What is the probability that the mean {this} taken from a sample of $\\var{thismany1}$ {these} is within $\\var{thismuch}$ {units} of the population mean?
\nInput the probability to 3 decimal places.
", "marks": 1}, {"showCorrectAnswer": true, "correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "prob2+0.001", "minValue": "prob2-0.001", "prompt": "What is the probability that the mean {this} taken from a sample of $\\var{thismany2}$ {these} is within $\\var{thismuch}$ {units} of the population mean?
\nInput the probability to 3 decimal places.
", "marks": 1}, {"showCorrectAnswer": true, "correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "prob3+0.001", "minValue": "prob3-0.001", "prompt": "What is the probability that the mean {this} taken from a sample of $\\var{thismany3}$ {these} is within $\\var{thismuch}$ {units} of the population mean?
\nInput the probability to 3 decimal places.
", "marks": 1}, {"displayType": "radiogroup", "choices": ["{thismany1}
", "{thismany2}
", "{thismany3}
", "Increase the sample size
"], "displayColumns": 0, "prompt": "Given the probabilities you have found, what sample size would you recommend to have at least a $0.95$ probability that the sample mean is within $\\var{thismuch}$ {units} of the population mean?
", "distractors": ["", "", "", ""], "shuffleChoices": false, "scripts": {}, "maxMarks": 0, "type": "1_n_2", "minMarks": 0, "showCorrectAnswer": true, "matrix": "mm", "marks": 0}], "statement": "The population mean of the {this} of {that} is $\\var{popmean}$ {units} with population standard deviation $\\var{popstdev}$ {units}.
\nFind the following probabilities.
\nInput all probabilities to 3 decimal places.
\n", "tags": ["ACE2013", "checked2015", "mean", "mean ", "normal distribution", "Normal distribution", "Probability", "probability", "sampling", "satisfy", "standard deviation", "statistics", "z score"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "There are variables set up to set up different scenarios. Present one is price of unleaded petrol in the UK.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Given data on population mean and population standard deviation and three sampling sizes, calculate the probabilities that the sample means are within a specified distance from the population mean.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a)
\nLet $M$ be the mean price of the sample
\nConverting to $N(0,1)$ we see that we require the probability that if $\\displaystyle Z=\\frac{|M-\\var{popmean}|}{\\frac{\\var{popstdev}}{\\sqrt{\\var{thismany1}}}}$ then we require the probability that \\[\\frac{-\\var{thismuch}}{\\frac{\\var{popstdev}}{\\sqrt{\\var{thismany1}}}} \\lt Z \\lt \\frac{\\var{thismuch}}{\\frac{\\var{popstdev}}{\\sqrt{\\var{thismany1}}}} \\implies \\var{-thismany1^0.5*thismuch/popstdev}\\lt Z \\lt \\var{thismany1^0.5*thismuch/popstdev}\\].
\nThe probability that $Z \\lt \\var{-thismany1^0.5*thismuch/popstdev}$ is $p=\\var{normalcdf(-thismany1^0.5*thismuch/popstdev,0,1)}$.
\nHence the probability we want is $P_1=1-2p=\\var{prob1}$ to 3 decimal places.
\nb)
\nSimilarly we have for sample size $\\var{thismany2}$ we want the probability that:
\n\\[\\frac{-\\var{thismuch}}{\\frac{\\var{popstdev}}{\\sqrt{\\var{thismany2}}}} \\lt Z \\lt \\frac{\\var{thismuch}}{\\frac{\\var{popstdev}}{\\sqrt{\\var{thismany2}}}} \\implies \\var{-thismany2^0.5*thismuch/popstdev}\\lt Z \\lt \\var{thismany2^0.5*thismuch/popstdev}\\].
\nThe probability that $Z \\lt \\var{-thismany2^0.5*thismuch/popstdev}$ is $p=\\var{normalcdf(-thismany2^0.5*thismuch/popstdev,0,1)}$.
\nHence the probability we want is $P_2=1-2p=\\var{prob2}$ to 3 decimal places.
\nc)
\nFor sample size $\\var{thismany3}$ we want the probability that:
\n\\[\\frac{-\\var{thismuch}}{\\frac{\\var{popstdev}}{\\sqrt{\\var{thismany3}}}} \\lt Z \\lt \\frac{\\var{thismuch}}{\\frac{\\var{popstdev}}{\\sqrt{\\var{thismany3}}}} \\implies \\var{-thismany3^0.5*thismuch/popstdev}\\lt Z \\lt \\var{thismany3^0.5*thismuch/popstdev}\\].
\nThe probability that $Z \\lt \\var{-thismany3^0.5*thismuch/popstdev}$ is $p=\\var{normalcdf(-thismany3^0.5*thismuch/popstdev,0,1)}$.
\nHence the probability we want is $P_3=1-2p=\\var{prob3}$ to 3 decimal places.
\nd) {message}
", "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/"}]}