// Numbas version: exam_results_page_options
{"name": "Find sample standard deviations of two samples", "extensions": ["stats"], "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", "description": "", "name": "tol"}, "r1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(round(normalsample(mu,sig1)),n)", "description": "", "name": "r1"}, "mean2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "mean(r1)", "description": "", "name": "mean2"}, "sig1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(9..15)", "description": "", "name": "sig1"}, "mean1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "mean(r0)", "description": "", "name": "mean1"}, "sig0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(6..12)", "description": "", "name": "sig0"}, "stdev1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(stdev(r0,true),1)", "description": "", "name": "stdev1"}, "stdev2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(stdev(r1,true),1)", "description": "", "name": "stdev2"}, "exam2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'Software Engineering'", "description": "", "name": "exam2"}, "stdevoverall": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(stdev(sscores,true),1)", "description": "", "name": "stdevoverall"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "10", "description": "", "name": "n"}, "mu": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(55..65)", "description": "", "name": "mu"}, "total": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'Total Score'", "description": "", "name": "total"}, "ssq2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sum(map(x^2,x,r1))", "description": "", "name": "ssq2"}, "exam1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'Programming'", "description": "", "name": "exam1"}, "var2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(variance(r1,true),3)", "description": "", "name": "var2"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "name": "m"}, "sscores": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(r0[x]+r1[x],x,0..n-1)", "description": "", "name": "sscores"}, "overallvar": {"templateType": "anything", "group": "Ungrouped variables", "definition": "variance(sscores,true)", "description": "", "name": "overallvar"}, "ssq1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sum(map(x^2,x,r0))", "description": "", "name": "ssq1"}, "s": {"templateType": "anything", "group": "Ungrouped variables", "definition": "2", "description": "", "name": "s"}, "overallmean": {"templateType": "anything", "group": "Ungrouped variables", "definition": "mean(sscores)", "description": "", "name": "overallmean"}, "r0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(random(25..100-m),n)", "description": "", "name": "r0"}, "var1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(variance(r0,true),3)", "description": "", "name": "var1"}}, "ungrouped_variables": ["overallmean", "mean1", "mean2", "overallvar", "ssq1", "ssq2", "total", "exam2", "tol", "exam1", "stdev1", "stdev2", "var1", "var2", "sig1", "sig0", "stdevoverall", "r0", "r1", "n", "mu", "s", "sscores", "m"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Find sample standard deviations of two samples", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{stdev1+tol}", "minValue": "{stdev1-tol}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n        \n        \n        <h4>{exam1}</h4>\n        \n        \n        \n        <p>Sample Standard Deviation = [[0]] (to one decimal place)</p>\n        \n        ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{stdev2+tol}", "minValue": "{stdev2-tol}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n        \n        \n        <h4>{exam2}</h4>\n        \n        \n        \n        <p>Sample Standard Deviation = [[0]] (to one decimal place)</p>\n        \n        ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{mean2+m+tol}", "minValue": "{mean2+m-tol}", "correctAnswerFraction": false, "marks": "0.5", "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "stdev2+tol", "minValue": "stdev2-tol", "correctAnswerFraction": false, "marks": "0.5", "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>Suppose that each student is awarded an extra $\\var{m}$ marks &nbsp;for Software Engineering. Find the new values for the sample mean and sample standard deviation.</p>\n<p>Sample mean =&nbsp;[[0]]&nbsp;<span>(to one decimal place)</span></p>\n<p>Sample Standard Deviation = [[1]] (to one decimal place)</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p><span>For a group of $n=10$ students, the following table gives the examination marks in Programming, $x_1,\\ldots, x_{10}$</span></p>\n<table border=\"1\">\n<tbody>\n<tr><th align=\"left\">{exam1}</th>\n<td>$\\var{r0[0]}$</td>\n<td>$\\var{r0[1]}$</td>\n<td>$\\var{r0[2]}$</td>\n<td>$\\var{r0[3]}$</td>\n<td>$\\var{r0[4]}$</td>\n<td>$\\var{r0[5]}$</td>\n<td>$\\var{r0[6]}$</td>\n<td>$\\var{r0[7]}$</td>\n<td>$\\var{r0[8]}$</td>\n<td>$\\var{r0[9]}$</td>\n<th align=\"left\">&nbsp;$\\bar{x}=\\var{mean1}$</th></tr>\n</tbody>\n</table>\n<p><span>For Software Engineering, their examination marks $y_1,...,y_{10}$ have been summarised as follows:</span><br/><br/></p>\n<table border=\"1\">\n<tbody>\n<tr><th align=\"left\">{exam2}</th>\n<td>$\\sum y^2 = \\var{ssq2}$</td>\n<td>$\\bar{y} = \\var{mean2}$</td>\n</tr>\n</tbody>\n</table>", "tags": ["checked2015", "cr1", "data analysis", "elementary statistics", "MAS8380", "mean", "mean ", "sample", "sample mean", "sample standard deviation", "sample variance", "standard deviation", "statistics", "stats", "tested1", "variance"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>11/07/2012:</strong></p>\n    <p>Added tags.</p>\n    <p>Set new variable tol=0 for all numeric input so that answers have to be accurate to 1 decimal place.</p>\n    <p>Testing calculation not yet possible due to stats extension unavailability.</p>\n    <p><strong>23/07/2012:</strong></p>\n    <p>Corrected error in calculation of variance of Total Score. The variable scores was not used and so mean and variance were not correct.</p>\n    <p>Checked calculations. OK.</p>\n    <p>Added description.</p>\n    <p><strong>1/08/2012:</strong></p>\n    <p>Added tags.</p>\n    <p>Question appears to be working correctly.</p>\n    <p><strong>19/12/2012:</strong></p>\n    <p>Changed stats functions to the ones from the new stats extension.</p>\n    <p>Checked calculations.</p>\n    <p>Added tested1 tag.</p>\n    <p><strong>21/12/2012:</strong></p>\n    <p>Checked rounding, OK. Added cr1 tag.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Two ordered data sets, each with 10 numbers. Find the sample standard deviation for each and for their sum.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>T<span>he solution to (a) is given below; (b) can be done in the same way.</span></p>\n<p>For {exam1} we have the mean is:</p>\n<p>\\[\\simplify[]{({r0[0]} + {r0[1]} + {r0[2]} + {r0[3]} + {r0[4]} + {r0[5]} + {r0[6]} + {r0[7]} + {r0[8]} + {r0[9]}) / {n} = {mean1}}\\]</p>\n<p>The sample variance is given by the formula:</p>\n<p>\\[\\textrm{Sample Variance} = \\frac{1}{n-1}\\left(\\sum_{j=1}^{n}x_j^2 -n\\bar{x}^2\\right)\\]</p>\n<p>where the $x_j$ are the exam scores for {exam1}, $n=\\var{n}$ the number of students and $\\bar{x}=\\var{mean1}$ the sample mean.</p>\n<p>We find that<br/>\\[\\begin{eqnarray*}\\sum_{j=1}^{n}x_j^2 &amp;=&amp; \\simplify[]{({r0[0]}^2 + {r0[1]}^2 + {r0[2]}^2 + {r0[3]}^2 + {r0[4]}^2 + {r0[5]}^2 + {r0[6]}^2 + {r0[7]}^2 + {r0[8]}^2 + {r0[9]}^2)}\\\\ &amp;=&amp; \\var{ssq1}\\\\ \\\\ \\\\ n\\bar{x}^2 &amp;=&amp;\\var{n} \\times\\var{mean1}^2\\\\ &amp;=&amp; \\var{n*mean1^2} \\end{eqnarray*} \\]<br/>Hence substituting these values into the formula we find that:</p>\n<p>\\[\\begin{eqnarray*} \\textrm{Sample Variance} &amp;=&amp; \\frac{1}{\\var{n-1}}\\left(\\var{ssq1}-\\var{n*mean1^2}\\right)\\\\ &amp;=&amp; \\var{var1} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<p>The Sample Standard Deviation is then the square root of the Sample Variance i.e.</p>\n<p>Sample Standard Deviation = $\\sqrt{\\var{var1}} = \\var{stdev1}$ to one decimal place.</p>\n<p><span>In part (c), adding a constant, $\\var{m}$, to each score shifts the mean by the same constant. The spread of the data is unaffected and so the sample standard deviation remains unchanged.</span></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/"}]}