What is $\\bar{x}$?
", "matrix": [0, 0, "1", 0], "shuffleChoices": true, "marks": 0, "variableReplacements": [], "choices": ["Sample variance
", "Population mean
", "Sample mean
", "A data value
"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "maxMarks": 0, "scripts": {}, "distractors": ["", "", "", ""], "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}, {"prompt": "What does $\\bar{x} - \\mu$ represent?
", "matrix": [0, "1", 0, 0], "shuffleChoices": true, "marks": 0, "variableReplacements": [], "choices": ["The difference between an observation value and the sample mean.
", "The difference between the sample mean and the population mean.
", "The difference between the sample mean and the sample deviation.
", "The difference between the sample mean and the population variance
"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "maxMarks": 0, "scripts": {}, "distractors": ["", "", "", ""], "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}, {"prompt": "You are given two samples A and B, with the same values of $\\bar{x}$, $\\mu$ and $\\sigma$. For sample A, $z = 5$. If the value of $n$ is {bigger} for sample B than sample A, then the value of $z$ for sample B is
", "matrix": "if(random=0,[1,0,0],[0,1,0])", "shuffleChoices": false, "marks": 0, "variableReplacements": [], "choices": ["more than $5$
", "less than $5$
", "$5$, the same as sample A.
"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "maxMarks": 0, "scripts": {}, "distractors": ["", "", ""], "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}], "statement": "Given the formula
\n$z = \\dfrac{(\\bar{x} -\\mu)}{\\sqrt{\\dfrac{\\sigma^2}{n}}}$,
\nanswer the following questions.
", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"random": {"definition": "random(0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "random", "description": ""}, "smaller": {"definition": "sizes[1-random]", "templateType": "anything", "group": "Ungrouped variables", "name": "smaller", "description": ""}, "bigger": {"definition": "sizes[random]", "templateType": "anything", "group": "Ungrouped variables", "name": "bigger", "description": ""}, "sizes": {"definition": "[ \"bigger\", \"smaller\" ]", "templateType": "list of strings", "group": "Ungrouped variables", "name": "sizes", "description": ""}}, "metadata": {"notes": "", "description": "Identify parts of the equation for a $z$-score.
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Identifying Parts of Equations: t", "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/"}, {"name": "Hayley Moore", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/495/"}], "functions": {}, "ungrouped_variables": ["changes", "r", "rev"], "tags": [], "advice": "", "rulesets": {}, "parts": [{"prompt": "If the number of observations, $n$, $\\var{changes[r]}$ what happens to the value of $t$?
", "matrix": [0, "1", 0], "shuffleChoices": false, "marks": 0, "variableReplacements": [], "choices": ["Stays the same
", "{changes[r]}
", "{rev[r]}
"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "maxMarks": 0, "scripts": {}, "distractors": ["", "", ""], "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}, {"prompt": "What is $s$?
", "matrix": ["1", 0, 0, 0, 0], "shuffleChoices": true, "marks": 0, "variableReplacements": [], "choices": ["The sample standard deviation.
", "The sample mean.
", "The sample variance.
", "The population variance.
", "The population mean.
"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "maxMarks": 0, "scripts": {}, "distractors": ["", "", "", "", ""], "displayColumns": "1", "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}], "statement": "Given
\n$t = \\dfrac{\\bar{x}-\\mu}{\\frac{s}{\\sqrt{n}}}$
\nanswer the following questions.
", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"rev": {"definition": "reverse(changes)", "templateType": "anything", "group": "Ungrouped variables", "name": "rev", "description": ""}, "r": {"definition": "random(0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "r", "description": ""}, "changes": {"definition": "[ \"increases\", \"decreases\" ]", "templateType": "list of strings", "group": "Ungrouped variables", "name": "changes", "description": ""}}, "metadata": {"notes": "", "description": "Identify parts of the equation for a $t$-score.
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Identifying Parts of Equations: R^2", "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/"}, {"name": "Hayley Moore", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/495/"}], "functions": {}, "ungrouped_variables": ["changes", "r", "wrong_answer"], "tags": [], "advice": "", "rulesets": {}, "parts": [{"prompt": "What is $\\hat{y}_i$?
", "matrix": ["1", 0, 0, 0], "shuffleChoices": true, "marks": 0, "variableReplacements": [], "choices": ["The model's prediction of $y_i$.
", "The mean of $y_i$.
", "The $i^{\\text{th}}$ $y$ value.
", "The variance of the $y$ values.
"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "maxMarks": 0, "scripts": {}, "distractors": ["", "", "", ""], "displayColumns": "2", "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}, {"prompt": "What can the formula be rearranged to?
", "matrix": ["1", 0, 0, 0], "shuffleChoices": true, "marks": 0, "variableReplacements": [], "choices": ["$R^2 = \\dfrac{\\sum(y_i-\\bar{y})^2-(y_i-\\hat{y}_i)^2}{\\sum(y_i-\\bar{y})^2}$
", "$R = 1 - \\dfrac{\\sum(y_i-\\hat{y}_i)}{\\sum(y_i - \\bar{y})}$
", "$R^2 = 1 - \\dfrac{\\sum(\\hat{y}_i-\\bar{y})^2}{\\sum(y_i-\\bar{y})^2}$
", "$R^2 = 1 - \\dfrac{\\sum\\hat{y}_i^2}{\\sum\\bar{y}^2}$
"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "maxMarks": 0, "scripts": {}, "distractors": ["", "", "", ""], "displayColumns": "1", "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}, {"prompt": "What is the largest value $R$ can take?
", "allowFractions": false, "variableReplacements": [], "maxValue": "1", "minValue": "1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": false, "scripts": {}, "marks": "1", "type": "numberentry", "showPrecisionHint": false}], "statement": "Given the formula for the coefficient of determination of a regression model,
\n\\[ R^2 = 1 - \\dfrac{\\sum(y_i-\\hat{y}_i)^2}{\\sum(y_i-\\bar{y})^2}, \\]
\nanswer the following questions
", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"wrong_answer": {"definition": "reverse(changes)", "templateType": "anything", "group": "Ungrouped variables", "name": "wrong_answer", "description": ""}, "r": {"definition": "random(0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "r", "description": ""}, "changes": {"definition": "[ \"increases\", \"decreases\" ]", "templateType": "list of strings", "group": "Ungrouped variables", "name": "changes", "description": ""}}, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Identifying Parts of Equations: b", "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/"}, {"name": "Hayley Moore", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/495/"}], "functions": {}, "ungrouped_variables": ["change", "changes", "r", "change2", "correct"], "tags": [], "advice": "", "rulesets": {}, "parts": [{"maxAnswers": 0, "prompt": "Which of the following is equivalent to the given equation for $b$?
", "matrix": ["1", "1", 0, 0, 0], "shuffleChoices": true, "marks": 0, "variableReplacements": [], "minAnswers": 0, "variableReplacementStrategy": "originalfirst", "displayType": "checkbox", "maxMarks": 0, "scripts": {}, "distractors": ["", "", "", "", ""], "warningType": "none", "displayColumns": "1", "showCorrectAnswer": true, "choices": ["$b = \\frac{ \\sum(xy)-\\frac{\\sum{x} \\sum{y}}{n}}{\\sum(x^2)-\\frac{(\\sum{x})^2}{n}}\\\\ $
", "$b = \\frac{n \\sum(xy)-\\sum{x} \\sum{y}}{n\\sum(x^2)-(\\sum{x})^2}\\\\ $
", "$b = \\dfrac{\\sum(y_i-\\bar{y})}{\\sum(x_i-\\bar{x})}\\\\ $
", "$b = \\dfrac{\\sum(x_iy_i-\\overline{xy})}{\\sum(x_i-\\bar{x})^2}\\\\ $
", "$b = \\dfrac{\\sum(x_iy_i-\\bar{x}\\bar{y})}{\\sum(x_i-\\bar{x})^2}\\\\ $
"], "type": "m_n_2", "minMarks": 0}, {"prompt": "What is $\\bar{y}$?
", "matrix": ["1", 0, 0, 0], "shuffleChoices": true, "marks": 0, "variableReplacements": [], "choices": ["Sample mean of the $y$ values
", "Population mean of the $y$ values
", "Sample variance of the $y$ values
", "Estimation of the $y$ values
"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "maxMarks": 0, "scripts": {}, "distractors": ["", "", "", ""], "displayColumns": "1", "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}, {"prompt": "If every value $y_i$ is the same, what is $b$?
", "allowFractions": false, "variableReplacements": [], "maxValue": "0", "minValue": "0", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": "1", "type": "numberentry", "showPrecisionHint": false}], "statement": "Given that
\n\\[ b = \\dfrac{\\sum(x_i-\\bar{x})(y_i-\\bar{y})}{\\sum(x_i-\\bar{x})^2}, \\]
\nanswer the following questions.
", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"r": {"definition": "random(0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "r", "description": ""}, "changes": {"definition": "change[r]", "templateType": "anything", "group": "Ungrouped variables", "name": "changes", "description": ""}, "change2": {"definition": "reverse(change)", "templateType": "anything", "group": "Ungrouped variables", "name": "change2", "description": ""}, "change": {"definition": "[ \"increases\", \"decreases\" ]", "templateType": "list of strings", "group": "Ungrouped variables", "name": "change", "description": ""}, "correct": {"definition": "change2[r]", "templateType": "anything", "group": "Ungrouped variables", "name": "correct", "description": ""}}, "metadata": {"notes": "", "description": "Identify parts of a statistical formula from a linear regression model
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Identifying Parts of Equations: r", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Hayley Moore", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/495/"}], "functions": {}, "ungrouped_variables": [], "tags": [], "advice": "", "rulesets": {}, "parts": [{"prompt": "What is $\\bar{x}$?
", "matrix": ["1", 0, 0, 0], "shuffleChoices": true, "marks": 0, "variableReplacements": [], "choices": ["Sample mean
", "Sample variance
", "Sum of the observation values
", "Population variance
"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "maxMarks": "1", "scripts": {}, "distractors": ["", "", "", ""], "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}, {"prompt": "The term $\\sum(x_i - \\bar{x})^2$ can be written in a different form. Select the alternative form in the list below.
", "matrix": ["1", 0, 0, 0], "shuffleChoices": true, "marks": 0, "variableReplacements": [], "choices": ["$\\sum({x_i}^2- 2x_i\\bar{x} + \\bar{x}^2)$
", "$\\sum(x_i-\\bar{x})\\times\\sum(x_i-\\bar{x})$
", "$\\sum({x_i}^2 + \\bar{x}^2)$
", "$\\sum({x_i}^2 - \\bar{x}^2)$
"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "maxMarks": 0, "scripts": {}, "distractors": ["", "", "", ""], "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}, {"prompt": "What does $x_i-\\bar{x}$ mean?
", "matrix": ["1", 0, 0], "shuffleChoices": true, "marks": 0, "variableReplacements": [], "choices": ["The amount observation $i$ deviated from the sample mean.
", "The difference between the population mean and the observation.
", "The amount observation are larger than the sample mean by.
"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "maxMarks": 0, "scripts": {}, "distractors": ["", "", ""], "displayColumns": "1", "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}], "statement": "Given the formula
\n$r = \\frac{\\sum(x_i-\\bar{x})(y_i-\\bar{y})}{\\sqrt{\\sum(x_i-\\bar{x})^2\\sum(y_i - \\bar{y})^2}}$
\nanswer the following questions.
", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {}, "metadata": {"notes": "", "description": "Identify the parts of a statistical formula
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}], "name": ""}], "pickQuestions": 0, "type": "exam", "name": "Identifying parts of statistical formulas", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Identify parts of statistical formulas, and answer questions about how changing one variable affects the statistic.
", "notes": ""}, "navigation": {"showresultspage": "oncompletion", "allowregen": true, "showfrontpage": true, "preventleave": true, "browse": true, "onleave": {"message": "", "action": "none"}, "reverse": true}, "questions": [], "percentPass": 0, "duration": 0, "timing": {"timeout": {"message": "", "action": "none"}, "allowPause": true, "timedwarning": {"message": "", "action": "none"}}, "allQuestions": true, "shuffleQuestions": false, "showQuestionGroupNames": false, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "extensions": [], "custom_part_types": [], "resources": []}