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<p><strong>Step 1: Null hypothesis</strong></p>\n      <p>$\\operatorname{H}_0$: The number of calls per minute follows a Poisson distribution.</p>\n      <p><strong>Step 2: Alternative hypothesis</strong></p>\n      <p>$\\operatorname{H}_1$:&nbsp;The number of calls per minute does not follow a Poisson distribution.</p>\n      ", "showCorrectAnswer": true, "scripts": {}, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "v", "minValue": "v", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "x[0]+tol1", "minValue": "x[0]-tol1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "x[1]+tol1", "minValue": "x[1]-tol1", "correctAnswerFraction": false, 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"allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ex5+tol", "minValue": "ex5-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "few", "minValue": "few", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ch+tol", "minValue": "ch-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n      <p><strong>Step 3: Test statistic</strong></p>\n        <p>(a) Calculate the Poisson rate parameter to 2 decimal places &nbsp;</p>\n        <p>$\\lambda=\\;$?[[0]]</p>\n        <p>(b) Calculate the Poisson probabilities for each category using the value for $\\lambda$ &nbsp;you have just calculated.</p>\n        <p>Calculate all probabilities to 3 decimal places.</p>\n        <table style=\"border: 1px solid black;\">\n        <tbody>\n        <tr>\n        <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>$P(X=0)$</strong></td>\n        <td>[[1]]</td>\n        </tr>\n        <tr>\n        <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>$P(X=1)$</strong></td>\n        <td>[[2]]</td>\n        </tr>\n        <tr>\n        <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>$P(X=2)$</strong></td>\n        <td>[[3]]</td>\n        </tr>\n        <tr>\n        <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>$P(X=3)$</strong></td>\n        <td>[[4]]</td>\n        </tr>\n        <tr>\n        <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>$P(X=4)$</strong></td>\n        <td>[[5]]</td>\n        </tr>\n        <tr>\n        <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>$P(X \\gt 4)$</strong></td>\n        <td>[[6]]</td>\n        </tr>\n        </tbody>\n        </table>\n        <p>&nbsp;</p>\n        <p>(Calculate $P(X \\gt 4)$ as $1-P(X=0)-P(X=1)-P(X=2)-P(X=3) - P(X=4)$ &nbsp;using the probabilities you have found).</p>\n        <p>c) Calculate the expected frequencies to 2 decimal places using the probabilities you have just calculated.</p>\n        <table style=\"border: 1px solid black;\">\n        <tbody>\n        <tr>\n        <td><strong>Observed Frequencies</strong></td>\n        <td><strong>Expected Frequencies</strong></td>\n        </tr>\n        <tr>\n        <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>$\\var{a}$</strong></td>\n        <td style=\"border-top: 1px solid;\">[[7]]</td>\n        </tr>\n        <tr>\n        <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>$\\var{b}$</strong></td>\n        <td style=\"border-top: 1px solid;\">[[8]]</td>\n        </tr>\n        <tr>\n        <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>$\\var{c}$</strong></td>\n        <td style=\"border-top: 1px solid;\">[[9]]</td>\n        </tr>\n        <tr>\n        <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>$\\var{d}$</strong></td>\n        <td style=\"border-top: 1px solid;\">[[10]]</td>\n        </tr>\n        <tr>\n        <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>$\\var{f}$</strong></td>\n        <td style=\"border-top: 1px solid;\">[[11]]</td>\n        </tr>\n        <tr>\n        <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>$\\var{t}$</strong></td>\n        <td style=\"border-top: 1px solid;\">[[12]]</td>\n        </tr>\n        </tbody>\n        </table>\n        <p>How many categories need to be pooled to make sure that the expected frequency for each category is $\\geq 5$ &nbsp;? &nbsp;[[13]]</p>\n        <p>Hence the test statistic is: $\\chi^2=\\;$?[[14]]</p>\n        <p>Input the test statistic to 2 decimal places.</p>\n      ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "5-{few}", "minValue": "5-{few}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"displayType": "radiogroup", "choices": ["<p>$p$ is bigger than $10 \\%$</p>", "<p>$p$ lies between $10 \\%$ and $5 \\%$</p>", "<p>$p$ lies between $5 \\%$ and $1 \\%$</p>", "<p>$p$ is less than $1 \\%$</p>"], "displayColumns": 0, "distractors": ["", "", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "Mm", "marks": 0}], "type": "gapfill", "prompt": "\n      <p><strong>Step 4: &nbsp;p-value range</strong></p>\n      <p>After pooling, calculate , the degrees of freedom, for this test: .&nbsp;</p>\n      <p>$\\nu =\\;$?[[0]]</p>\n      <p>Use tables to find a range for your -value. &nbsp;Choose the correct choice below.&nbsp;</p>\n      <p>[[1]]</p>\n      <p>&nbsp;</p>\n      ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["None", "Slight", "Moderate", "Strong"], "displayColumns": 0, "distractors": ["", "", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "Mm", "marks": 0}, {"displayType": "radiogroup", "choices": ["Retain $\\operatorname{H}_0$", "Reject  $\\operatorname{H}_0$"], "displayColumns": 0, "distractors": ["", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "Mc", "marks": 0}, {"displayType": "radiogroup", "choices": ["No evidence to suggest that the number of calls does not follow a Poisson distribution", "There is evidence to suggest that the number of calls does not follow a Poisson distribution"], "displayColumns": 0, "distractors": ["", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "Mc", "marks": 0}], "type": "gapfill", "prompt": "\n      <p><strong>Step 5: Conclusion</strong></p>\n      <p>&nbsp;</p>\n      <p>Given the &nbsp;$p$- value and the range you have found what is the strength of evidence against the null hypothesis?</p>\n      <p>[[0]]</p>\n      <p>Your Decision:</p>\n      <p>&nbsp;[[1]]</p>\n      <p>Conclusion:</p>\n      <p>&nbsp;[[2]]</p>\n      ", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>You are employed as a business analyst for the NHS European Health Insurance Card (EHIC) division.</p>\n<p>Before launching a new automatic telephone application system, you are required to find out whether the number of calls made to the department, per minute, follows a Poisson distribution.&nbsp;</p>\n<p>Over a period of {thismany} minutes, the following information was obtained:</p>\n<p>{table1([['0',{a}],['1',{b}],['2',{c}],['3',{d}],['4',{f}],['5+',{t}]],['No. of Calls (X)','Frequency per minute'],true,true,false,false)}</p>", "tags": ["checked2015", "MAS1403"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find out whether the data presented in this question follows a Poisson distribution. Uses the $\\chi^2$ test.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p><strong>Step 3.</strong></p>\n  <p>a) $\\lambda$ &nbsp;is the mean of the data i.e.</p>\n  <p>\\[\\lambda = \\frac{0 \\times \\var{a}+1\\times \\var{b}+2\\times\\var{c}+3\\times\\var{d}+4\\times\\var{f}+5\\times \\var{t}}{\\var{thismany}} = \\var{v}\\] to 2 decimal places.</p>\n  <p>b)&nbsp;This is the value of $\\lambda$ we use to calculate the probabilities, to 3 decimal places, &nbsp;assuming that the data is from a Poisson distribution with this parameter.</p>\n  <p>For example, $ \\displaystyle P(X=2)=\\frac{e^{-\\lambda}\\lambda^2}{2!}=\\frac{e^{-\\var{v}}\\times \\var{v}^2}{2}=\\var{x[2]}$</p>\n  <p>to 3 decimal places.</p>\n  <p>The next table shows the values to 3 decimal places you should have obtained:</p>\n  <p>&nbsp;</p>\n  <table style=\"border: 1px solid black;\">\n  <tbody>\n  <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>$P(X=0)$</strong></td>\n  <td style=\"border-bottom: 1px solid;\">$\\var{x[0]}$</td>\n  </tr>\n  <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>$P(X=1)$</strong></td>\n  <td style=\"border-bottom: 1px solid;\">$\\var{x[1]}$</td>\n  </tr>\n  <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>$P(X=2)$</strong></td>\n  <td style=\"border-bottom: 1px solid;\">$\\var{x[2]}$</td>\n  </tr>\n  <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>$P(X=3)$</strong></td>\n  <td style=\"border-bottom: 1px solid;\">$\\var{x[3]}$</td>\n  </tr>\n  <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>$P(X=4)$</strong></td>\n  <td style=\"border-bottom: 1px solid;\">$\\var{x[4]}$</td>\n  </tr>\n  <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>$P(X \\gt 4)$</strong></td>\n  <td style=\"border-bottom: 1px solid;\">$\\var{x5}$</td>\n  </tr>\n  </tbody>\n  </table>\n  <p>c) Using this probability then the expected number of occurences of 2 calls in a minute is \\[P(X=2) \\times \\var{thismany} = \\var{x[2]}\\times \\var{thismany}=\\var{ex[2]}\\] &nbsp;to 2 decimal places.</p>\n  <p>In the same way we find all the expected frequencies assuming that it is a Poisson distribution with parameter $\\lambda=\\var{v}$.</p>\n  <p>The following table shows the actual observed frequencies and the expected frequencies, to 2 decimal places, under this assumption.</p>\n  <p>{table1([[{a},{ex[0]}],[{b},{ex[1]}],[{c},{ex[2]}],[{d},{ex[3]}],[{f},{ex[4]}],[{t},{ex5}]],['No. of Calls (X) Observed','Expected Frequency '],true,true,false,false)}</p>\n  <p>The expected frequencies in the last {few} categories need to be pooled as the {sumofthe} &nbsp;expected {freq} for the last {few1} {cat} is less than 5.</p>\n  <p>Hence we obtain the following table used to calculate the test statistic:</p>\n  <p>{table1([[{a},{ex[0]}],[{b},{ex[1]}],[{c},{ex[2]}],[{dp},{ep4}],[{fq},{eq5}]],['No. of Calls (X) Observed','Expected Frequency '],true,true,false,false)}</p>\n  <p>We now use the last table to calculate the $\\chi^2$ statistic to see if there is a reasonable match between the observed and expected frequencies.&nbsp;</p>\n  <p>We have:</p>\n  <p>\\[\\chi ^ 2 = \\simplify[all,!collectNumbers]{({a} -{ex[0]}) ^ 2 / {ex[0]} + ({b} -{ex[1]}) ^ 2 / {ex[1]} + ({c} -{ex[2]}) ^ 2 / {ex[2]} + ({dp} -{ep4}) ^ 2 / {ep4} + {z} * (({fp} -{ep5}) ^ 2 / {ep5}) = {ch}}\\] to 2 decimal places.</p>\n  <p><strong>Step 4:</strong></p>\n  <p>The degrees of freedom is given by: &nbsp;$\\nu=\\;$no. of pooled categories $- 2 =\\var{6-few+1}-2=\\var{5-few}$</p>\n  <p>(We have to take away $2$ from the number of pooled categories as we have estimated the parameter $\\lambda$ .)</p>\n  <p>The following are the critical values for $\\nu=\\var{5-few}$:</p>\n  <p>{table1([['Critical Value',{ch90},{ch95},{ch99}]],['p-Value','10%','5%','1%'],false,false,false,false)}</p>\n  <p>Comparing these values with the the test statistic we see that the &nbsp;$p$-value {Correct}.</p>\n  <p><strong>Step 5: Conclusion</strong></p>\n  <p>Hence there is {evi[pval]} evidence against and so we {cho} the null hypothesis that the number of calls per minute follows a Poisson distribution.</p>\n  <p>&nbsp;</p>\n  <p>&nbsp;</p>", "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/"}]}