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Sample Standard Deviation = [[0]] (to one decimal place)
\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 \nSample Standard Deviation = [[0]] (to one decimal place)
\n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{stdevoverall+tol}", "minValue": "{stdevoverall-tol}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n \n \nSample Standard Deviation = [[0]] (to one decimal place)
\n \n ", "showCorrectAnswer": true, "marks": 0}], "statement": "\n \n \nThe following table gives the examination marks in {exam1} and in {exam2} and their total for a group of $n=\\var{n}$ students.
\n \n \n \n{exam1} | $\\var{r0[0]}$ | $\\var{r0[1]}$ | $\\var{r0[2]}$ | $\\var{r0[3]}$ | $\\var{r0[4]}$ | $\\var{r0[5]}$ | $\\var{r0[6]}$ | $\\var{r0[7]}$ | $\\var{r0[8]}$ | $\\var{r0[9]}$ | Mean = $\\var{mean1}$ |
---|---|---|---|---|---|---|---|---|---|---|---|
{exam2} | $\\var{r1[0]}$ | $\\var{r1[1]}$ | $\\var{r1[2]}$ | $\\var{r1[3]}$ | $\\var{r1[4]}$ | $\\var{r1[5]}$ | $\\var{r1[6]}$ | $\\var{r1[7]}$ | $\\var{r1[8]}$ | $\\var{r1[9]}$ | Mean = $\\var{mean2}$ |
{total} | $\\var{sscores[0]}$ | $\\var{sscores[1]}$ | $\\var{sscores[2]}$ | $\\var{sscores[3]}$ | $\\var{sscores[4]}$ | $\\var{sscores[5]}$ | $\\var{sscores[6]}$ | $\\var{sscores[7]}$ | $\\var{sscores[8]}$ | $\\var{sscores[9]}$ | Mean = $\\var{overallmean}$ |
Find the sample standard deviation for each of {exam1}, {exam2} and Total Score.
\n \n ", "tags": ["checked2015", "cr1", "data analysis", "elementary statistics", "mean ", "sample", "sample mean", "sample standard deviation", "sample variance", "standard deviation", "statistics", "stats", "tested1", "unused", "variance"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "11/07/2012:
\nAdded tags.
\nSet new variable tol=0 for all numeric input so that answers have to be accurate to 1 decimal place.
\nTesting calculation not yet possible due to stats extension unavailability.
\n23/07/2012:
\nCorrected error in calculation of variance of Total Score. The variable scores was not used and so mean and variance were not correct.
\nChecked calculations. OK.
\nAdded description.
\n1/08/2012:
\nAdded tags.
\nQuestion appears to be working correctly.
\n19/12/2012:
\nChanged stats functions to the ones from the new stats extension.
\nChecked calculations.
\nAdded tested1 tag.
\n21/12/2012:
\nChecked rounding, OK. Added cr1 tag.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Two ordered data sets, each with 10 numbers. Find the sample standard deviation for each and for their sum.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n \n \nThe solution to the first part is here – the other parts can be done in the same way.
\n \n \n \nFor {exam1} we have the mean is:
\n \n \n \n\\[\\simplify[]{({r0[0]} + {r0[1]} + {r0[2]} + {r0[3]} + {r0[4]} + {r0[5]} + {r0[6]} + {r0[7]} + {r0[8]} + {r0[9]}) / {n} = {mean1}}\\]
\n \n \n \nThe sample variance is given by the formula:
\n \n \n \n\\[\\textrm{Sample Variance} = \\frac{1}{n-1}\\left(\\sum_{j=1}^{n}x_j^2 -n\\mu^2\\right)\\]
\n \n \n \nwhere the $x_j$ are the exam scores for {exam1}, $n=\\var{n}$ the number of students and $\\mu=\\var{mean1}$ the sample mean.
\n \n \n \nWe find that
\\[\\begin{eqnarray*}\\sum_{j=1}^{n}x_j^2 &=& \\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)}\\\\ \n \n &=& \\var{ssq1}\\\\\n \n \\\\\n \n \\\\\n \n n\\mu^2 &=&\\var{n} \\times\\var{mean1}^2\\\\\n \n &=& \\var{n*mean1^2}\n \n \\end{eqnarray*}\n \n \\]
Hence substituting these values into the formula we find that:
\\[\\begin{eqnarray*}\n \n \\textrm{Sample Variance} &=& \\frac{1}{\\var{n-1}}\\left(\\var{ssq1}-\\var{n*mean1^2}\\right)\\\\\n \n &=& \\var{var1}\n \n \\end{eqnarray*}\n \n \\] to 3 decimal places.
\n \n \n \nThe Sample Standard Deviation is then the square root of the Sample Variance i.e.
\n \n \n \nSample Standard Deviation = $\\sqrt{\\var{var1}} = \\var{stdev1}$ to one decimal place.
\n \n ", "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/"}]}