>{r2[9]}</td>\n</tr>\n</tbody>\n</table>\n<p>Carry out a two-sample <em>t</em>-test to decide&nbsp;if there is evidence of a difference in the average&nbsp;test scores for the two sets of students.</p>", "preamble": {"js": "", "css": ""}, "metadata": {"description": "<p>Two sample t-test to see if there is a difference between scores on questions between two groups when the questions are asked in a different order.</p>", "notes": "\n                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t<p><strong>11/07/2012:</strong></p>\n                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t        <p><strong><br /></strong>Added tags.</p>\n                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t        <p>Calculation not yet tested.</p>\n                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t        <p><strong>23/07/2012:</strong></p>\n                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t        <p>Added description.</p>\n                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t        <p>Checked calculation.</p>\n                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t        <p>Changed display slightly in Advice.</p>\n                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t        <p><strong>3/08/2012:</strong></p>\n                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t        <p>Added tags.</p>\n                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t        <p>Question appears to be working correctly.</p>\n                                                                                                                                                                                                                                                                                                                            \t\t                        \n                                                                                                                                                                                                                                                                                                                            \t\t", "licence": "None specified"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "advice": "<p>We test the following hypothesis,</p>\n<p style=\"text-align: center;\">$H_0:\\; \\mu_1=\\mu_2$ versus $H_1:\\; \\mu_1 \\neq \\mu_2$</p>\n<p>We find that the mean score of Group 1 is $\\overline{x}_1=\\var{mean1}$ with standard deviation $s_1=\\var{sd1}$ and the mean score of Group 2 is $\\overline{x}_2=\\var{mean2}$ with standard deviation $s_2=\\var{sd2}$.</p>\n<p>(All calculated to 3 decimal places.)</p>\n<p>Using the formula for the two-sample $t$-statistic as &nbsp;shown above with $n_1=n_2=10$:</p>\n<p>The estimate of the pooled variance is calculated to be:</p>\n<p>\\[s^2=\\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}= \\frac{\\var{n1-1}\\times \\var{sd1}^2+\\var{n2-1}\\times \\var{sd2}^2}{\\var{n1+n2-2}}=\\var{s^2}.\\]&nbsp;</p>\n<p>Hence $s = \\sqrt{\\var{s^2}}=\\var{s}$ to 3 decimal places.</p>\n<p>We find that the t-statistic has value:</p>\n<p>\\begin{align}<br/>T &amp;= \\frac{(\\overline{x}_1-\\overline{x}_2)-(\\mu_1-\\mu_2)}{s\\sqrt{\\frac{1}{n_1}+\\frac{1}{n_2}}} \\\\<br/>&amp;= \\frac{(\\var{mean1}-\\var{mean2})-(0)}{\\var{s}\\sqrt{\\frac{1}{\\var{n1}}+\\frac{1}{\\var{n2}}}} \\\\<br/>&amp;= \\var{t_statistic}<br/>\\end{align}</p>\n<p>Our test statistic is $|T|=\\var{abs(t_statistic)}$.</p>\n<p>Given that we have $n_1+n_2-2=18$ degrees of freedom, we look up this value on the T-distribution table for $t_{18}$</p>\n<p>\\[\\begin{array}{r|rrrrr}&amp;0.10&amp;0.05&amp;0.01&amp;0.001\\\\\\hline18&amp;1.734&amp;2.101&amp;2.878&amp;3.922\\end{array}\\]</p>\n<p>We see that the t-statistic {t_statistic_range} and the table tells us that the $p$ value {p_value_range}.</p>\n<p>Hence we conclude that we {reject} the null hypothesis. There is {evidence_strength} evidence of a difference between the average scores of the two groups.</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}}], "extensions": ["stats"], "percentPass": 0, "navigation": {"showresultspage": "oncompletion", "showfrontpage": false, "browse": true, "reverse": true, "onleave": {"action": "none", "message": ""}, "preventleave": false, "allowregen": true}, "showQuestionGroupNames": false, "shuffleQuestions": false, "pickQuestions": 0, "metadata": {"description": "<p>Some questions which demonstrate the adaptive marking feature.</p>", "notes": "", "licence": "Creative Commons Attribution 4.0 International"}, "question_groups": [{"pickQuestions": 0, "name": "", "pickingStrategy": "all-ordered", "questions": [{"name": "Adaptive marking: quadratic equation", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "variable_groups": [{"name": "Quadratic equation", "variables": ["roots", "a", "b", "xs", "ys"]}], "parts": [{"unitTests": [], "marks": 0, "sortAnswers": true, "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>A quadratic function&nbsp;$f(x)$ goes through the points $(\\var{xs[0]},\\var{ys[0]})$,&nbsp;$(\\var{xs[1]},\\var{ys[1]})$ and&nbsp;$(\\var{xs[2]},\\var{ys[2]})$.</p>\n<p>Find the two roots of $f$.</p>\n<p>First root:&nbsp;[[0]]</p>\n<p>Second root:&nbsp;[[1]]</p>", "gaps": [{"maxValue": "a", "notationStyles": ["plain", "en", "si-en"], "unitTests": [], "variableReplacements": [], "scripts": {}, "marks": "1", "correctAnswerStyle": "plain", "correctAnswerFraction": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "type": "numberentry", "mustBeReduced": false, "allowFractions": false, "customMarkingAlgorithm": "", "minValue": "a", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"maxValue": "b", "notationStyles": ["plain", "en", "si-en"], "unitTests": [], "variableReplacements": [], "scripts": {}, "marks": "1", "correctAnswerStyle": "plain", "correctAnswerFraction": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "type": "numberentry", "mustBeReduced": false, "allowFractions": false, "customMarkingAlgorithm": "", "minValue": "b", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"unitTests": [], "marks": 0, "sortAnswers": false, "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>Write&nbsp;a quadratic function $g(x)$ whose roots are the values you entered above, with $x^2$ coefficient $1$.</p>\n<p>$g(x) =&nbsp;$&nbsp;[[0]]</p>", "gaps": [{"vsetRange": [0, 1], "unitTests": [], "variableReplacements": [{"part": "p0g0", "variable": "a", "must_go_first": false}, {"part": "p0g1", "variable": "b", "must_go_first": false}], "marks": 1, "checkVariableNames": false, "showPreview": true, "answer": "(x-{a})(x-{b})", "showCorrectAnswer": true, "vsetRangePoints": 5, "expectedVariableNames": [], "failureRate": 1, "checkingAccuracy": 0.001, "type": "jme", "scripts": {}, "checkingType": "absdiff", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"maxValue": "mean(roots)", "notationStyles": ["plain", "en", "si-en"], "unitTests": [], "correctAnswerFraction": false, "scripts": {}, "marks": 1, "correctAnswerStyle": "plain", "variableReplacements": [{"part": "p0", "variable": "roots", "must_go_first": false}], "showCorrectAnswer": true, "prompt": "<p>What is the midpoint of the two roots of $f$?</p>", "mustBeReducedPC": 0, "type": "numberentry", "mustBeReduced": false, "allowFractions": false, "customMarkingAlgorithm": "", "minValue": "mean(roots)", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}], "statement": "<p>(This question doesn't make a lot of pedagogic sense, it just shows off how adaptive marking works)</p>", "ungrouped_variables": [], "tags": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "functions": {}, "preamble": {"js": "", "css": ""}, "metadata": {"description": "<p data-bind=\"html: 'hi'\">Give the student three points lying on a quadratic, and ask them to find the roots.</p>\n<p data-bind=\"html: 'hi'\">Then ask them to find the equation of the quadratic, using their roots. Error in calculating the roots is carried forward.</p>\n<p data-bind=\"html: 'hi'\">Finally, ask them to find the midpoint of the roots (just for fun). Error is carried forward again.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "advice": "", "variables": {"roots": {"name": "roots", "description": "", "templateType": "anything", "definition": "sort(shuffle(list(-5..5))[0..2])", "group": "Quadratic equation"}, "b": {"name": "b", "description": "<p>Greatest root of the quadratic</p>", "templateType": "anything", "definition": "roots[1]", "group": "Quadratic equation"}, "xs": {"name": "xs", "description": "<p>Three given x coordinates</p>", "templateType": "anything", "definition": "shuffle(-5..5 except [a,b])[0..3]", "group": "Quadratic equation"}, "a": {"name": "a", "description": "<p>Least root of the quadratic equation</p>", "templateType": "anything", "definition": "roots[0]", "group": "Quadratic equation"}, "ys": {"name": "ys", "description": "<p>y coordinates corresponding to the given x coordinates</p>", "templateType": "anything", "definition": "map((x-a)*(x-b),x,xs)", "group": "Quadratic equation"}}, "rulesets": {}}, {"name": "Adaptive marking: rotation matrix", "extensions": [], "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/"}], "variables": {"rad_rotation": {"name": "rad_rotation", "description": "<p>Angle of rotation in radians</p>", "templateType": "anything", "group": "Rotation matrix", "definition": "radians(rotation)"}, "det_m": {"name": "det_m", "description": "", "templateType": "anything", "group": "Rotation matrix", "definition": "det(mat)"}, "mat": {"name": "mat", "description": "<p>Rotation matrix for the given rotation</p>", "templateType": "anything", "group": "Rotation matrix", "definition": "matrix([\n  [cos(rad_rotation),-sin(rad_rotation)],\n  [sin(rad_rotation),cos(rad_rotation)]\n])"}, "rotation": {"name": "rotation", "description": "<p>Angle of rotation in degrees</p>", "templateType": "anything", "group": "Rotation matrix", "definition": "random(10..90)"}}, "variable_groups": [{"name": "Rotation matrix", "variables": ["rotation", "rad_rotation", "mat", "det_m"]}], "parts": [{"gaps": [{"correctAnswerFractions": false, "numRows": "2", "scripts": {}, "marks": "4", "precisionType": "dp", "precisionPartialCredit": 0, "variableReplacements": [], "showCorrectAnswer": true, "markPerCell": true, "allowResize": false, "type": "matrix", "strictPrecision": false, "allowFractions": false, "correctAnswer": "mat", "precision": "2", "numColumns": "2", "tolerance": 0, "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst"}], "marks": 0, "scripts": {}, "showCorrectAnswer": true, "variableReplacements": [], "type": "gapfill", "prompt": "<p>Write a matrix $\\mathrm{M}$ corresponding to a rotation of $\\var{rotation}^{\\circ}$ anti-clockwise about the origin.</p>\n<p>Round numbers to 2 decimal places.</p>\n<p>$\\mathrm{M} = $&nbsp;[[0]]</p>", "variableReplacementStrategy": "originalfirst"}, {"gaps": [{"maxValue": "det(mat)", "correctAnswerFraction": false, "precision": "2", "showPrecisionHint": false, "marks": 1, "precisionType": "dp", "showCorrectAnswer": true, "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacements": [{"part": "p0g0", "variable": "mat"}], "strictPrecision": false, "type": "numberentry", "scripts": {}, "allowFractions": false, "minValue": "det(mat)", "precisionPartialCredit": 0, "variableReplacementStrategy": "originalfirst"}], "marks": 0, "scripts": {}, "showCorrectAnswer": true, "variableReplacements": [], "type": "gapfill", "prompt": "<p>$\\operatorname{det}\\mathrm{M} = $&nbsp;[[0]] (Round your answer to 2 decimal places)</p>", "variableReplacementStrategy": "originalfirst"}, {"gaps": [{"correctAnswerFractions": false, "numRows": "2", "scripts": {}, "marks": 1, "precisionType": "dp", "precisionPartialCredit": 0, "variableReplacements": [{"part": "p0g0", "variable": "mat"}, {"part": "p1g0", "variable": "det_m"}], "showCorrectAnswer": true, "markPerCell": false, "allowResize": false, "type": "matrix", "strictPrecision": false, "allowFractions": false, "correctAnswer": "matrix([[mat[1][1],-mat[0][1]],[-mat[1][0],mat[0][0]]])/det_m", "precision": "2", "numColumns": "2", "tolerance": 0, "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst"}], "marks": 0, "scripts": {}, "showCorrectAnswer": true, "variableReplacements": [], "type": "gapfill", "prompt": "<p>$\\mathrm{M}^{-1} = $&nbsp;[[0]]</p>\n<p></p>", "variableReplacementStrategy": "originalfirst"}], "functions": {}, "showQuestionGroupNames": false, "tags": [], "ungrouped_variables": [], "preamble": {"js": "", "css": ""}, "metadata": {"description": "<p data-bind=\"html: 'hi'\">Ask the student to find a&nbsp;matrix corresponding to a given rotation about the origin.</p>\n<p data-bind=\"html: 'hi'\">Then ask them to find the determinant. Their answer is marked against the matrix they gave, not just the correct one.</p>\n<p data-bind=\"html: 'hi'\">Finally, ask them to find the inverse of their matrix. Marking is&nbsp;against the matrix and determinant they gave.</p>", "notes": "", "licence": "Creative Commons Attribution 4.0 International"}, "question_groups": [{"pickQuestions": 0, "pickingStrategy": "all-ordered", "questions": [], "name": ""}], "type": "question", "statement": "", "variablesTest": {"maxRuns": 100, "condition": ""}, "advice": "", "rulesets": {}}, {"name": "Adaptive marking: Independent two sample t-test", "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/"}], "tags": ["average", "data analysis", "differences", "elementary statistics", "hypothesis testing", "mean", "standard deviation", "statistics", "stats", "t-test", "two sample t-test", "variance"], "metadata": {"description": "<p>Two sample t-test to see if there is a difference between scores on questions between two groups when the questions are asked in a different order.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>An educational psychologist claimed that the order in which questions were asked affected the&nbsp;student&rsquo;s ability to answer them correctly and hence their total score. In order to test this, $20$ students were randomly divided into two groups of $10$. The first group were given questions&nbsp;in increasing order of difficulty and the second group in decreasing order of difficulty. The&nbsp;ordered test scores obtained were:</p>\n<table border=\"1\">\n<tbody>\n<tr><th align=\"left\">Group 1</th>\n<td>{r1[0]}</td>\n<td>{r1[1]}</td>\n<td>{r1[2]}</td>\n<td>{r1[3]}</td>\n<td>{r1[4]}</td>\n<td>{r1[5]}</td>\n<td>{r1[6]}</td>\n<td>{r1[7]}</td>\n<td>{r1[8]}</td>\n<td>{r1[9]}</td>\n</tr>\n<tr><th align=\"left\">Group 2</th>\n<td>{r2[0]}</td>\n<td>{r2[1]}</td>\n<td>{r2[2]}</td>\n<td>{r2[3]}</td>\n<td>{r2[4]}</td>\n<td>{r2[5]}</td>\n<td>{r2[6]}</td>\n<td>{r2[7]}</td>\n<td>{r2[8]}</td>\n<td>{r2[9]}</td>\n</tr>\n</tbody>\n</table>\n<p>Carry out a two-sample <em>t</em>-test to decide&nbsp;if there is evidence of a difference in the average&nbsp;test scores for the two sets of students.</p>", "advice": "<p>We test the following hypothesis,</p>\n<p style=\"text-align: center;\">$H_0:\\; \\mu_1=\\mu_2$ versus $H_1:\\; \\mu_1 \\neq \\mu_2$</p>\n<p>We find that the mean score of Group 1 is $\\overline{x}_1=\\var{mean1}$ with standard deviation $s_1=\\var{sd1}$ and the mean score of Group 2 is $\\overline{x}_2=\\var{mean2}$ with standard deviation $s_2=\\var{sd2}$.</p>\n<p>(All calculated to 3 decimal places.)</p>\n<p>Using the formula for the two-sample $t$-statistic as &nbsp;shown above with $n_1=n_2=10$:</p>\n<p>The estimate of the pooled variance is calculated to be:</p>\n<p>\\[s^2=\\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}= \\frac{\\var{n1-1}\\times \\var{sd1}^2+\\var{n2-1}\\times \\var{sd2}^2}{\\var{n1+n2-2}}=\\var{s^2}.\\]&nbsp;</p>\n<p>Hence $s = \\sqrt{\\var{s^2}}=\\var{s}$ to 3 decimal places.</p>\n<p>We find that the t-statistic has value:</p>\n<p>\\begin{align}<br/>T &amp;= \\frac{(\\overline{x}_1-\\overline{x}_2)-(\\mu_1-\\mu_2)}{s\\sqrt{\\frac{1}{n_1}+\\frac{1}{n_2}}} \\\\<br/>&amp;= \\frac{(\\var{mean1}-\\var{mean2})-(0)}{\\var{s}\\sqrt{\\frac{1}{\\var{n1}}+\\frac{1}{\\var{n2}}}} \\\\<br/>&amp;= \\var{t_statistic}<br/>\\end{align}</p>\n<p>Our test statistic is $|T|=\\var{abs(t_statistic)}$.</p>\n<p>Given that we have $n_1+n_2-2=18$ degrees of freedom, we look up this value on the T-distribution table for $t_{18}$</p>\n<p>\\[\\begin{array}{r|rrrrr}&amp;0.10&amp;0.05&amp;0.01&amp;0.001\\\\\\hline18&amp;1.734&amp;2.101&amp;2.878&amp;3.922\\end{array}\\]</p>\n<p>We see that the t-statistic {t_statistic_range} and the table tells us that the $p$ value {p_value_range}.</p>\n<p>Hence we conclude that we {reject} the null hypothesis. There is {evidence_strength} evidence of a difference between the average scores of the two groups.</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"p_value_range": {"name": "p_value_range", "group": "Advice messages", "definition": "['is less than $0.001$','lies between $0.001$ and $0.01$','lies between $0.01$ and $0.05$','lies between $0.05$ and $0.10$','is greater than $0.10$'][scenario]", "description": "<p>Describe where the p-value lies in relation to the critical values</p>", "templateType": "anything", "can_override": false}, "sigma2": {"name": "sigma2", "group": "Setup", "definition": "random(8..10#0.2)", "description": "<p>Population standard deviation of sample 2</p>", "templateType": "anything", "can_override": false}, "t99": {"name": "t99", "group": "Critical t-values", "definition": "2.878", "description": "", "templateType": "anything", "can_override": false}, "n2": {"name": "n2", "group": "Setup", "definition": "10", "description": "<p>Size of sample 2</p>", "templateType": "anything", "can_override": false}, "t999": {"name": "t999", "group": "Critical t-values", "definition": "3.922", "description": "", "templateType": "anything", "can_override": false}, "t_statistic_range": {"name": "t_statistic_range", "group": "Advice messages", "definition": "['is greater than $\\\\var{t999}$','lies between $\\\\var{t99}$ and $\\\\var{t999}$','lies between $\\\\var{t95}$ and $\\\\var{t99}$','lies between $\\\\var{t90}$ and $\\\\var{t95}$','is less than $\\\\var{t90}$'][scenario]", "description": "<p>Describe where the t-statistic lies in relation to the critical values</p>", "templateType": "anything", "can_override": false}, "mu2": {"name": "mu2", "group": "Setup", "definition": "random(65..75#0.5)", "description": "<p>Population mean of sample 2</p>", "templateType": "anything", "can_override": false}, "evidence_strength": {"name": "evidence_strength", "group": "Advice messages", "definition": "['very strong','strong','slight','no','no'][scenario]", "description": "<p>How much evidence is there against the null hypothesis?</p>", "templateType": "anything", "can_override": false}, "n1": {"name": "n1", "group": "Setup", "definition": "10", "description": "<p>Size of sample 1</p>", "templateType": "anything", "can_override": false}, "mu1": {"name": "mu1", "group": "Setup", "definition": "random(55..75#0.5)", "description": "<p>Population mean of sample 1 (we'll generate samples from different distributions to produce different outcomes)</p>", "templateType": "anything", "can_override": false}, "reject": {"name": "reject", "group": "Advice messages", "definition": "if(scenario<2,'do reject','do not reject')", "description": "<p>Do we reject the null hypothesis?</p>", "templateType": "anything", "can_override": false}, "p_value": {"name": "p_value", "group": "Stats", "definition": "ttest(abs(t_statistic),19,2)", "description": "<p>p-value&nbsp;corresponding to the t-statistic</p>", "templateType": "anything", "can_override": false}, "t_statistic": {"name": "t_statistic", "group": "Stats", "definition": "(mean1-mean2)*sqrt(n1*n2)/(s*sqrt(n1+n2))", "description": "", "templateType": "anything", "can_override": false}, "t90": {"name": "t90", "group": "Critical t-values", "definition": "1.734", "description": "", "templateType": "anything", "can_override": false}, "decision_marking_matrix": {"name": "decision_marking_matrix", "group": "Advice messages", "definition": "[\n  [1,0,0,0,0],\n  [0,1,0,0,0],\n  [0,0,1,0,0],\n  [0,0,0,1,0],\n  [0,0,0,0,1]\n][scenario]", "description": "<p>Marking matrix for the multiple choice questions</p>", "templateType": "anything", "can_override": false}, "sd2": {"name": "sd2", "group": "Stats", "definition": "precround(pstdev(r2),3)", "description": "<p>Sample standard deviation of sample 2</p>", "templateType": "anything", "can_override": false}, "t95": {"name": "t95", "group": "Critical t-values", "definition": "2.101", "description": "", "templateType": "anything", "can_override": false}, "r1": {"name": "r1", "group": "Samples", "definition": "repeat(round(normalsample(mu1,sigma1)),n1)", "description": "<p>Sample 1</p>", "templateType": "anything", "can_override": false}, "mean1": {"name": "mean1", "group": "Stats", "definition": "mean(r1)", "description": "<p>Sample mean of sample 1</p>", "templateType": "anything", "can_override": false}, "mean2": {"name": "mean2", "group": "Stats", "definition": "mean(r2)", "description": "<p>Sample&nbsp;mean of sample 1</p>", "templateType": "anything", "can_override": false}, "r2": {"name": "r2", "group": "Samples", "definition": "repeat(round(normalsample(mu2,sigma2)),n2)", "description": "<p>Sample 2</p>", "templateType": "anything", "can_override": false}, "scenario": {"name": "scenario", "group": "Advice messages", "definition": "sum(map(award(1,abs(t_statistic)<x),x,[t90,t95,t99,t999]))", "description": "<p>Which scenario are we in - how many critical values of the t distribution does t_statistic exceed?</p>", "templateType": "anything", "can_override": false}, "sigma1": {"name": "sigma1", "group": "Setup", "definition": "random(8..10#0.2)", "description": "<p>Population standard deviation of sample 1</p>", "templateType": "anything", "can_override": false}, "sd1": {"name": "sd1", "group": "Stats", "definition": "precround(pstdev(r1),3)", "description": "<p>Sample standard deviation of sample 1</p>", "templateType": "anything", "can_override": false}, "s": {"name": "s", "group": "Stats", "definition": "precround(sqrt(((n1-1)*sd1^2+(n2-1)*sd2^2)/(n1+n2-2)),3)", "description": "<p>Used in the&nbsp;formula for the t statistic</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Setup", "variables": ["n1", "n2", "mu1", "sigma1", "mu2", "sigma2"]}, {"name": "Samples", "variables": ["r1", "r2"]}, {"name": "Stats", "variables": ["mean1", "sd1", "mean2", "sd2", "s", "t_statistic", "p_value"]}, {"name": "Advice messages", "variables": ["scenario", "decision_marking_matrix", "reject", "evidence_strength", "t_statistic_range", "p_value_range"]}, {"name": "Critical t-values", "variables": ["t90", "t95", "t99", "t999"]}], "functions": {"pstdev": {"parameters": [["l", "list"]], "type": "number", "language": "jme", "definition": "sqrt(len(l)/(len(l)-1))*stdev(l)"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the mean and standard deviations of the scores of the two groups.&nbsp;Round your answers to 3 decimal places.</p>\n<table>\n<thead>\n<tr>\n<th></th>\n<th>Mean</th>\n<th>Standard deviation</th>\n</tr>\n</thead>\n<tbody>\n<tr>\n<th>Group 1</th>\n<td>[[0]]</td>\n<td>[[1]]</td>\n</tr>\n<tr>\n<th>Group 2</th>\n<td>[[2]]</td>\n<td>[[3]]</td>\n</tr>\n</tbody>\n</table>\n<p>Now find the two sample <em>t</em>-test statistic $T$ using the values you have just calculated and enter it here: [[4]]</p>", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The 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"plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [{"variable": "mean1", "part": "p0g0", "must_go_first": false}, {"variable": "sd1", "part": "p0g1", "must_go_first": false}, {"variable": "mean2", "part": "p0g2", "must_go_first": false}, {"variable": "sd2", "part": "p0g3", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "t_statistic", "maxValue": "t_statistic", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [{"variable": "t_statistic", "part": "p0g4", "must_go_first": false}], "variableReplacementStrategy": "alwaysreplace", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Given the value $|T|$ of the <em>t</em>-statistic you have found, choose the range for the $p$ value by looking up the <em>t</em>&nbsp;tables:</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": "1", "showCellAnswerState": true, "choices": ["<p>$p$ is less than $0.1\\%$</p>", "<p>$p$ lies between $0.1\\%$ and $1\\%$</p>", "<p>$p$ lies between  $1 \\%$ and $5\\%$</p>", "<p>$p$ lies between  $5 \\%$ and $10\\%$</p>", "<p>$p$ is greater than $10\\%$</p>"], "matrix": "decision_marking_matrix"}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [{"variable": "scenario", "part": "p1", "must_go_first": false}], "variableReplacementStrategy": "alwaysreplace", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Given the $p$-value and the range you have found, what is the strength of evidence against the null hypothesis that there is no difference in the average times for the left and right hands?</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": 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