For a sample of size n from a normal distribution, given mean of the sample mean and the standard deviation , find the t-statistic corresponding to a null hypothesis $\\mu=m$ and a given confidence level. Check if the result is significant at this level.
", "notes": "\n \t\t27/01/2013:
\n \t\tFirst draft completed.
\n \t\t"}, "ungrouped_variables": ["le", "that", "mm", "r11", "tr", "n", "mu", "isthis", "tol", "pert", "test", "st", "sd"], "parts": [{"scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "prompt": "Calculate the t-statistic you will need to test the null hypothesis:
\nTest statistic $t=\\;$?[[0]] (to 3 decimal places).
\n", "gaps": [{"showPrecisionHint": false, "showCorrectAnswer": true, "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "maxValue": "test+tol", "minValue": "test-tol", "scripts": {}, "marks": 1}], "marks": 0}, {"scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "prompt": "
What is the critical value with which to compare $|t|$
\nCritical value $=\\;$[[0]] (to 3 decimal places, the answer should be positive)
", "gaps": [{"showPrecisionHint": false, "showCorrectAnswer": true, "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "maxValue": "st+tol", "minValue": "st-tol", "scripts": {}, "marks": 1}], "marks": 0}, {"scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "prompt": "Is the test significant?
\n[[0]]
", "gaps": [{"distractors": ["", ""], "displayColumns": 0, "showCorrectAnswer": true, "maxMarks": 0, "scripts": {}, "marks": 0, "displayType": "radiogroup", "matrix": "mm", "choices": ["Yes
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\n\\[t = \\frac{\\bar{x}-\\mu}{\\frac{s}{\\sqrt{n}}}\\]
\nand in this case we have:
\n\\[t = \\frac{\\var{r11}-\\var{tr}}{\\frac{\\var{sd}}{\\sqrt{\\var{n}}}}=\\var{test}\\] to 3 decimal places.
\nNow look up in the tables the critical value at $\\var{le}$% for $\\var{n}-1=\\var{n-1}$ degrees of freedom and a two-sided test.
\nNote that as we are using one-sided tables we have to look at the $1-\\var{le}/200=\\var{1-le/200}$ critical value and find the corresponding value.
\nIn this case it is $\\var{st}$ to 3 decimal places.
\nSince $\\var{abs(test)}$ {isthis} $\\var{st}$ we see that the result is {that}.
\n", "statement": "
$X_1,\\;X_2,\\;\\dots, X_n$ is a random sample from $\\operatorname{N}(\\mu,\\sigma^2)$.
\nThe sample size is $n=\\var{n}$, with sample mean $\\bar{x}=\\var{r11}$ and $s=\\var{sd}$, the sample standard deviation.
\nSuppose we wish to test at the $\\var{le}$% level the null hypothesis $H_0: \\mu=\\var{tr}$ versus a two sided alternative hypothesis.
\n", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}