The 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 ", "statement": "The following table gives the number of participants in two gym classes; {exam1} and {exam2}, and the total attendance for both classes over each week ($n=\\var{n}$).
\n| \n | Week 1 | \nWeek 2 | \nWeek 3 | \nWeek 4 | \nWeek 5 | \nWeek 6 | \nWeek 7 | \nWeek 8 | \nWeek 9 | \nWeek 10 | \n\n |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Pilates | \n$\\var{r0[0]}$ | \n$\\var{r0[1]}$ | \n$\\var{r0[2]}$ | \n$\\var{r0[3]}$ | \n$\\var{r0[4]}$ | \n$\\var{r0[5]}$ | \n$\\var{r0[6]}$ | \n$\\var{r0[7]}$ | \n$\\var{r0[8]}$ | \n$\\var{r0[9]}$ | \nMean = $\\var{mean1}$ | \n
| Circuits | \n$\\var{r1[0]}$ | \n$\\var{r1[1]}$ | \n$\\var{r1[2]}$ | \n$\\var{r1[3]}$ | \n$\\var{r1[4]}$ | \n$\\var{r1[5]}$ | \n$\\var{r1[6]}$ | \n$\\var{r1[7]}$ | \n$\\var{r1[8]}$ | \n$\\var{r1[9]}$ | \nMean = $\\var{mean2}$ | \n
| {total} | \n$\\var{sscores[0]}$ | \n$\\var{sscores[1]}$ | \n$\\var{sscores[2]}$ | \n$\\var{sscores[3]}$ | \n$\\var{sscores[4]}$ | \n$\\var{sscores[5]}$ | \n$\\var{sscores[6]}$ | \n$\\var{sscores[7]}$ | \n$\\var{sscores[8]}$ | \n$\\var{sscores[9]}$ | \nMean = $\\var{overallmean}$ | \n
Find the sample standard deviation for each of Pilates, Circuits and Total Score.
", "variable_groups": [], "metadata": {"description": "Two ordered data sets, each with 10 numbers. Find the sample standard deviation for each and for their sum.
", "licence": "Creative Commons Attribution 4.0 International"}, "parts": [{"showCorrectAnswer": true, "marks": 0, "customMarkingAlgorithm": "", "scripts": {}, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "gaps": [{"showCorrectAnswer": true, "marks": "3", "customMarkingAlgorithm": "", "correctAnswerStyle": "plain", "minValue": "{stdev1-tol}", "maxValue": "{stdev1+tol}", "scripts": {}, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "allowFractions": false, "mustBeReducedPC": 0, "type": "numberentry", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "unitTests": [], "correctAnswerFraction": false}], "type": "gapfill", "unitTests": [], "sortAnswers": false, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "prompt": "Sample Standard Deviation = [[0]] (to one decimal place)
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"}], "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"], "preamble": {"js": "", "css": ""}, "functions": {}, "extensions": ["stats"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "type": "question", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Paul Finley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2232/"}, {"name": "Charlie Earle", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3054/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Paul Finley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2232/"}, {"name": "Charlie Earle", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3054/"}]}