$\\var{no_people}$ track and field athletes were asked whether they perform static stretches before they competed. $\\var{stretch}$ athletes said that they performed static stretches.

\nWhat is the relative frequency of athletes who perform static stretches before competing?

\n", "scripts": {}, "correctAnswerFraction": false, "precisionType": "dp", "correctAnswerStyle": "plain", "showFeedbackIcon": true, "showCorrectAnswer": true, "type": "numberentry", "precisionPartialCredit": 0, "minValue": "{stretch}/{no_people}", "unitTests": [], "precisionMessage": "You must give your answer as a decimal to 2 decimal places.

", "allowFractions": false, "precision": "2", "maxValue": "{stretch}/{no_people}", "marks": 1, "mustBeReduced": false}, {"variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "strictPrecision": false, "showPrecisionHint": true, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "customMarkingAlgorithm": "", "prompt": "The force required to break a femur during a simulated lateral fall was measured on multiple samples. The relative frequency of a femur breaking when $4000N$ was applied was estimated to be $\\var{rel_freq}$.

\nEstimate the number of femurs that would break in a sample of $\\var{no_students}$ when a $4000N$ force is applied to them.

", "scripts": {}, "correctAnswerFraction": false, "precisionType": "dp", "correctAnswerStyle": "plain", "showFeedbackIcon": true, "showCorrectAnswer": true, "type": "numberentry", "precisionPartialCredit": 0, "minValue": "{no_students}*{rel_freq}", "unitTests": [], "precisionMessage": "Round your answer to the nearest integer.

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", "allowFractions": false, "precision": "2", "maxValue": "{Vicon}/({Vicon} + {qualysis} + {Optotrak})", "marks": 1, "mustBeReduced": false}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "sortAnswers": false, "customMarkingAlgorithm": "", "prompt": "A survey was conducted to find out what type of $3D$ motion analysis system is the most commonly used. The results are given in the table below.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nType of $3D$ motion anaylsis system | Frequency |

Vicon | $\\var{vicon}$ |

Qualysis | $\\var{qualysis}$ |

Optotrak | $\\var{optotrak}$ |

i)* *

What was the sample size for the survey?

\n[[0]]

\nii)

\nWhat is the relative frequency of using Vicon as a $3D$ motion analysis system?

\n[[1]]

", "scripts": {}, "marks": 0, "showFeedbackIcon": true}], "type": "question", "tags": [], "functions": {}, "variables": {"no_students": {"templateType": "anything", "description": "Number of students in the class for part b

", "name": "no_students", "group": "Ungrouped variables", "definition": "random(20..40 #10)"}, "Vicon": {"templateType": "anything", "description": "", "name": "Vicon", "group": "Ungrouped variables", "definition": "random(20..40)"}, "qualysis": {"templateType": "anything", "description": "Frequency of cat in part c.

", "name": "qualysis", "group": "Ungrouped variables", "definition": "random(5..15 except vicon)"}, "Optotrak": {"templateType": "anything", "description": "Frequency of hamster in part c

", "name": "Optotrak", "group": "Ungrouped variables", "definition": "random(1..10 except vicon except qualysis) "}, "rel_freq": {"templateType": "anything", "description": "Relative frequency for part b

", "name": "rel_freq", "group": "Ungrouped variables", "definition": "random(0.3..0.5 # 0.01)"}, "stretch": {"templateType": "anything", "description": "Number of people who prefer free-range eggs in part a)

", "name": "stretch", "group": "Ungrouped variables", "definition": "random(10..40)"}, "n": {"templateType": "anything", "description": "Sample size for part c

", "name": "n", "group": "Ungrouped variables", "definition": "Vicon + qualysis + Optotrak"}, "no_people": {"templateType": "anything", "description": "Number of people asked in part a)

", "name": "no_people", "group": "Ungrouped variables", "definition": "random(50..150 #10)"}}, "rulesets": {}, "statement": "The relative frequency of an outcome is the frequency of the outcome divided by the number of trials.

", "contributors": [{"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}, {"name": "Lauren Frances Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2490/"}], "extensions": [], "name": "Relative Frequency", "advice": "The relative frequency of an outcome is the frequency of the outcome divided by the number of trials.

\nTo calculate the relative frequency of people who performed static stretches we need the number of trials, which is $\\var{no_people}$, and the number of people who said that they they performed static stretches, which is $\\var{stretch}$.

\n\nTherefore, the relative frequency of people who perform static stretches is

\n\\[

\\frac{\\var{stretch}}{\\var{no_people}} = \\var{dpformat({stretch/no_people}, 2)} \\; (\\text{rounded to $2$ decimal places}).

\\]

We are told that the relative frequency of the number of femurs that break when $4000N$ is applied to them is $\\var{rel_freq}$, so we assume that the relative frequency is the same across all samples of femurs exposed to the same $4000N$ force. Therefore, in our sample of $\\var{no_students}$ femurs we expect that

\n\\[

\\begin{align}

\\var{rel_freq} \\times \\var{no_students} = \\var{{rel_freq}*{no_students}}

\\end{align}

\\]

femurs will break.

\nAs this result represents a number of femurs, if the answer is not an integer, we must round it to the nearest integer.

\n* *

i)

\nUsing the frequency table given in the question, we can calculate the sample size of the survey by adding together the frequencies of each of the different types of $3D$ motion analysis systems.

\nSo, the sample size of the survey is

\n\\[

\\var{Vicon}+\\var{qualysis}+\\var{Optotrak} = \\var{Vicon + qualysis + Optotrak}.

\\]

ii)

\nTo calculate the relative frequency of using Vicon as a type of $3D$ motion anaylsis system, we divide the frequency of Vicon by the sample size of the survey.

\nSo, the relative frequency is

\n\\[

\\frac{\\var{Vicon}}{\\var{n}} = \\var{dpformat({Vicon/n}, 2)} \\; (\\text{rounded to $2$ decimal places}).

\\]

Calculate relative frequencies in a variety of scenarios.

", "licence": "Creative Commons Attribution 4.0 International"}}], "pickingStrategy": "all-ordered"}], "contributors": [{"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}, {"name": "Lauren Frances Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2490/"}]}