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1. Converting to $\\operatorname{N}(0,1)$
\n$\\simplify[all,!collectNumbers]{P(X < {lower}) = P(Z < ({lower} -{m}) / {s}) = P(Z < {lower-m}/{s}) = 1 -P(Z < {m-lower}/{s})} = 1 -\\var{p} = \\var{precround(1 -p,4)}$ to 4 decimal places.
\n2. Converting to $\\operatorname{N}(0,1)$
\n$\\simplify[all,!collectNumbers]{P(X > {upper}) = P(Z > ({upper} -{m}) / {s}) = P(Z > {upper-m}/{s}) = 1 -P(Z < {upper-m}/{s})} = 1-\\var{p1} = \\var{precround(1 -p1,4)}$ to 4 decimal places.
\n ", "rulesets": {}, "parts": [{"prompt": "Find the probability that in a particular week the {amount} is less than {lower} {units1}:
\nProbability = [[0]] (to 4 decimal places)
\nFind the probability that in a particular week the {amount} is greater than {upper} {units1}:
\nProbability = [[1]] (to 4 decimal places)
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\ni.e. \\[X \\sim \\operatorname{N}(\\var{m},\\var{s}^2)\\]
\n\n ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"units1": {"definition": "\"k Wh\"", "templateType": "anything", "group": "Ungrouped variables", "name": "units1", "description": ""}, "upper": {"definition": "random(m+0.5s..m+1.5*s#5)", "templateType": "anything", "group": "Ungrouped variables", "name": "upper", "description": ""}, "lower": {"definition": "random(m-1.5*s..m-0.5s#5)", "templateType": "anything", "group": "Ungrouped variables", "name": "lower", "description": ""}, "p1": {"definition": "normalcdf((upper-m)/s,0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "p1", "description": ""}, "m": {"definition": "random(750..1250#50)", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "s": {"definition": "random(60..100#10)", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "p": {"definition": "normalcdf((m-lower)/s,0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "amount": {"definition": "\"electricity consumption\"", "templateType": "anything", "group": "Ungrouped variables", "name": "amount", "description": ""}, "stuff": {"definition": "\"a frozen foods warehouse each week in the summer months \"", "templateType": "anything", "group": "Ungrouped variables", "name": "stuff", "description": ""}, "tol": {"definition": "0.0001", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "prob2": {"definition": "precround(1-normalcdf(upper,m,s),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "prob2", "description": ""}, "prob1": {"definition": "precround(normalcdf(lower,m,s),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "prob1", "description": ""}}, "metadata": {"notes": "\n \t\t
1/1/2012:
\n \t\tCan be configured to other applications using the string variables suppplied. Included tag sc.
\n \t\t", "description": "Given a random variable $X$ normally distributed as $\\operatorname{N}(m,\\sigma^2)$ find probabilities $P(X \\gt a),\\; a \\gt m;\\;\\;P(X \\lt b),\\;b \\lt m$.
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "showQuestionGroupNames": false, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "\n \t\t6 questions on standard statistical distributions.
\n \t\tBinomial, Poisson, Normal, Uniform, Exponential.
\n \t\t"}, "name": "The normal distribution", "type": "exam", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Lauren Frances Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2490/"}], "extensions": ["stats"], "custom_part_types": [], "resources": []}