File: maxima_248.html

package info (click to toggle)
maxima 5.47.0-9
links: PTS
area: main
in suites: forky, sid
size: 193,104 kB
sloc: lisp: 434,678; fortran: 14,665; tcl: 10,990; sh: 4,577; makefile: 2,763; ansic: 447; java: 328; python: 262; perl: 201; xml: 60; awk: 28; sed: 15; javascript: 2
file content (1146 lines) | stat: -rw-r--r-- 45,794 bytes
parent folder | download | duplicates (2)
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Created by GNU Texinfo 5.1, http://www.gnu.org/software/texinfo/ -->
<head>
<title>Maxima Manual: Functions and Variables for stats</title>

<meta name="description" content="Maxima Manual: Functions and Variables for stats">
<meta name="keywords" content="Maxima Manual: Functions and Variables for stats">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<link href="maxima_toc.html#Top" rel="start" title="Top">
<link href="maxima_264.html#g_t_0423_043a_0430_0437_0430_0442_0435_043b_044c-_0444_0443_043d_043a_0446_0438_0439-_0438-_043f_0435_0440_0435_043c_0435_043d_043d_044b_0445" rel="index" title="Указатель функций и переменных">
<link href="maxima_toc.html#SEC_Contents" rel="contents" title="Table of Contents">
<link href="maxima_245.html#stats_002dpkg" rel="up" title="stats-pkg">
<link href="maxima_249.html#Functions-and-Variables-for-special-distributions" rel="next" title="Functions and Variables for special distributions">
<link href="maxima_247.html#Functions-and-Variables-for-inference_005fresult" rel="previous" title="Functions and Variables for inference_result">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
blockquote.smallquotation {font-size: smaller}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
div.indentedblock {margin-left: 3.2em}
div.lisp {margin-left: 3.2em}
div.smalldisplay {margin-left: 3.2em}
div.smallexample {margin-left: 3.2em}
div.smallindentedblock {margin-left: 3.2em; font-size: smaller}
div.smalllisp {margin-left: 3.2em}
kbd {font-style:oblique}
pre.display {font-family: inherit}
pre.format {font-family: inherit}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: inherit; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: inherit; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.nocodebreak {white-space:nowrap}
span.nolinebreak {white-space:nowrap}
span.roman {font-family:serif; font-weight:normal}
span.sansserif {font-family:sans-serif; font-weight:normal}
ul.no-bullet {list-style: none}
body {color: black; background: white;  margin-left: 8%; margin-right: 13%;
      font-family: "FreeSans", sans-serif}
h1 {font-size: 150%; font-family: "FreeSans", sans-serif}
h2 {font-size: 125%; font-family: "FreeSans", sans-serif}
h3 {font-size: 100%; font-family: "FreeSans", sans-serif}
a[href] {color: rgb(0,0,255); text-decoration: none;}
a[href]:hover {background: rgb(220,220,220);}
div.textbox {border: solid; border-width: thin; padding-top: 1em;
    padding-bottom: 1em; padding-left: 2em; padding-right: 2em}
div.titlebox {border: none; padding-top: 1em; padding-bottom: 1em;
    padding-left: 2em; padding-right: 2em; background: rgb(200,255,255);
    font-family: sans-serif}
div.synopsisbox {
    border: none; padding-top: 1em; padding-bottom: 1em; padding-left: 2em;
    padding-right: 2em; background: rgb(255,220,255);}
pre.example {border: 1px solid rgb(180,180,180); padding-top: 1em;
    padding-bottom: 1em; padding-left: 1em; padding-right: 1em;
    background-color: rgb(238,238,255)}
div.spacerbox {border: none; padding-top: 2em; padding-bottom: 2em}
div.image {margin: 0; padding: 1em; text-align: center}
div.categorybox {border: 1px solid gray; padding-top: 1em; padding-bottom: 1em;
    padding-left: 1em; padding-right: 1em; background: rgb(247,242,220)}
img {max-width:80%; max-height: 80%; display: block; margin-left: auto; margin-right: auto}

-->
</style>

<link rel="icon" href="figures/favicon.ico">
<script src="https://polyfill.io/v3/polyfill.min.js?features=es6>"></script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
</head>

<body lang="ru" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="Functions-and-Variables-for-stats"></a>
<div class="header">
<p>
Next: <a href="maxima_249.html#Functions-and-Variables-for-special-distributions" accesskey="n" rel="next">Functions and Variables for special distributions</a>, Previous: <a href="maxima_247.html#Functions-and-Variables-for-inference_005fresult" accesskey="p" rel="previous">Functions and Variables for inference_result</a>, Up: <a href="maxima_245.html#stats_002dpkg" accesskey="u" rel="up">stats-pkg</a> &nbsp; [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_264.html#g_t_0423_043a_0430_0437_0430_0442_0435_043b_044c-_0444_0443_043d_043a_0446_0438_0439-_0438-_043f_0435_0440_0435_043c_0435_043d_043d_044b_0445" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Functions-and-Variables-for-stats-1"></a>
<h3 class="section">69.3 Functions and Variables for stats</h3>


<a name="stats_005fnumer"></a><a name="Item_003a-stats_002fdefvr_002fstats_005fnumer"></a><dl>
<dt><a name="index-stats_005fnumer"></a>Option variable: <strong>stats_numer</strong></dt>
<dd><p>Default value: <code>true</code>
</p>
<p>If <code>stats_numer</code> is <code>true</code>, inference statistical functions 
return their results in floating point numbers. If it is <code>false</code>,
results are given in symbolic and rational format.
</p>





</dd></dl>



<a name="test_005fmean"></a><a name="Item_003a-stats_002fdeffn_002ftest_005fmean"></a><dl>
<dt><a name="index-test_005fmean"></a>Function: <strong>test_mean</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>test_mean</tt> (<var>x</var>) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>test_mean</tt> (<var>x</var>, <var>options</var> ...)</em></dt>
<dd>
<p>This is the mean <var>t</var>-test. Argument <var>x</var> is a list or a column matrix
containing an one dimensional sample. It also performs an asymptotic test
based on the <i>Central Limit Theorem</i> if option <code>'asymptotic</code> is
<code>true</code>.
</p>
<p>Options:
</p>
<ul>
<li> <code>'mean</code>, default <code>0</code>, is the mean value to be checked.

</li><li> <code>'alternative</code>, default <code>'twosided</code>, is the alternative hypothesis;
valid values are: <code>'twosided</code>, <code>'greater</code> and <code>'less</code>.

</li><li> <code>'dev</code>, default <code>'unknown</code>, this is the value of the standard deviation when it is 
known; valid values are: <code>'unknown</code> or a positive expression.

</li><li> <code>'conflevel</code>, default <code>95/100</code>, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).

</li><li> <code>'asymptotic</code>, default <code>false</code>, indicates whether it performs an exact <var>t</var>-test or
an asymptotic one based on the <i>Central Limit Theorem</i>;
valid values are <code>true</code> and <code>false</code>.

</li></ul>

<p>The output of function <code>test_mean</code> is an <code>inference_result</code> Maxima object
showing the following results:
</p>
<ol>
<li> <code>'mean_estimate</code>: the sample mean.

</li><li> <code>'conf_level</code>: confidence level selected by the user.

</li><li> <code>'conf_interval</code>: confidence interval for the population mean.

</li><li> <code>'method</code>: inference procedure.

</li><li> <code>'hypotheses</code>: null and alternative hypotheses to be tested.

</li><li> <code>'statistic</code>: value of the sample statistic used for testing the null hypothesis.

</li><li> <code>'distribution</code>: distribution of the sample statistic, together with its parameter(s).

</li><li> <code>'p_value</code>: <em>p</em>-value of the test.

</li></ol>

<p>Examples:
</p>
<p>Performs an exact <var>t</var>-test with unknown variance. The null hypothesis
is <em>H_0: mean=50</em> against the one sided alternative <em>H_1: mean&lt;50</em>;
according to the results, the <em>p</em>-value is too great, there are no
evidence for rejecting <em>H_0</em>.
</p>
<div class="example">
<pre class="example">(%i1) load(&quot;stats&quot;)$
(%i2) data: [78,64,35,45,45,75,43,74,42,42]$
(%i3) test_mean(data,'conflevel=0.9,'alternative='less,'mean=50);
          |                 MEAN TEST
          |
          |            mean_estimate = 54.3
          |
          |              conf_level = 0.9
          |
          | conf_interval = [minf, 61.51314273502712]
          |
(%o3)     |  method = Exact t-test. Unknown variance.
          |
          | hypotheses = H0: mean = 50 , H1: mean &lt; 50
          |
          |       statistic = .8244705235071678
          |
          |       distribution = [student_t, 9]
          |
          |        p_value = .7845100411786889
</pre></div>

<p>This time Maxima performs an asymptotic test, based on the <i>Central Limit Theorem</i>.
The null hypothesis is <em>H_0: equal(mean, 50)</em> against the two sided alternative <em>H_1: not equal(mean, 50)</em>;
according to the results, the <em>p</em>-value is very small, <em>H_0</em> should be rejected in
favor of the alternative <em>H_1</em>. Note that, as indicated by the <code>Method</code> component,
this procedure should be applied to large samples.
</p>
<div class="example">
<pre class="example">(%i1) load(&quot;stats&quot;)$
(%i2) test_mean([36,118,52,87,35,256,56,178,57,57,89,34,25,98,35,
              98,41,45,198,54,79,63,35,45,44,75,42,75,45,45,
              45,51,123,54,151],
              'asymptotic=true,'mean=50);
          |                       MEAN TEST
          |
          |           mean_estimate = 74.88571428571429
          |
          |                   conf_level = 0.95
          |
          | conf_interval = [57.72848600856194, 92.04294256286663]
          |
(%o2)     |    method = Large sample z-test. Unknown variance.
          |
          |       hypotheses = H0: mean = 50 , H1: mean # 50
          |
          |             statistic = 2.842831192874313
          |
          |             distribution = [normal, 0, 1]
          |
          |             p_value = .004471474652002261
</pre></div>





</dd></dl>







<a name="test_005fmeans_005fdifference"></a><a name="Item_003a-stats_002fdeffn_002ftest_005fmeans_005fdifference"></a><dl>
<dt><a name="index-test_005fmeans_005fdifference"></a>Function: <strong>test_means_difference</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>test_means_difference</tt> (<var>x1</var>, <var>x2</var>) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>test_means_difference</tt> (<var>x1</var>, <var>x2</var>, <var>options</var> ...)</em></dt>
<dd>
<p>This is the difference of means <var>t</var>-test for two samples.
Arguments <var>x1</var> and <var>x2</var> are lists or column matrices
containing two independent samples. In case of different unknown variances
(see options <code>'dev1</code>, <code>'dev2</code> and <code>'varequal</code> bellow),
the degrees of freedom are computed by means of the Welch approximation.
It also performs an asymptotic test
based on the <i>Central Limit Theorem</i> if option <code>'asymptotic</code> is
set to <code>true</code>.
</p>
<p>Options:
</p>
<ul>
<li> <code>'alternative</code>, default <code>'twosided</code>, is the alternative hypothesis;
valid values are: <code>'twosided</code>, <code>'greater</code> and <code>'less</code>.

</li><li> <code>'dev1</code>, default <code>'unknown</code>, this is the value of the standard deviation
of the <var>x1</var> sample when it is known; valid values are: <code>'unknown</code> or a positive expression.

</li><li> <code>'dev2</code>, default <code>'unknown</code>, this is the value of the standard deviation
of the <var>x2</var> sample when it is known; valid values are: <code>'unknown</code> or a positive expression.

</li><li> <code>'varequal</code>, default <code>false</code>, whether variances should be considered to be equal or not;
this option takes effect only when <code>'dev1</code> and/or <code>'dev2</code> are  <code>'unknown</code>.

</li><li> <code>'conflevel</code>, default <code>95/100</code>, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).

</li><li> <code>'asymptotic</code>, default <code>false</code>, indicates whether it performs an exact <var>t</var>-test or
an asymptotic one based on the <i>Central Limit Theorem</i>;
valid values are <code>true</code> and <code>false</code>.

</li></ul>

<p>The output of function <code>test_means_difference</code> is an <code>inference_result</code> Maxima object
showing the following results:
</p>
<ol>
<li> <code>'diff_estimate</code>: the difference of means estimate.

</li><li> <code>'conf_level</code>: confidence level selected by the user.

</li><li> <code>'conf_interval</code>: confidence interval for the difference of means.

</li><li> <code>'method</code>: inference procedure.

</li><li> <code>'hypotheses</code>: null and alternative hypotheses to be tested.

</li><li> <code>'statistic</code>: value of the sample statistic used for testing the null hypothesis.

</li><li> <code>'distribution</code>: distribution of the sample statistic, together with its parameter(s).

</li><li> <code>'p_value</code>: <em>p</em>-value of the test.

</li></ol>

<p>Examples:
</p>
<p>The equality of means is tested with two small samples <var>x</var> and <var>y</var>,
against the alternative <em>H_1: m_1&gt;m_2</em>, being <em>m_1</em> and <em>m_2</em>
the populations means; variances are unknown and supposed to be different.
</p>

<div class="example">
<pre class="example">(%i1) load(&quot;stats&quot;)$
(%i2) x: [20.4,62.5,61.3,44.2,11.1,23.7]$
(%i3) y: [1.2,6.9,38.7,20.4,17.2]$
(%i4) test_means_difference(x,y,'alternative='greater);
            |              DIFFERENCE OF MEANS TEST
            |
            |         diff_estimate = 20.31999999999999
            |
            |                 conf_level = 0.95
            |
            |    conf_interval = [- .04597417812882298, inf]
            |
(%o4)       |        method = Exact t-test. Welch approx.
            |
            | hypotheses = H0: mean1 = mean2 , H1: mean1 &gt; mean2
            |
            |           statistic = 1.838004300728477
            |
            |    distribution = [student_t, 8.62758740184604]
            |
            |            p_value = .05032746527991905
</pre></div>

<p>The same test as before, but now variances are supposed to be
equal.
</p>

<div class="example">
<pre class="example">(%i1) load(&quot;stats&quot;)$
(%i2) x: [20.4,62.5,61.3,44.2,11.1,23.7]$
(%i3) y: matrix([1.2],[6.9],[38.7],[20.4],[17.2])$
(%i4) test_means_difference(x,y,'alternative='greater,
                                                 'varequal=true);
            |              DIFFERENCE OF MEANS TEST
            |
            |         diff_estimate = 20.31999999999999
            |
            |                 conf_level = 0.95
            |
            |     conf_interval = [- .7722627696897568, inf]
            |
(%o4)       |   method = Exact t-test. Unknown equal variances
            |
            | hypotheses = H0: mean1 = mean2 , H1: mean1 &gt; mean2
            |
            |           statistic = 1.765996124515009
            |
            |           distribution = [student_t, 9]
            |
            |            p_value = .05560320992529344
</pre></div>





</dd></dl>

<a name="test_005fvariance"></a><a name="Item_003a-stats_002fdeffn_002ftest_005fvariance"></a><dl>
<dt><a name="index-test_005fvariance"></a>Function: <strong>test_variance</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>test_variance</tt> (<var>x</var>) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>test_variance</tt> (<var>x</var>, <var>options</var>, ...)</em></dt>
<dd>
<p>This is the variance <var>chi^2</var>-test. Argument <var>x</var> is a list or a column matrix
containing an one dimensional sample taken from a normal population.
</p>
<p>Options:
</p>
<ul>
<li> <code>'mean</code>, default <code>'unknown</code>, is the population&rsquo;s mean, when it is known.

</li><li> <code>'alternative</code>, default <code>'twosided</code>, is the alternative hypothesis;
valid values are: <code>'twosided</code>, <code>'greater</code> and <code>'less</code>.

</li><li> <code>'variance</code>, default <code>1</code>, this is the variance value (positive) to be checked.

</li><li> <code>'conflevel</code>, default <code>95/100</code>, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).

</li></ul>

<p>The output of function <code>test_variance</code> is an <code>inference_result</code> Maxima object
showing the following results:
</p>
<ol>
<li> <code>'var_estimate</code>: the sample variance.

</li><li> <code>'conf_level</code>: confidence level selected by the user.

</li><li> <code>'conf_interval</code>: confidence interval for the population variance.

</li><li> <code>'method</code>: inference procedure.

</li><li> <code>'hypotheses</code>: null and alternative hypotheses to be tested.

</li><li> <code>'statistic</code>: value of the sample statistic used for testing the null hypothesis.

</li><li> <code>'distribution</code>: distribution of the sample statistic, together with its parameter.

</li><li> <code>'p_value</code>: <em>p</em>-value of the test.

</li></ol>

<p>Examples:
</p>
<p>It is tested whether the variance of a population with unknown mean
is equal to or greater than 200.
</p>
<div class="example">
<pre class="example">(%i1) load(&quot;stats&quot;)$
(%i2) x: [203,229,215,220,223,233,208,228,209]$
(%i3) test_variance(x,'alternative='greater,'variance=200);
             |                  VARIANCE TEST
             |
             |              var_estimate = 110.75
             |
             |                conf_level = 0.95
             |
             |     conf_interval = [57.13433376937479, inf]
             |
(%o3)        | method = Variance Chi-square test. Unknown mean.
             |
             |    hypotheses = H0: var = 200 , H1: var &gt; 200
             |
             |                 statistic = 4.43
             |
             |             distribution = [chi2, 8]
             |
             |           p_value = .8163948512777689
</pre></div>





</dd></dl>







<a name="test_005fvariance_005fratio"></a><a name="Item_003a-stats_002fdeffn_002ftest_005fvariance_005fratio"></a><dl>
<dt><a name="index-test_005fvariance_005fratio"></a>Function: <strong>test_variance_ratio</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>test_variance_ratio</tt> (<var>x1</var>, <var>x2</var>) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>test_variance_ratio</tt> (<var>x1</var>, <var>x2</var>, <var>options</var> ...)</em></dt>
<dd>
<p>This is the variance ratio <var>F</var>-test for two normal populations.
Arguments <var>x1</var> and <var>x2</var> are lists or column matrices
containing two independent samples.
</p>
<p>Options:
</p>
<ul>
<li> <code>'alternative</code>, default <code>'twosided</code>, is the alternative hypothesis;
valid values are: <code>'twosided</code>, <code>'greater</code> and <code>'less</code>.

</li><li> <code>'mean1</code>, default <code>'unknown</code>, when it is known, this is the mean of
the population from which <var>x1</var> was taken.

</li><li> <code>'mean2</code>, default <code>'unknown</code>, when it is known, this is the mean of
the population from which <var>x2</var> was taken.

</li><li> <code>'conflevel</code>, default <code>95/100</code>, confidence level for the confidence interval of the
ratio; it must be an expression which takes a value in (0,1).

</li></ul>

<p>The output of function <code>test_variance_ratio</code> is an <code>inference_result</code> Maxima object
showing the following results:
</p>
<ol>
<li> <code>'ratio_estimate</code>: the sample variance ratio.

</li><li> <code>'conf_level</code>: confidence level selected by the user.

</li><li> <code>'conf_interval</code>: confidence interval for the variance ratio.

</li><li> <code>'method</code>: inference procedure.

</li><li> <code>'hypotheses</code>: null and alternative hypotheses to be tested.

</li><li> <code>'statistic</code>: value of the sample statistic used for testing the null hypothesis.

</li><li> <code>'distribution</code>: distribution of the sample statistic, together with its parameters.

</li><li> <code>'p_value</code>: <em>p</em>-value of the test.

</li></ol>


<p>Examples:
</p>
<p>The equality of the variances of two normal populations is checked
against the alternative that the first is greater than the second.
</p>

<div class="example">
<pre class="example">(%i1) load(&quot;stats&quot;)$
(%i2) x: [20.4,62.5,61.3,44.2,11.1,23.7]$
(%i3) y: [1.2,6.9,38.7,20.4,17.2]$
(%i4) test_variance_ratio(x,y,'alternative='greater);
              |              VARIANCE RATIO TEST
              |
              |       ratio_estimate = 2.316933391522034
              |
              |               conf_level = 0.95
              |
              |    conf_interval = [.3703504689507268, inf]
              |
(%o4)         | method = Variance ratio F-test. Unknown means.
              |
              | hypotheses = H0: var1 = var2 , H1: var1 &gt; var2
              |
              |         statistic = 2.316933391522034
              |
              |            distribution = [f, 5, 4]
              |
              |          p_value = .2179269692254457
</pre></div>





</dd></dl>




<a name="test_005fproportion"></a><a name="Item_003a-stats_002fdeffn_002ftest_005fproportion"></a><dl>
<dt><a name="index-test_005fproportion"></a>Function: <strong>test_proportion</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>test_proportion</tt> (<var>x</var>, <var>n</var>) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>test_proportion</tt> (<var>x</var>, <var>n</var>, <var>options</var> ...)</em></dt>
<dd>
<p>Inferences on a proportion. Argument <var>x</var> is the number of successes
in <var>n</var> trials in a Bernoulli experiment with unknown probability.
</p>
<p>Options:
</p>
<ul>
<li> <code>'proportion</code>, default <code>1/2</code>, is the value of the proportion to be checked.

</li><li> <code>'alternative</code>, default <code>'twosided</code>, is the alternative hypothesis;
valid values are: <code>'twosided</code>, <code>'greater</code> and <code>'less</code>.

</li><li> <code>'conflevel</code>, default <code>95/100</code>, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).

</li><li> <code>'asymptotic</code>, default <code>false</code>, indicates whether it performs an exact test
based on the binomial distribution, or an asymptotic one based on the <i>Central Limit Theorem</i>;
valid values are <code>true</code> and <code>false</code>.

</li><li> <code>'correct</code>, default <code>true</code>, indicates whether Yates correction is applied or not.

</li></ul>

<p>The output of function <code>test_proportion</code> is an <code>inference_result</code> Maxima object
showing the following results:
</p>
<ol>
<li> <code>'sample_proportion</code>: the sample proportion.

</li><li> <code>'conf_level</code>: confidence level selected by the user.

</li><li> <code>'conf_interval</code>: Wilson confidence interval for the proportion.

</li><li> <code>'method</code>: inference procedure.

</li><li> <code>'hypotheses</code>: null and alternative hypotheses to be tested.

</li><li> <code>'statistic</code>: value of the sample statistic used for testing the null hypothesis.

</li><li> <code>'distribution</code>: distribution of the sample statistic, together with its parameters.

</li><li> <code>'p_value</code>: <em>p</em>-value of the test.

</li></ol>

<p>Examples:
</p>
<p>Performs an exact test. The null hypothesis
is <em>H_0: p=1/2</em> against the one sided alternative <em>H_1: p&lt;1/2</em>.
</p>
<div class="example">
<pre class="example">(%i1) load(&quot;stats&quot;)$
(%i2) test_proportion(45, 103, alternative = less);
         |            PROPORTION TEST              
         |                                         
         | sample_proportion = .4368932038834951   
         |                                         
         |           conf_level = 0.95             
         |                                         
         | conf_interval = [0, 0.522714149150231]  
         |                                         
(%o2)    |     method = Exact binomial test.       
         |                                         
         | hypotheses = H0: p = 0.5 , H1: p &lt; 0.5  
         |                                         
         |             statistic = 45              
         |                                         
         |  distribution = [binomial, 103, 0.5]    
         |                                         
         |      p_value = .1184509388901454 
</pre></div>

<p>A two sided asymptotic test. Confidence level is 99/100.
</p>
<div class="example">
<pre class="example">(%i1) load(&quot;stats&quot;)$
(%i2) fpprintprec:7$
(%i3) test_proportion(45, 103, 
                  conflevel = 99/100, asymptotic=true);
      |                 PROPORTION TEST                  
      |                                                  
      |           sample_proportion = .43689             
      |                                                  
      |                conf_level = 0.99                 
      |                                                  
      |        conf_interval = [.31422, .56749]          
      |                                                  
(%o3) | method = Asympthotic test with Yates correction.
      |                                                  
      |     hypotheses = H0: p = 0.5 , H1: p # 0.5       
      |                                                  
      |               statistic = .43689                 
      |                                                  
      |      distribution = [normal, 0.5, .048872]       
      |                                                  
      |                p_value = .19662
</pre></div>





</dd></dl>





<a name="test_005fproportions_005fdifference"></a><a name="Item_003a-stats_002fdeffn_002ftest_005fproportions_005fdifference"></a><dl>
<dt><a name="index-test_005fproportions_005fdifference"></a>Function: <strong>test_proportions_difference</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>test_proportions_difference</tt> (<var>x1</var>, <var>n1</var>, <var>x2</var>, <var>n2</var>) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>test_proportions_difference</tt> (<var>x1</var>, <var>n1</var>, <var>x2</var>, <var>n2</var>, <var>options</var> &hellip;)</em></dt>
<dd>
<p>Inferences on the difference of two proportions. Argument <var>x1</var> is the number of successes
in <var>n1</var> trials in a Bernoulli experiment in the first population, and <var>x2</var> and <var>n2</var>
are the corresponding values in the second population. Samples are independent and the test
is asymptotic.
</p>
<p>Options:
</p>
<ul>
<li> <code>'alternative</code>, default <code>'twosided</code>, is the alternative hypothesis;
valid values are: <code>'twosided</code> (<code>p1 # p2</code>), <code>'greater</code> (<code>p1 &gt; p2</code>)
and <code>'less</code> (<code>p1 &lt; p2</code>).

</li><li> <code>'conflevel</code>, default <code>95/100</code>, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).

</li><li> <code>'correct</code>, default <code>true</code>, indicates whether Yates correction is applied or not.

</li></ul>

<p>The output of function <code>test_proportions_difference</code> is an <code>inference_result</code> Maxima object
showing the following results:
</p>
<ol>
<li> <code>'proportions</code>: list with the two sample proportions.

</li><li> <code>'conf_level</code>: confidence level selected by the user.

</li><li> <code>'conf_interval</code>: Confidence interval for the difference of proportions <code>p1 - p2</code>.

</li><li> <code>'method</code>: inference procedure and warning message in case of any of the samples sizes
is less than 10.

</li><li> <code>'hypotheses</code>: null and alternative hypotheses to be tested.

</li><li> <code>'statistic</code>: value of the sample statistic used for testing the null hypothesis.

</li><li> <code>'distribution</code>: distribution of the sample statistic, together with its parameters.

</li><li> <code>'p_value</code>: <em>p</em>-value of the test.

</li></ol>

<p>Examples:
</p>
<p>A machine produced 10 defective articles in a batch of 250.
After some maintenance work, it produces 4 defective in a batch of 150.
In order to know if the machine has improved, we test the null
hypothesis <code>H0:p1=p2</code>, against the alternative <code>H0:p1&gt;p2</code>,
where <code>p1</code> and <code>p2</code> are the probabilities for one produced
article to be defective before and after maintenance. According to
the p value, there is not enough evidence to accept the alternative.
</p>
<div class="example">
<pre class="example">(%i1) load(&quot;stats&quot;)$
(%i2) fpprintprec:7$
(%i3) test_proportions_difference(10, 250, 4, 150,
                                alternative = greater);
      |       DIFFERENCE OF PROPORTIONS TEST         
      |                                              
      |       proportions = [0.04, .02666667]        
      |                                              
      |              conf_level = 0.95               
      |                                              
      |      conf_interval = [- .02172761, 1]        
      |                                              
(%o3) | method = Asymptotic test. Yates correction.  
      |                                              
      |   hypotheses = H0: p1 = p2 , H1: p1 &gt; p2     
      |                                              
      |            statistic = .01333333             
      |                                              
      |    distribution = [normal, 0, .01898069]     
      |                                              
      |             p_value = .2411936 
</pre></div>

<p>Exact standard deviation of the asymptotic normal
distribution when the data are unknown.
</p>
<div class="example">
<pre class="example">(%i1) load(&quot;stats&quot;)$
(%i2) stats_numer: false$
(%i3) sol: test_proportions_difference(x1,n1,x2,n2)$
(%i4) last(take_inference('distribution,sol));
               1    1                  x2 + x1
              (-- + --) (x2 + x1) (1 - -------)
               n2   n1                 n2 + n1
(%o4)    sqrt(---------------------------------)
                           n2 + n1
</pre></div>





</dd></dl>

<a name="test_005fsign"></a><a name="Item_003a-stats_002fdeffn_002ftest_005fsign"></a><dl>
<dt><a name="index-test_005fsign"></a>Function: <strong>test_sign</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>test_sign</tt> (<var>x</var>) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>test_sign</tt> (<var>x</var>, <var>options</var> &hellip;)</em></dt>
<dd>
<p>This is the non parametric sign test for the median of a continuous population.
Argument <var>x</var> is a list or a column matrix containing an one dimensional sample.
</p>
<p>Options:
</p>
<ul>
<li> <code>'alternative</code>, default <code>'twosided</code>, is the alternative hypothesis;
valid values are: <code>'twosided</code>, <code>'greater</code> and <code>'less</code>.

</li><li> <code>'median</code>, default <code>0</code>, is the median value to be checked.

</li></ul>

<p>The output of function <code>test_sign</code> is an <code>inference_result</code> Maxima object
showing the following results:
</p>
<ol>
<li> <code>'med_estimate</code>: the sample median.

</li><li> <code>'method</code>: inference procedure.

</li><li> <code>'hypotheses</code>: null and alternative hypotheses to be tested.

</li><li> <code>'statistic</code>: value of the sample statistic used for testing the null hypothesis.

</li><li> <code>'distribution</code>: distribution of the sample statistic, together with its parameter(s).

</li><li> <code>'p_value</code>: <em>p</em>-value of the test.

</li></ol>

<p>Examples:
</p>
<p>Checks whether the population from which the sample was taken has median 6, 
against the alternative <em>H_1: median &gt; 6</em>.
</p>
<div class="example">
<pre class="example">(%i1) load(&quot;stats&quot;)$
(%i2) x: [2,0.1,7,1.8,4,2.3,5.6,7.4,5.1,6.1,6]$
(%i3) test_sign(x,'median=6,'alternative='greater);
               |                  SIGN TEST
               |
               |              med_estimate = 5.1
               |
               |      method = Non parametric sign test.
               |
(%o3)          | hypotheses = H0: median = 6 , H1: median &gt; 6
               |
               |                statistic = 7
               |
               |      distribution = [binomial, 10, 0.5]
               |
               |         p_value = .05468749999999989
</pre></div>





</dd></dl>

<a name="test_005fsigned_005frank"></a><a name="Item_003a-stats_002fdeffn_002ftest_005fsigned_005frank"></a><dl>
<dt><a name="index-test_005fsigned_005frank"></a>Function: <strong>test_signed_rank</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>test_signed_rank</tt> (<var>x</var>) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>test_signed_rank</tt> (<var>x</var>, <var>options</var> &hellip;)</em></dt>
<dd>
<p>This is the Wilcoxon signed rank test to make inferences about the median of a
continuous population. Argument <var>x</var> is a list or a column matrix
containing an one dimensional sample. Performs normal approximation if the
sample size is greater than 20, or if there are zeroes or ties.
</p>
<p>See also <code>pdf_rank_test</code> and <code>cdf_rank_test</code>
</p>
<p>Options:
</p>
<ul>
<li> <code>'median</code>, default <code>0</code>, is the median value to be checked.

</li><li> <code>'alternative</code>, default <code>'twosided</code>, is the alternative hypothesis;
valid values are: <code>'twosided</code>, <code>'greater</code> and <code>'less</code>.

</li></ul>

<p>The output of function <code>test_signed_rank</code> is an <code>inference_result</code> Maxima object
with the following results:
</p>
<ol>
<li> <code>'med_estimate</code>: the sample median.

</li><li> <code>'method</code>: inference procedure.

</li><li> <code>'hypotheses</code>: null and alternative hypotheses to be tested.

</li><li> <code>'statistic</code>: value of the sample statistic used for testing the null hypothesis.

</li><li> <code>'distribution</code>: distribution of the sample statistic, together with its parameter(s).

</li><li> <code>'p_value</code>: <em>p</em>-value of the test.

</li></ol>

<p>Examples:
</p>
<p>Checks the null hypothesis <em>H_0: median = 15</em> against the 
alternative <em>H_1: median &gt; 15</em>. This is an exact test, since
there are no ties.
</p>

<div class="example">
<pre class="example">(%i1) load(&quot;stats&quot;)$
(%i2) x: [17.1,15.9,13.7,13.4,15.5,17.6]$
(%i3) test_signed_rank(x,median=15,alternative=greater);
                 |             SIGNED RANK TEST
                 |
                 |           med_estimate = 15.7
                 |
                 |           method = Exact test
                 |
(%o3)            | hypotheses = H0: med = 15 , H1: med &gt; 15
                 |
                 |              statistic = 14
                 |
                 |     distribution = [signed_rank, 6]
                 |
                 |            p_value = 0.28125
</pre></div>

<p>Checks the null hypothesis <em>H_0: equal(median, 2.5)</em> against the 
alternative <em>H_1: not equal(median, 2.5)</em>. This is an approximated test,
since there are ties.
</p>

<div class="example">
<pre class="example">(%i1) load(&quot;stats&quot;)$
(%i2) y:[1.9,2.3,2.6,1.9,1.6,3.3,4.2,4,2.4,2.9,1.5,3,2.9,4.2,3.1]$
(%i3) test_signed_rank(y,median=2.5);
             |                 SIGNED RANK TEST
             |
             |                med_estimate = 2.9
             |
             |          method = Asymptotic test. Ties
             |
(%o3)        |    hypotheses = H0: med = 2.5 , H1: med # 2.5
             |
             |                 statistic = 76.5
             |
             | distribution = [normal, 60.5, 17.58195097251724]
             |
             |           p_value = .3628097734643669
</pre></div>





</dd></dl>

<a name="test_005frank_005fsum"></a><a name="Item_003a-stats_002fdeffn_002ftest_005frank_005fsum"></a><dl>
<dt><a name="index-test_005frank_005fsum"></a>Function: <strong>test_rank_sum</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>test_rank_sum</tt> (<var>x1</var>, <var>x2</var>) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>test_rank_sum</tt> (<var>x1</var>, <var>x2</var>, <var>option</var>)</em></dt>
<dd>
<p>This is the Wilcoxon-Mann-Whitney test for comparing the medians of two
continuous populations. The first two arguments <var>x1</var> and <var>x2</var> are lists
or column matrices with the data of two independent samples. Performs normal
approximation if any of the sample sizes is greater than 10, or if there are ties.
</p>
<p>Option:
</p>
<ul>
<li> <code>'alternative</code>, default <code>'twosided</code>, is the alternative hypothesis;
valid values are: <code>'twosided</code>, <code>'greater</code> and <code>'less</code>.

</li></ul>

<p>The output of function <code>test_rank_sum</code> is an <code>inference_result</code> Maxima object
with the following results:
</p>
<ol>
<li> <code>'method</code>: inference procedure.

</li><li> <code>'hypotheses</code>: null and alternative hypotheses to be tested.

</li><li> <code>'statistic</code>: value of the sample statistic used for testing the null hypothesis.

</li><li> <code>'distribution</code>: distribution of the sample statistic, together with its parameters.

</li><li> <code>'p_value</code>: <em>p</em>-value of the test.

</li></ol>

<p>Examples:
</p>
<p>Checks whether populations have similar medians. Samples sizes
are small and an exact test is made.
</p>

<div class="example">
<pre class="example">(%i1) load(&quot;stats&quot;)$
(%i2) x:[12,15,17,38,42,10,23,35,28]$
(%i3) y:[21,18,25,14,52,65,40,43]$
(%i4) test_rank_sum(x,y);
              |                 RANK SUM TEST
              |
              |              method = Exact test
              |
              | hypotheses = H0: med1 = med2 , H1: med1 # med2
(%o4)         |
              |                 statistic = 22
              |
              |        distribution = [rank_sum, 9, 8]
              |
              |          p_value = .1995886466474702
</pre></div>

<p>Now, with greater samples and ties, the procedure makes 
normal approximation. The alternative hypothesis is
<em>H_1: median1 &lt; median2</em>.
</p>

<div class="example">
<pre class="example">(%i1) load(&quot;stats&quot;)$
(%i2) x: [39,42,35,13,10,23,15,20,17,27]$
(%i3) y: [20,52,66,19,41,32,44,25,14,39,43,35,19,56,27,15]$
(%i4) test_rank_sum(x,y,'alternative='less);
             |                  RANK SUM TEST
             |
             |          method = Asymptotic test. Ties
             |
             |  hypotheses = H0: med1 = med2 , H1: med1 &lt; med2
(%o4)        |
             |                 statistic = 48.5
             |
             | distribution = [normal, 79.5, 18.95419580097078]
             |
             |           p_value = .05096985666598441
</pre></div>





</dd></dl>







<a name="test_005fnormality"></a><a name="Item_003a-stats_002fdeffn_002ftest_005fnormality"></a><dl>
<dt><a name="index-test_005fnormality"></a>Function: <strong>test_normality</strong> <em>(<var>x</var>)</em></dt>
<dd>
<p>Shapiro-Wilk test for normality. Argument <var>x</var> is a list of numbers, and sample
size must be greater than 2 and less or equal than 5000, otherwise, function
<code>test_normality</code> signals an error message.
</p>
<p>Reference:
</p>
<p>[1] Algorithm AS R94, Applied Statistics (1995), vol.44, no.4, 547-551
</p>
<p>The output of function <code>test_normality</code> is an <code>inference_result</code> Maxima object
with the following results:
</p>
<ol>
<li> <code>'statistic</code>: value of the <var>W</var> statistic.

</li><li> <code>'p_value</code>: <em>p</em>-value under normal assumption.

</li></ol>

<p>Examples:
</p>
<p>Checks for the normality of a population, based on a sample of size 9.
</p>

<div class="example">
<pre class="example">(%i1) load(&quot;stats&quot;)$
(%i2) x:[12,15,17,38,42,10,23,35,28]$
(%i3) test_normality(x);
                       |      SHAPIRO - WILK TEST
                       |
(%o3)                  | statistic = .9251055695162436
                       |
                       |  p_value = .4361763918860381
</pre></div>





</dd></dl>

<a name="linear_005fregression"></a><a name="Item_003a-stats_002fdeffn_002flinear_005fregression"></a><dl>
<dt><a name="index-linear_005fregression"></a>Function: <strong>linear_regression</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>linear_regression</tt> (<var>x</var>) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>linear_regression</tt> (<var>x</var> <var>option</var>)</em></dt>
<dd>
<p>Multivariate linear regression, 
<em>y_i = b0 + b1*x_1i + b2*x_2i + ... + bk*x_ki + u_i</em>,
where <em>u_i</em> are <em>N(0,sigma)</em> independent random variables.
Argument <var>x</var> must be a matrix with more than one column. The
last column is considered as the responses (<em>y_i</em>).
</p>
<p>Option:
</p>
<ul>
<li> <code>'conflevel</code>, default <code>95/100</code>, confidence level for the
confidence intervals; it must be an expression which takes a value
in (0,1).
</li></ul>

<p>The output of function <code>linear_regression</code> is an 
<code>inference_result</code> Maxima object with the following results:
</p>
<ol>
<li> <code>'b_estimation</code>: regression coefficients estimates.

</li><li> <code>'b_covariances</code>: covariance matrix of the regression 
coefficients estimates.

</li><li> <code>b_conf_int</code>: confidence intervals of the regression coefficients.

</li><li> <code>b_statistics</code>: statistics for testing coefficient.

</li><li> <code>b_p_values</code>: p-values for coefficient tests.

</li><li> <code>b_distribution</code>: probability distribution for coefficient tests.

</li><li> <code>v_estimation</code>: unbiased variance estimator.

</li><li> <code>v_conf_int</code>: variance confidence interval.

</li><li> <code>v_distribution</code>: probability distribution for variance test.

</li><li> <code>residuals</code>: residuals.

</li><li> <code>adc</code>: adjusted determination coefficient.

</li><li> <code>aic</code>: Akaike&rsquo;s information criterion.

</li><li> <code>bic</code>: Bayes&rsquo;s information criterion.

</li></ol>

<p>Only items 1, 4, 5, 6, 7, 8, 9 and 11 above, in this order, 
are shown by default. The rest remain hidden until the user
makes use of functions <code>items_inference</code> and <code>take_inference</code>.
</p>
<p>Example:
</p>
<p>Fitting a linear model to a trivariate sample. The
last column is considered as the responses (<em>y_i</em>).
</p>
<div class="example">
<pre class="example">(%i2) load(&quot;stats&quot;)$
(%i3) X:matrix(
    [58,111,64],[84,131,78],[78,158,83],
    [81,147,88],[82,121,89],[102,165,99],
    [85,174,101],[102,169,102])$
(%i4) fpprintprec: 4$
(%i5) res: linear_regression(X);
             |       LINEAR REGRESSION MODEL         
             |                                       
             | b_estimation = [9.054, .5203, .2397]  
             |                                       
             | b_statistics = [.6051, 2.246, 1.74]   
             |                                       
             | b_p_values = [.5715, .07466, .1423]   
             |                                       
(%o5)        |   b_distribution = [student_t, 5]     
             |                                       
             |         v_estimation = 35.27          
             |                                       
             |     v_conf_int = [13.74, 212.2]       
             |                                       
             |      v_distribution = [chi2, 5]       
             |                                       
             |             adc = .7922               
(%i6) items_inference(res);
(%o6) [b_estimation, b_covariances, b_conf_int, b_statistics, 
b_p_values, b_distribution, v_estimation, v_conf_int, 
v_distribution, residuals, adc, aic, bic]
(%i7) take_inference('b_covariances, res);
                  [  223.9    - 1.12   - .8532  ]
                  [                             ]
(%o7)             [ - 1.12    .05367   - .02305 ]
                  [                             ]
                  [ - .8532  - .02305   .01898  ]
(%i8) take_inference('bic, res);
(%o8)                          30.98
(%i9) load(&quot;draw&quot;)$
(%i10) draw2d(
    points_joined = true,
    grid = true,
    points(take_inference('residuals, res)) )$
</pre></div>






</dd></dl>






<a name="Item_003a-stats_002fnode_002fFunctions-and-Variables-for-special-distributions"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_249.html#Functions-and-Variables-for-special-distributions" accesskey="n" rel="next">Functions and Variables for special distributions</a>, Previous: <a href="maxima_247.html#Functions-and-Variables-for-inference_005fresult" accesskey="p" rel="previous">Functions and Variables for inference_result</a>, Up: <a href="maxima_245.html#stats_002dpkg" accesskey="u" rel="up">stats-pkg</a> &nbsp; [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_264.html#g_t_0423_043a_0430_0437_0430_0442_0435_043b_044c-_0444_0443_043d_043a_0446_0438_0439-_0438-_043f_0435_0440_0435_043c_0435_043d_043d_044b_0445" title="Index" rel="index">Index</a>]</p>
</div>



</body>
</html>