// Numbas version: exam_results_page_options
{"name": "mathcentre: Hypothesis testing with the Student t-distribution", "type": "exam", "duration": 0, "metadata": {"notes": "", "description": "<p>Three questions on using the Student $t$ distribution to perform&nbsp;hypothesis tests.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "allQuestions": true, "shuffleQuestions": false, "questions": [], "percentPass": 75, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "pickQuestions": 0, "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "preventleave": false, "browse": true, "showfrontpage": false, "showresultspage": "never"}, "feedback": {"showtotalmark": true, "advicethreshold": 0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Independent two sample t-test.", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {"pstdev": {"definition": "sqrt(abs(l)/(abs(l)-1))*stdev(l)", "type": "number", "language": "jme", "parameters": [["l", "list"]]}}, "tags": ["average", "data analysis", "differences", "elementary statistics", "hypothesis testing", "mean", "standard deviation", "statistics", "stats", "t-test", "two sample t-test", "variance"], "advice": "\n  <p>We test the following hypothesis,</p>\n  <p style=\"text-align: center;\">$H_0:\\; \\mu_1=\\mu_2$ versus $H_1:\\; \\mu_1 \\neq \\mu_2$</p>\n  <p>We find that the mean score of Group 1 is $\\overline{x}_1=\\var{m1}$ with standard deviation $s_1=\\var{sd1}$ and the mean score of Group 2 is $\\overline{x}_2=\\var{m2}$ with standard deviation $s_2=\\var{sd2}$.</p>\n  <p>All calculated to 3 decimal places.</p>\n  <p>Using the formula for the two-sample t-statistic as &nbsp;shown above with $n_1=n_2=10$:</p>\n  <p>The estimate of the pooled variance is calculated to be:</p>\n  <p>\\[s^2=\\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}= \\frac{\\var{n1-1}\\times \\var{sd1}^2+\\var{n2-1}\\times \\var{sd2}^2}{\\var{n1+n2-2}}=\\var{s^2}.\\]&nbsp;</p>\n  <p>Hence $s= \\sqrt{\\var{s^2}}=\\var{s}$ to 3 decimal places.</p>\n  <p>We find that the t-statistic has value:</p>\n  <p>\\[\\begin{eqnarray*}T&amp;=&amp; \\frac{(\\overline{x}_1-\\overline{x}_2)-(\\mu_1-\\mu_2)}{s\\sqrt{\\frac{1}{n_1}+\\frac{1}{n_2}}}\\\\&amp;=&amp;\\frac{(\\var{m1}-\\var{m2})-(0)}{\\var{s}\\sqrt{\\frac{1}{\\var{n1}}+\\frac{1}{\\var{n2}}}}\\\\&amp;=&amp;\\var{tvalue}\\end{eqnarray*}\\] to 3 decimal places.</p>\n  <p>Our test statistic is $|T|=\\var{abs(tvalue)}$.</p>\n  <p>Given that we have $n_1+n_2-2=18$ degrees of freedom, we look up this value on the T-distribution table for $t_{18}$</p>\n  <p>\\[\\begin{array}{r|rrrrr}&amp;0.20&amp;0.10&amp;0.05&amp;0.01&amp;0.001\\\\\\hline18&amp;1.330&amp;1.734&amp;2.101&amp;2.878&amp;3.922\\end{array}\\]</p>\n  <p>We see that the t-statistic {msg[t]} and the table tells us that the $p$ value {pmsg[t]}.</p>\n  <p>Hence we conclude that we {cmsg[t]} the null hypothesis. There is {cmsg1[t]} evidence of a difference between the average scores of the two groups.</p>\n  <p>&nbsp;</p>\n  ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 0.0, "prompt": "\n      <p>Find the mean and standard deviations of the scores of the two groups:</p>\n        <p>Mean scores of Group 1= [[0]] (input to 3 decimal places)</p>\n        <p>Standard deviation of scores for Group 1 = [[1]] (input to 3 decimal places)</p>\n        <p>Mean scores of Group 2= [[2]] (input to 3 decimal places)</p>\n        <p>Standard deviation of scores for Group 2 = [[3]] (input to 3 decimal places)</p>\n        <p>Now find the two sample t-test statistic $T$ using the values you have just calculated to 3 decimal places and input $|T|$ here: [[4]] (3 decimal places)</p>\n      ", "gaps": [{"minvalue": "{m1}", "type": "numberentry", "maxvalue": "{m1}", "marks": 0.5, "showPrecisionHint": false}, {"minvalue": "{sd1}", "type": "numberentry", "maxvalue": "{sd1}", "marks": 0.5, "showPrecisionHint": false}, {"minvalue": "{m2}", "type": "numberentry", "maxvalue": "{m2}", "marks": 0.5, "showPrecisionHint": false}, {"minvalue": "{sd2}", "type": "numberentry", "maxvalue": "{sd2}", "marks": 0.5, "showPrecisionHint": false}, {"minvalue": "{abs(tvalue)-tol}", "type": "numberentry", "maxvalue": "{abs(tvalue)+tol}", "marks": 1.0, "showPrecisionHint": false}], "steps": [{"prompt": "\n          <p>The two-sample t-statistic for two independent sets of data where one set has $n_1$ datapoints and the other set $n_2$ datapoints is calculated as follows:</p>\n            <p>\\[T=\\frac{(\\overline{x}_1-\\overline{x}_2)-(\\mu_1-\\mu_2)}{s\\times\\sqrt{\\frac{1}{n_1}+\\frac{1}{n_2}}}\\;\\;\\;\\]</p>\n            <p>where $\\overline{x}_1,\\;\\overline{x}_2$ are the sample means and&nbsp;</p>\n            <p>\\[s^2=\\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}\\]</p>\n            <p>where $s_1,\\;s_2$ are the sample standard deviations.</p>\n            <p>Use the values you calculated to 3 decimal places in order to find $T$.</p>\n          ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}, {"maxanswers": 0.0, "displaycolumns": 0.0, "prompt": "\n      <p>Give the value $|T|$ of the t-statistic you have found, choose the range for the $p$ value by looking up the t tables:</p>\n      <p>&nbsp;</p>\n      ", "matrix": "v", "shufflechoices": false, "minanswers": 0.0, "choices": ["<p>$p$ less than $0.1\\%$</p>", "<p>$p$ lies between $0.1\\%$ and $1%$</p>", "<p>$p$ lies between  $1 \\%$ and $5\\%$</p>", "<p>$p$ lies between  $5 \\%$ and $10\\%$</p>", "<p>$p$ is greater than $10\\%$</p>"], "displaytype": "radiogroup", "maxmarks": 0.0, "distractors": ["", "", "", "", ""], "marks": 0.0, "type": "1_n_2", "minmarks": 0.0}, {"maxanswers": 0.0, "displaycolumns": 0.0, "prompt": "<p>Given the $p$-value and the range you have found, what is the strength of evidence against the null hypothesis that there is no difference in the average times for the left and right hands?</p>", "matrix": "v", "shufflechoices": false, "minanswers": 0.0, "choices": ["Very Strong", "Strong", "Moderate", "Slight", "None"], "displaytype": "radiogroup", "maxmarks": 0.0, "distractors": ["", "", "", "", ""], "marks": 0.0, "type": "1_n_2", "minmarks": 0.0}], "statement": "\n  <p>An educational psychologist claimed that the order in which questions were asked affected the&nbsp;student&rsquo;s ability to answer them correctly and hence their total score. In order to test this, $20$ students were randomly divided into two groups of $10$. The first group were given questions&nbsp;in increasing order of difficulty and the second group in decreasing order of difficulty. The&nbsp;ordered test scores obtained were:</p>\n    <table border=\"1\">\n    <tbody>\n    <tr><th align=\"left\">Group 1</th>\n    <td>$\\var{r1[0]}$</td>\n    <td>$\\var{r1[1]}$</td>\n    <td>$\\var{r1[2]}$</td>\n    <td>$\\var{r1[3]}$</td>\n    <td>$\\var{r1[4]}$</td>\n    <td>$\\var{r1[5]}$</td>\n    <td>$\\var{r1[6]}$</td>\n    <td>$\\var{r1[7]}$</td>\n    <td>$\\var{r1[8]}$</td>\n    <td>$\\var{r1[9]}$</td>\n    </tr>\n    <tr><th align=\"left\">Group 2</th>\n    <td>$\\var{r2[0]}$</td>\n    <td>$\\var{r2[1]}$</td>\n    <td>$\\var{r2[2]}$</td>\n    <td>$\\var{r2[3]}$</td>\n    <td>$\\var{r2[4]}$</td>\n    <td>$\\var{r2[5]}$</td>\n    <td>$\\var{r2[6]}$</td>\n    <td>$\\var{r2[7]}$</td>\n    <td>$\\var{r2[8]}$</td>\n    <td>$\\var{r2[9]}$</td>\n    </tr>\n    </tbody>\n    </table>\n    <p>Carry out by hand a two-sample t-test to test if there is evidence of a difference in the average&nbsp;test scores for the two sets of students.</p>\n  ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"sd1": {"definition": "precround(pstdev(r1),3)", "name": "sd1"}, "sd2": {"definition": "precround(pstdev(r2),3)", "name": "sd2"}, "m1": {"definition": "mean(r1)", "name": "m1"}, "tol": {"definition": 0.001, "name": "tol"}, "m2": {"definition": "mean(r2)", "name": "m2"}, "msg": {"definition": "['is greater than $\\\\var{t999}$','lies between $\\\\var{t99}$ and $\\\\var{t999}$','lies between $\\\\var{t95}$ and $\\\\var{t99}$','lies between $\\\\var{t90}$ and $\\\\var{t95}$','is less than $\\\\var{t90}$']", "name": "msg"}, "cmsg1": {"definition": "['very strong','strong','slight','no','no']", "name": "cmsg1"}, "t999": {"definition": 3.922, "name": "t999"}, "atvalue": {"definition": "abs(tvalue)", "name": "atvalue"}, "sig1": {"definition": "random(8..10#0.2)", "name": "sig1"}, "tvalue": {"definition": "precround((m1-m2)*sqrt(n1*n2)/(s*sqrt(n1+n2)),3)", "name": "tvalue"}, "sig2": {"definition": "random(8..10#0.2)", "name": "sig2"}, "t99": {"definition": 2.878, "name": "t99"}, "t95": {"definition": 2.101, "name": "t95"}, "t90": {"definition": 1.734, "name": "t90"}, "cmsg": {"definition": "['do reject','do reject','do not reject','do not reject','do not reject']", "name": "cmsg"}, "mu1": {"definition": "random(55..75#0.5)", "name": "mu1"}, "r1": {"definition": "repeat(round(normalsample(mu1,sig1)),10)", "name": "r1"}, "r2": {"definition": "repeat(round(normalsample(mu2,sig2)),10)", "name": "r2"}, "mu2": {"definition": "random(65..75#0.5)", "name": "mu2"}, "pmsg": {"definition": "[' is less than $0.001$',' lies between $0.001$ and $0.01$',' lies between $0.01$ and $0.05$',' lies between $0.05$ and $0.10$',' is greater than $0.10$']", "name": "pmsg"}, "s": {"definition": "precround(sqrt(((n1-1)*sd1^2+(n2-1)*sd2^2)/(n1+n2-2)),3)", "name": "s"}, "t": {"definition": "switch(v[0]=1,0,v[1]=1,1,v[2]=1,2,v[3]=1,3,4)", "name": "t"}, "v": {"definition": "if(atvalue>=t999,[1,0,0,0,0],if(atvalue>=t99,[0,1,0,0,0],if(atvalue>=t95,[0,0,1,0,0],if(atvalue>=t90,[0,0,0,1,0],[0,0,0,0,1]))))", "name": "v"}, "n1": {"definition": 10.0, "name": "n1"}, "n2": {"definition": 10.0, "name": "n2"}, "pvalue": {"definition": "ttest(atvalue,19,2)", "name": "pvalue"}}, "metadata": {"notes": "\n    \t\t                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t<p><strong>11/07/2012:</strong></p>\n    \t\t                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t        <p><strong><br /></strong>Added tags.</p>\n    \t\t                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t        <p>Calculation not yet tested.</p>\n    \t\t                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t        <p><strong>23/07/2012:</strong></p>\n    \t\t                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t        <p>Added description.</p>\n    \t\t                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t        <p>Checked calculation.</p>\n    \t\t                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t        <p>Changed display slightly in Advice.</p>\n    \t\t                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t        <p><strong>3/08/2012:</strong></p>\n    \t\t                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t        <p>Added tags.</p>\n    \t\t                                                                                                                                                                                                                                                                                                                            \t\t                        \t\t        <p>Question appears to be working correctly.</p>\n    \t\t                                                                                                                                                                                                                                                                                                                            \t\t                        \n    \t\t                                                                                                                                                                                                                                                                                                                            \n    \t\t", "description": "<p>Two sample t-test to see if there is a difference between scores on questions between two groups when the questions are asked in a different order.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Paired sample t-test.", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {"pstdev": {"definition": "sqrt(abs(l)/(abs(l)-1))*stdev(l)", "type": "number", "language": "jme", "parameters": [["l", "list"]]}}, "tags": ["average", "data analysis", "differences", "elementary statistics", "hypothesis testing", "mean", "mean of differences", "paired t-test", "standard deviation", "standard deviation of differences", "statistics", "stats", "t-test", "variance"], "advice": "\n  <p>The table of differences is given by:</p>\n    <table border=\"1\">\n    <tbody>\n    <tr><th>{object}</th><th>A</th><th>B</th><th>C</th><th>D</th><th>E</th><th>F</th><th>G</th><th>H</th><th>I</th><th>J</th><th>K</th><th>L</th></tr>\n    <tr><th align=\"left\">Right</th>\n    <td>$\\var{r1[0]}$</td>\n    <td>$\\var{r1[1]}$</td>\n    <td>$\\var{r1[2]}$</td>\n    <td>$\\var{r1[3]}$</td>\n    <td>$\\var{r1[4]}$</td>\n    <td>$\\var{r1[5]}$</td>\n    <td>$\\var{r1[6]}$</td>\n    <td>$\\var{r1[7]}$</td>\n    <td>$\\var{r1[8]}$</td>\n    <td>$\\var{r1[9]}$</td>\n    <td>$\\var{r1[10]}$</td>\n    <td>$\\var{r1[11]}$</td>\n    </tr>\n    <tr><th align=\"left\">Left</th>\n    <td>$\\var{r2[0]}$</td>\n    <td>$\\var{r2[1]}$</td>\n    <td>$\\var{r2[2]}$</td>\n    <td>$\\var{r2[3]}$</td>\n    <td>$\\var{r2[4]}$</td>\n    <td>$\\var{r2[5]}$</td>\n    <td>$\\var{r2[6]}$</td>\n    <td>$\\var{r2[7]}$</td>\n    <td>$\\var{r2[8]}$</td>\n    <td>$\\var{r2[9]}$</td>\n    <td>$\\var{r2[10]}$</td>\n    <td>$\\var{r2[11]}$</td>\n    </tr>\n    <tr><th align=\"left\">Differences</th>\n    <td>$\\var{d[0]}$</td>\n    <td>$\\var{d[1]}$</td>\n    <td>$\\var{d[2]}$</td>\n    <td>$\\var{d[3]}$</td>\n    <td>$\\var{d[4]}$</td>\n    <td>$\\var{d[5]}$</td>\n    <td>$\\var{d[6]}$</td>\n    <td>$\\var{d[7]}$</td>\n    <td>$\\var{d[8]}$</td>\n    <td>$\\var{d[9]}$</td>\n    <td>$\\var{d[10]}$</td>\n    <td>$\\var{d[11]}$</td>\n    </tr>\n    </tbody>\n    </table>\n    <p>We test the following hypothesis:</p>\n    <p style=\"text-align: center;\">$H_0:\\;\\mu_d=0$ versus $H_1:\\;\\mu_d\\neq 0$</p>\n    <p>$n=\\var{n}$ and the mean of the differences is $\\overline{d}=\\var{meandiff}$.</p>\n    <p>The variance $V$ of the differences is calculated to be $\\var{pstdev(d)^2}$<br/><br/>Hence we have the standard deviation $s_d= \\sqrt{V}=\\var{stdiff}$ to 3 decimal places.</p>\n    <p>The paired t-statistic is given by:</p>\n    <p>\\[\\begin{eqnarray*} T&amp;=&amp;\\frac{\\overline{d}-\\mu_d}{\\frac{s_d}{\\sqrt{n}}}\\\\&amp;=&amp;\\frac{\\var{meandiff}-0}{\\frac{\\var{stdiff}}{\\sqrt{\\var{n}}}}\\\\&amp;=&amp;\\var{tvalue}\\end{eqnarray*}\\]</p>\n    <p>(Using the&nbsp;null hypothesis that the means are the same i.e. $\\mu_d=0$.)</p>\n    <p>Hence our test statistic &nbsp;$|T|=\\var{tvalue}$.</p>\n    <p>Looking up this value on the T-distribution table for $t_{11}$</p>\n    <p>\\[\\begin{array}{r|rrrrr}&amp;0.20&amp;0.10&amp;0.05&amp;0.01&amp;0.001\\\\\\hline11&amp;1.363&amp;1.796&amp;2.201&amp;3.106&amp;4.437\\end{array}\\]</p>\n    <p>We see that the t-statistic {msg[t]} and the table tells us that the $p$ value {pmsg[t]}.</p>\n    <p>Hence we conclude that we {cmsg[t]} the null hypothesis. There is {cmsg1[t]} evidence of a difference between the average scores of the two groups.</p>\n    <p>&nbsp;</p>\n  ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n      <p>Find the mean and standard deviations of the difference between left and right {attempt}s.</p>\n        <p>Calculate differences for <strong>left</strong> {attempt} times &ndash; <strong>right</strong> {attempt} times. Make sure you take the differences this way round.</p>\n        <p>Mean of difference = [[0]] (input&nbsp; to 3 decimal places&nbsp;)</p>\n        <p>Standard deviation of difference = [[1]] (input to 3 decimal places)</p>\n        <p>Now find the t-test statistic $T$ using the values you have just calculated and &nbsp;input the absolute value $|T|$ here: [[2]] (3 decimal places).&nbsp;</p>\n        <p>&nbsp;</p>\n      ", "gaps": [{"minvalue": "{meandiff}", "type": "numberentry", "maxvalue": "{meandiff}", "marks": 0.5, "showPrecisionHint": false}, {"minvalue": "{stdiff}", "type": "numberentry", "maxvalue": "{stdiff}", "marks": 0.5, "showPrecisionHint": false}, {"minvalue": "{tvalue}", "type": "numberentry", "maxvalue": "{tvalue}", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}, {"maxanswers": 0.0, "distractors": ["", "", "", "", ""], "prompt": "<p>Give the value of the t-statistic you have found, choose the range for the $p$ value by looking up the t-statistic tables:</p>", "matrix": "v", "shufflechoices": false, "minanswers": 0.0, "choices": ["<p>$p$ less than $0.1 \\%$</p>", "<p>$p$ lies between  $0.1\\%$ and $1 \\%$</p>", "<p>$p$ lies between  $1 \\%$ and $5\\%$</p>", "<p>$p$ lies between  $5 \\%$ and $10\\%$</p>", "<p>$p$ is greater than $10\\%$</p>"], "displaytype": "radiogroup", "maxmarks": 0.0, "marks": 0.0, "displaycolumns": 0.0, "type": "1_n_2", "minmarks": 0.0}, {"maxanswers": 0.0, "distractors": ["", "", "", "", ""], "prompt": "<p>Given the $p$-value and the range you have found, what is the strength of evidence against the null hypothesis that there is no difference in the average times for the left and right hands?</p>", "matrix": "v", "shufflechoices": false, "minanswers": 0.0, "choices": ["Very strong", "Strong", "Moderate", "Slight", "None"], "displaytype": "radiogroup", "maxmarks": 0.0, "marks": 0.0, "displaycolumns": 0.0, "type": "1_n_2", "minmarks": 0.0}], "statement": "\n  <p>The following data was obtained from $12$ individuals. The observations consist of the time taken to complete a dexterity task using their left and right hands.</p>\n  <table border=\"1\">\n  <tbody>\n  <tr><th>{object}</th><th>A</th><th>B</th><th>C</th><th>D</th><th>E</th><th>F</th><th>G</th><th>H</th><th>I</th><th>J</th><th>K</th><th>L</th></tr>\n  <tr><th align=\"left\">Right</th>\n  <td>$\\var{r1[0]}$</td>\n  <td>$\\var{r1[1]}$</td>\n  <td>$\\var{r1[2]}$</td>\n  <td>$\\var{r1[3]}$</td>\n  <td>$\\var{r1[4]}$</td>\n  <td>$\\var{r1[5]}$</td>\n  <td>$\\var{r1[6]}$</td>\n  <td>$\\var{r1[7]}$</td>\n  <td>$\\var{r1[8]}$</td>\n  <td>$\\var{r1[9]}$</td>\n  <td>$\\var{r1[10]}$</td>\n  <td>$\\var{r1[11]}$</td>\n  </tr>\n  <tr><th align=\"left\">Left</th>\n  <td>$\\var{r2[0]}$</td>\n  <td>$\\var{r2[1]}$</td>\n  <td>$\\var{r2[2]}$</td>\n  <td>$\\var{r2[3]}$</td>\n  <td>$\\var{r2[4]}$</td>\n  <td>$\\var{r2[5]}$</td>\n  <td>$\\var{r2[6]}$</td>\n  <td>$\\var{r2[7]}$</td>\n  <td>$\\var{r2[8]}$</td>\n  <td>$\\var{r2[9]}$</td>\n  <td>$\\var{r2[10]}$</td>\n  <td>$\\var{r2[11]}$</td>\n  </tr>\n  </tbody>\n  </table>\n  <p>Carry out by hand a paired t-test to test whether there is evidence of a difference in the average times for the left and right hands.</p>\n  ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"pmsg": {"definition": "[' is less than $0.001$',' lies between $0.001$ and $0.01$',' lies between $0.01$ and $0.05$',' lies between $0.05$ and $0.10$',' is greater than $0.10$']", "name": "pmsg"}, "msg": {"definition": "['is greater than $\\\\var{t999}$','lies between $\\\\var{t99}$ and $\\\\var{t999}$','lies between $\\\\var{t95}$ and $\\\\var{t99}$','lies between $\\\\var{t90}$ and $\\\\var{t95}$','is less than $\\\\var{t90}$']", "name": "msg"}, "cmsg1": {"definition": "['very strong','strong','slight','no','no']", "name": "cmsg1"}, "t999": {"definition": 4.437, "name": "t999"}, "meandiff": {"definition": "precround(mean(d),3)", "name": "meandiff"}, "object": {"definition": "'Individual'", "name": "object"}, "sig1": {"definition": "random(2..3#0.2)", "name": "sig1"}, "tvalue": {"definition": "precround(abs(meandiff*sqrt(n)/stdiff),3)", "name": "tvalue"}, "stdiff": {"definition": "precround(pstdev(d),3)", "name": "stdiff"}, "sig2": {"definition": "random(2..4#0.2)", "name": "sig2"}, "t99": {"definition": 3.106, "name": "t99"}, "t95": {"definition": 2.201, "name": "t95"}, "t90": {"definition": 1.796, "name": "t90"}, "cmsg": {"definition": "['do reject','do reject','do not reject','do not reject','do not reject']", "name": "cmsg"}, "d": {"definition": "list(vector(r2)-vector(r1))", "name": "d"}, "mu1": {"definition": "random(18..25#0.5)", "name": "mu1"}, "r1": {"definition": "repeat(round(normalsample(mu1,sig1)),12)", "name": "r1"}, "r2": {"definition": "repeat(round(normalsample(mu2,sig2)),12)", "name": "r2"}, "mu2": {"definition": "mu1+random(2..4#0.1)", "name": "mu2"}, "n": {"definition": 12.0, "name": "n"}, "t": {"definition": "switch(v[0]=1,0,v[1]=1,1,v[2]=1,2,v[3]=1,3,4)", "name": "t"}, "v": {"definition": "if(tvalue>=t999,[1,0,0,0,0],if(tvalue>=t99,[0,1,0,0,0],if(tvalue>=t95,[0,0,1,0,0],if(tvalue>=t90,[0,0,0,1,0],[0,0,0,0,1]))))", "name": "v"}, "pvalue": {"definition": "precround(ttest(0,d,2),3)", "name": "pvalue"}}, "metadata": {"notes": "\n    \t\t                                                                                                                                \t\t                                                            \t\t<p><strong>11/07/2012:</strong></p>\n    \t\t                                                                                                                                \t\t                                                            \t\t        <p><strong><br /></strong>Added tags.</p>\n    \t\t                                                                                                                                \t\t                                                            \t\t        <p>Calculation not yet tested.</p>\n    \t\t                                                                                                                                \t\t                                                            \t\t        <p><strong>23/07/2012:</strong></p>\n    \t\t                                                                                                                                \t\t                                                            \t\t        <p>Added description.</p>\n    \t\t                                                                                                                                \t\t                                                            \t\t        <p>Checked calculation.</p>\n    \t\t                                                                                                                                \t\t                                                            \t\t        <p>Changed display slightly in Advice.</p>\n    \t\t                                                                                                                                \t\t                                                            \t\t        <p><strong>3/08/2012:</strong></p>\n    \t\t                                                                                                                                \t\t                                                            \t\t        <p>Added tags.</p>\n    \t\t                                                                                                                                \t\t                                                            \t\t        <p>Question appears to be working correctly.</p>\n    \t\t                                                                                                                                \t\t                                                            \n    \t\t                                                                                                                                \n    \t\t", "description": "<p>Paired t-test to see if there is a difference between times take in a task.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Paired t-test after treatment.", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "functions": {"pstdev": {"definition": "sqrt(abs(l)/(abs(l)-1))*stdev(l)", "type": "number", "language": "jme", "parameters": [["l", "list"]]}}, "ungrouped_variables": ["attempt", "cmsg", "d", "meandiff", "msg", "mu1", "mu2", "object", "objects", "pmsg", "pvalue", "r1", "r2", "sig1", "sig2", "stdiff", "t", "t90", "t95", "t99", "t999", "thismany", "tvalue", "v"], "tags": ["average", "data analysis", "differences", "elementary statistics", "hypothesis testing", "mean", "mean of differences", "paired t-test", "standard deviation", "standard deviation of differences", "statistics", "stats", "t-test", "variance"], "preamble": {"css": "", "js": ""}, "advice": "<p>The table of differences is given by:</p>\n<table border=\"1\">\n<tbody>\n<tr><th>{object}</th><th>A</th><th>B</th><th>C</th><th>D</th><th>E</th><th>F</th><th>G</th><th>H</th><th>I</th><th>J</th><th>K</th><th>L</th><th>M</th><th>N</th><th>O</th></tr>\n<tr><th align=\"left\">Before</th>\n<td>$\\var{r1[0]}$</td>\n<td>$\\var{r1[1]}$</td>\n<td>$\\var{r1[2]}$</td>\n<td>$\\var{r1[3]}$</td>\n<td>$\\var{r1[4]}$</td>\n<td>$\\var{r1[5]}$</td>\n<td>$\\var{r1[6]}$</td>\n<td>$\\var{r1[7]}$</td>\n<td>$\\var{r1[8]}$</td>\n<td>$\\var{r1[9]}$</td>\n<td>$\\var{r1[10]}$</td>\n<td>$\\var{r1[11]}$</td>\n<td>$\\var{r1[12]}$</td>\n<td>$\\var{r1[13]}$</td>\n<td>$\\var{r1[14]}$</td>\n</tr>\n<tr><th align=\"left\">After</th>\n<td>$\\var{r2[0]}$</td>\n<td>$\\var{r2[1]}$</td>\n<td>$\\var{r2[2]}$</td>\n<td>$\\var{r2[3]}$</td>\n<td>$\\var{r2[4]}$</td>\n<td>$\\var{r2[5]}$</td>\n<td>$\\var{r2[6]}$</td>\n<td>$\\var{r2[7]}$</td>\n<td>$\\var{r2[8]}$</td>\n<td>$\\var{r2[9]}$</td>\n<td>$\\var{r2[10]}$</td>\n<td>$\\var{r2[11]}$</td>\n<td>$\\var{r2[12]}$</td>\n<td>$\\var{r2[13]}$</td>\n<td>$\\var{r2[14]}$</td>\n</tr>\n<tr><th align=\"left\">Differences</th>\n<td>$\\var{d[0]}$</td>\n<td>$\\var{d[1]}$</td>\n<td>$\\var{d[2]}$</td>\n<td>$\\var{d[3]}$</td>\n<td>$\\var{d[4]}$</td>\n<td>$\\var{d[5]}$</td>\n<td>$\\var{d[6]}$</td>\n<td>$\\var{d[7]}$</td>\n<td>$\\var{d[8]}$</td>\n<td>$\\var{d[9]}$</td>\n<td>$\\var{d[10]}$</td>\n<td>$\\var{d[11]}$</td>\n<td>$\\var{d[12]}$</td>\n<td>$\\var{d[13]}$</td>\n<td>$\\var{d[14]}$</td>\n</tr>\n</tbody>\n</table>\n<p>The mean of the differences is $\\overline{x_d}=\\var{meandiff}$.</p>\n<p>The variance $V$ of the differences is <br/>\\[\\begin{eqnarray*} V&amp;=&amp;\\frac{1}{14}\\left(\\simplify[]{{d[0]^2}+{d[1]^2}+{d[2]^2}+{d[3]^2}+{d[4]^2}+{d[5]^2}+{d[6]^2}+{d[7]^2}+{d[8]^2}+{d[9]^2}+{d[10]^2}+{d[11]^2}+{d[12]^2}+{d[13]^2}+{d[14]^2}}-15\\times \\var{meandiff}^2\\right)\\\\ &amp;=&amp;\\var{variance(d)} \\end{eqnarray*} \\]<br/>Hence we have the standard deviation $s_d= \\sqrt{V}=\\var{stdiff}$ to 3 decimal places.</p>\n<p>The paired t-statistic is given by:</p>\n<p>\\[t_d=\\frac{\\overline{x_d}-\\mu_d}{\\frac{s_d}{\\sqrt{n}}}\\]</p>\n<p>In this example $n=15$ and the null hypothesis is that the means are the same i.e. $\\mu_d=0$.</p>\n<p>On calculating we find $t_d=\\var{tvalue}$.</p>\n<p>Looking up this value on the T-distribution table for $t_{14}$</p>\n<p>\\[\\begin{array}{r|rrrrr}&amp;0.20&amp;0.10&amp;0.05&amp;0.01&amp;0.001\\\\\\hline14&amp;1.345&amp;1.761&amp;2.145&amp;2.977&amp;4.140\\end{array}\\]</p>\n<p>We see that the t-statistic {msg[t]} and the table tells us that the $p$ value {pmsg[t]}.</p>\n<p>Hence we conclude that {cmsg[t]}.</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n        <p>Find the mean and standard deviations of the difference between the before and after responses</p>\n      <p>Calculate differences for after response&ndash; before response.</p>\n      <p>Mean of difference = [[0]] (input&nbsp; to 3 decimal places&nbsp;)</p>\n      <p>Standard deviation of difference = [[1]] (input to 3 decimal places)</p>\n      <p>Now find the paired t-test statistic &nbsp;using the values you have just calculated =[[2]] (3 decimal places)</p>\n      <p>&nbsp;</p>\n        ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 0.5, "maxValue": "{meandiff}", "minValue": "{meandiff}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.5, "maxValue": "{stdiff}", "minValue": "{stdiff}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "{tvalue}", "minValue": "{tvalue}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"displayColumns": 0, "prompt": "<p>Give the value of the t-statistic you have found, choose the range for the $p$ value by looking up the t-statistic tables:</p>", "matrix": "v", "shuffleChoices": false, "maxMarks": 0, "distractors": ["", "", "", "", ""], "choices": ["<p>$p$ is greater than $10\\%$</p>", "<p>$p$ lies between  $5 \\%$ and $10\\%$</p>", "<p>$p$ lies between  $1 \\%$ and $5\\%$</p>", "<p>$p$ lies between $0.1 \\%$ and $1\\%$</p>", "<p>$p$ is less than $0.1 \\%$</p>"], "displayType": "radiogroup", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "1_n_2", "minMarks": 0}, {"displayColumns": 0, "prompt": "<p>Given the $p$-value and the range you have found, what is the strength of evidence against the null hypothesis that there is a difference in the average responses of the two groups?</p>", "matrix": "v", "shuffleChoices": false, "maxMarks": 0, "distractors": ["", "", "", "", ""], "choices": ["None", "Slight", "Moderate", "Strong", "Very strong"], "displayType": "radiogroup", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "1_n_2", "minMarks": 0}], "statement": "<p>Suppose that 15 individuals, diagnosed with bipolar disorder take part in an experiment that grades their happiness on a scale from 1 to 25. They take the test before treatment and then again after a specific drug has been prescribed. The data is shown below:</p>\n<table border=\"1\">\n<tbody>\n<tr><th>{object}</th><th>A</th><th>B</th><th>C</th><th>D</th><th>E</th><th>F</th><th>G</th><th>H</th><th>I</th><th>J</th><th>K</th><th>L</th><th>M</th><th>N</th><th>O</th></tr>\n<tr><th align=\"left\">Before</th>\n<td>$\\var{r1[0]}$</td>\n<td>$\\var{r1[1]}$</td>\n<td>$\\var{r1[2]}$</td>\n<td>$\\var{r1[3]}$</td>\n<td>$\\var{r1[4]}$</td>\n<td>$\\var{r1[5]}$</td>\n<td>$\\var{r1[6]}$</td>\n<td>$\\var{r1[7]}$</td>\n<td>$\\var{r1[8]}$</td>\n<td>$\\var{r1[9]}$</td>\n<td>$\\var{r1[10]}$</td>\n<td>$\\var{r1[11]}$</td>\n<td>$\\var{r1[12]}$</td>\n<td>$\\var{r1[13]}$</td>\n<td>$\\var{r1[14]}$</td>\n</tr>\n<tr><th align=\"left\">After</th>\n<td>$\\var{r2[0]}$</td>\n<td>$\\var{r2[1]}$</td>\n<td>$\\var{r2[2]}$</td>\n<td>$\\var{r2[3]}$</td>\n<td>$\\var{r2[4]}$</td>\n<td>$\\var{r2[5]}$</td>\n<td>$\\var{r2[6]}$</td>\n<td>$\\var{r2[7]}$</td>\n<td>$\\var{r2[8]}$</td>\n<td>$\\var{r2[9]}$</td>\n<td>$\\var{r2[10]}$</td>\n<td>$\\var{r2[11]}$</td>\n<td>$\\var{r2[12]}$</td>\n<td>$\\var{r2[13]}$</td>\n<td>$\\var{r2[14]}$</td>\n</tr>\n</tbody>\n</table>\n<p><strong>Is there a difference between the average responses of the two groups? </strong></p>\n<p>In order to&nbsp;answer this, complete the following questions.</p>", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"t95": {"definition": "2.145", "templateType": "anything", "group": "Ungrouped variables", "name": "t95", "description": ""}, "tvalue": {"definition": "precround(abs(tscore(0,d)),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "tvalue", "description": ""}, "msg": {"definition": "['is greater than $\\\\var{t999}$','lies between $\\\\var{t99}$ and $\\\\var{t999}$','lies between $\\\\var{t95}$ and $\\\\var{t99}$','lies between $\\\\var{t90}$ and $\\\\var{t95}$','is less than $\\\\var{t90}$']", "templateType": "anything", "group": "Ungrouped variables", "name": "msg", "description": ""}, "t999": {"definition": "4.14", "templateType": "anything", "group": "Ungrouped variables", "name": "t999", "description": ""}, "meandiff": {"definition": "precround(mean(d),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "meandiff", "description": ""}, "object": {"definition": "'Individual'", "templateType": "anything", "group": "Ungrouped variables", "name": "object", "description": ""}, "sig1": {"definition": "random(2..3#0.2)", "templateType": "anything", "group": "Ungrouped variables", "name": "sig1", "description": ""}, "thismany": {"definition": "15", "templateType": "anything", "group": "Ungrouped variables", "name": "thismany", "description": ""}, "stdiff": {"definition": "precround(pstdev(d),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "stdiff", "description": ""}, "sig2": {"definition": "random(2..3#0.2)", "templateType": "anything", "group": "Ungrouped variables", "name": "sig2", "description": ""}, "t99": {"definition": "2.977", "templateType": "anything", "group": "Ungrouped variables", "name": "t99", "description": ""}, "attempt": {"definition": "'hand'", "templateType": "anything", "group": "Ungrouped variables", "name": "attempt", "description": ""}, "t90": {"definition": "1.761", "templateType": "anything", "group": "Ungrouped variables", "name": "t90", "description": ""}, "cmsg": {"definition": "[ \"there is very strong evidence against the null hypothesis\", \"there is strong evidence against the null hypothesis\", \"there is moderate evidence against the null hypothesis\", \"there is slight evidence against the null hypothesis\", \"there is no evidence against the null hypothesis\" ]", "templateType": "list of strings", "group": "Ungrouped variables", "name": "cmsg", "description": ""}, "r1": {"definition": "repeat(min(round(normalsample(mu1,sig1)),25),15)", "templateType": "anything", "group": "Ungrouped variables", "name": "r1", "description": ""}, "mu1": {"definition": "random(10..14#0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "mu1", "description": ""}, "d": {"definition": "list(vector(r2)-vector(r1))", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "r2": {"definition": "repeat(min(round(normalsample(mu2,sig2)),25),15)", "templateType": "anything", "group": "Ungrouped variables", "name": "r2", "description": ""}, "mu2": {"definition": "mu1+random(2..4#0.1)", "templateType": "anything", "group": "Ungrouped variables", "name": "mu2", "description": ""}, "pmsg": {"definition": "[' is less than $0.001$',' lies between $0.001$ and $0.01$',' lies between $0.01$ and $0.05$',' lies between $0.05$ and $0.10$',' is greater than $0.10$']", "templateType": "anything", "group": "Ungrouped variables", "name": "pmsg", "description": ""}, "objects": {"definition": "'individuals'", "templateType": "anything", "group": "Ungrouped variables", "name": "objects", "description": ""}, "t": {"definition": "switch(v[0]=1,0,v[1]=1,1,v[2]=1,2,v[3]=1,3,4)", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}, "v": {"definition": "if(tvalue>=t999,[1,0,0,0,0],if(tvalue>=t99,[0,1,0,0,0],if(tvalue>=t95,[0,0,1,0,0],if(tvalue>=t90,[0,0,0,1,0],[0,0,0,0,1]))))", "templateType": "anything", "group": "Ungrouped variables", "name": "v", "description": ""}, "pvalue": {"definition": "precround(ttest(0,d,2),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "pvalue", "description": ""}}, "metadata": {"notes": "<p><strong>11/07/2012:</strong></p>\n    <p><strong><br /></strong>Added tags.</p>\n    <p>Calculation not yet tested.</p>\n    <p><strong>23/07/2012:</strong></p>\n    <p>Added description.</p>\n    <p>Checked calculation.</p>\n    <p>Changed display slightly in Advice.</p>\n    <p><strong>3/08/2012:</strong></p>\n    <p>Added tags.</p>\n    <p>Question appears to be working correctly.</p>\n    <p><strong>26/01/2013:</strong></p>\n    <p>Advice needs to be finished.</p>", "description": "<p>Paired t-test to see if there is a difference between responses after treatment.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "extensions": ["stats"], "custom_part_types": [], "resources": []}