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<HTML> <BODY BGCOLOR="white"> <PRE> <FONT color="green">001</FONT> /*<a name="line.1"></a> <FONT color="green">002</FONT> * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a> <FONT color="green">003</FONT> * contributor license agreements. See the NOTICE file distributed with<a name="line.3"></a> <FONT color="green">004</FONT> * this work for additional information regarding copyright ownership.<a name="line.4"></a> <FONT color="green">005</FONT> * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a> <FONT color="green">006</FONT> * (the "License"); you may not use this file except in compliance with<a name="line.6"></a> <FONT color="green">007</FONT> * the License. You may obtain a copy of the License at<a name="line.7"></a> <FONT color="green">008</FONT> *<a name="line.8"></a> <FONT color="green">009</FONT> * http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a> <FONT color="green">010</FONT> *<a name="line.10"></a> <FONT color="green">011</FONT> * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a> <FONT color="green">012</FONT> * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a> <FONT color="green">013</FONT> * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a> <FONT color="green">014</FONT> * See the License for the specific language governing permissions and<a name="line.14"></a> <FONT color="green">015</FONT> * limitations under the License.<a name="line.15"></a> <FONT color="green">016</FONT> */<a name="line.16"></a> <FONT color="green">017</FONT> package org.apache.commons.math.stat.inference;<a name="line.17"></a> <FONT color="green">018</FONT> <a name="line.18"></a> <FONT color="green">019</FONT> import org.apache.commons.math.MathException;<a name="line.19"></a> <FONT color="green">020</FONT> import org.apache.commons.math.stat.descriptive.StatisticalSummary;<a name="line.20"></a> <FONT color="green">021</FONT> <a name="line.21"></a> <FONT color="green">022</FONT> /**<a name="line.22"></a> <FONT color="green">023</FONT> * An interface for Student's t-tests.<a name="line.23"></a> <FONT color="green">024</FONT> * <p><a name="line.24"></a> <FONT color="green">025</FONT> * Tests can be:<ul><a name="line.25"></a> <FONT color="green">026</FONT> * <li>One-sample or two-sample</li><a name="line.26"></a> <FONT color="green">027</FONT> * <li>One-sided or two-sided</li><a name="line.27"></a> <FONT color="green">028</FONT> * <li>Paired or unpaired (for two-sample tests)</li><a name="line.28"></a> <FONT color="green">029</FONT> * <li>Homoscedastic (equal variance assumption) or heteroscedastic<a name="line.29"></a> <FONT color="green">030</FONT> * (for two sample tests)</li><a name="line.30"></a> <FONT color="green">031</FONT> * <li>Fixed significance level (boolean-valued) or returning p-values.<a name="line.31"></a> <FONT color="green">032</FONT> * </li></ul></p><a name="line.32"></a> <FONT color="green">033</FONT> * <p><a name="line.33"></a> <FONT color="green">034</FONT> * Test statistics are available for all tests. Methods including "Test" in<a name="line.34"></a> <FONT color="green">035</FONT> * in their names perform tests, all other methods return t-statistics. Among<a name="line.35"></a> <FONT color="green">036</FONT> * the "Test" methods, <code>double-</code>valued methods return p-values;<a name="line.36"></a> <FONT color="green">037</FONT> * <code>boolean-</code>valued methods perform fixed significance level tests.<a name="line.37"></a> <FONT color="green">038</FONT> * Significance levels are always specified as numbers between 0 and 0.5<a name="line.38"></a> <FONT color="green">039</FONT> * (e.g. tests at the 95% level use <code>alpha=0.05</code>).</p><a name="line.39"></a> <FONT color="green">040</FONT> * <p><a name="line.40"></a> <FONT color="green">041</FONT> * Input to tests can be either <code>double[]</code> arrays or<a name="line.41"></a> <FONT color="green">042</FONT> * {@link StatisticalSummary} instances.</p><a name="line.42"></a> <FONT color="green">043</FONT> *<a name="line.43"></a> <FONT color="green">044</FONT> *<a name="line.44"></a> <FONT color="green">045</FONT> * @version $Revision: 811786 $ $Date: 2009-09-06 05:36:08 -0400 (Sun, 06 Sep 2009) $<a name="line.45"></a> <FONT color="green">046</FONT> */<a name="line.46"></a> <FONT color="green">047</FONT> public interface TTest {<a name="line.47"></a> <FONT color="green">048</FONT> /**<a name="line.48"></a> <FONT color="green">049</FONT> * Computes a paired, 2-sample t-statistic based on the data in the input<a name="line.49"></a> <FONT color="green">050</FONT> * arrays. The t-statistic returned is equivalent to what would be returned by<a name="line.50"></a> <FONT color="green">051</FONT> * computing the one-sample t-statistic {@link #t(double, double[])}, with<a name="line.51"></a> <FONT color="green">052</FONT> * <code>mu = 0</code> and the sample array consisting of the (signed)<a name="line.52"></a> <FONT color="green">053</FONT> * differences between corresponding entries in <code>sample1</code> and<a name="line.53"></a> <FONT color="green">054</FONT> * <code>sample2.</code><a name="line.54"></a> <FONT color="green">055</FONT> * <p><a name="line.55"></a> <FONT color="green">056</FONT> * <strong>Preconditions</strong>: <ul><a name="line.56"></a> <FONT color="green">057</FONT> * <li>The input arrays must have the same length and their common length<a name="line.57"></a> <FONT color="green">058</FONT> * must be at least 2.<a name="line.58"></a> <FONT color="green">059</FONT> * </li></ul></p><a name="line.59"></a> <FONT color="green">060</FONT> *<a name="line.60"></a> <FONT color="green">061</FONT> * @param sample1 array of sample data values<a name="line.61"></a> <FONT color="green">062</FONT> * @param sample2 array of sample data values<a name="line.62"></a> <FONT color="green">063</FONT> * @return t statistic<a name="line.63"></a> <FONT color="green">064</FONT> * @throws IllegalArgumentException if the precondition is not met<a name="line.64"></a> <FONT color="green">065</FONT> * @throws MathException if the statistic can not be computed do to a<a name="line.65"></a> <FONT color="green">066</FONT> * convergence or other numerical error.<a name="line.66"></a> <FONT color="green">067</FONT> */<a name="line.67"></a> <FONT color="green">068</FONT> double pairedT(double[] sample1, double[] sample2)<a name="line.68"></a> <FONT color="green">069</FONT> throws IllegalArgumentException, MathException;<a name="line.69"></a> <FONT color="green">070</FONT> /**<a name="line.70"></a> <FONT color="green">071</FONT> * Returns the <i>observed significance level</i>, or<a name="line.71"></a> <FONT color="green">072</FONT> * <i> p-value</i>, associated with a paired, two-sample, two-tailed t-test<a name="line.72"></a> <FONT color="green">073</FONT> * based on the data in the input arrays.<a name="line.73"></a> <FONT color="green">074</FONT> * <p><a name="line.74"></a> <FONT color="green">075</FONT> * The number returned is the smallest significance level<a name="line.75"></a> <FONT color="green">076</FONT> * at which one can reject the null hypothesis that the mean of the paired<a name="line.76"></a> <FONT color="green">077</FONT> * differences is 0 in favor of the two-sided alternative that the mean paired<a name="line.77"></a> <FONT color="green">078</FONT> * difference is not equal to 0. For a one-sided test, divide the returned<a name="line.78"></a> <FONT color="green">079</FONT> * value by 2.</p><a name="line.79"></a> <FONT color="green">080</FONT> * <p><a name="line.80"></a> <FONT color="green">081</FONT> * This test is equivalent to a one-sample t-test computed using<a name="line.81"></a> <FONT color="green">082</FONT> * {@link #tTest(double, double[])} with <code>mu = 0</code> and the sample<a name="line.82"></a> <FONT color="green">083</FONT> * array consisting of the signed differences between corresponding elements of<a name="line.83"></a> <FONT color="green">084</FONT> * <code>sample1</code> and <code>sample2.</code></p><a name="line.84"></a> <FONT color="green">085</FONT> * <p><a name="line.85"></a> <FONT color="green">086</FONT> * <strong>Usage Note:</strong><br><a name="line.86"></a> <FONT color="green">087</FONT> * The validity of the p-value depends on the assumptions of the parametric<a name="line.87"></a> <FONT color="green">088</FONT> * t-test procedure, as discussed<a name="line.88"></a> <FONT color="green">089</FONT> * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"><a name="line.89"></a> <FONT color="green">090</FONT> * here</a></p><a name="line.90"></a> <FONT color="green">091</FONT> * <p><a name="line.91"></a> <FONT color="green">092</FONT> * <strong>Preconditions</strong>: <ul><a name="line.92"></a> <FONT color="green">093</FONT> * <li>The input array lengths must be the same and their common length must<a name="line.93"></a> <FONT color="green">094</FONT> * be at least 2.<a name="line.94"></a> <FONT color="green">095</FONT> * </li></ul></p><a name="line.95"></a> <FONT color="green">096</FONT> *<a name="line.96"></a> <FONT color="green">097</FONT> * @param sample1 array of sample data values<a name="line.97"></a> <FONT color="green">098</FONT> * @param sample2 array of sample data values<a name="line.98"></a> <FONT color="green">099</FONT> * @return p-value for t-test<a name="line.99"></a> <FONT color="green">100</FONT> * @throws IllegalArgumentException if the precondition is not met<a name="line.100"></a> <FONT color="green">101</FONT> * @throws MathException if an error occurs computing the p-value<a name="line.101"></a> <FONT color="green">102</FONT> */<a name="line.102"></a> <FONT color="green">103</FONT> double pairedTTest(double[] sample1, double[] sample2)<a name="line.103"></a> <FONT color="green">104</FONT> throws IllegalArgumentException, MathException;<a name="line.104"></a> <FONT color="green">105</FONT> /**<a name="line.105"></a> <FONT color="green">106</FONT> * Performs a paired t-test evaluating the null hypothesis that the<a name="line.106"></a> <FONT color="green">107</FONT> * mean of the paired differences between <code>sample1</code> and<a name="line.107"></a> <FONT color="green">108</FONT> * <code>sample2</code> is 0 in favor of the two-sided alternative that the<a name="line.108"></a> <FONT color="green">109</FONT> * mean paired difference is not equal to 0, with significance level<a name="line.109"></a> <FONT color="green">110</FONT> * <code>alpha</code>.<a name="line.110"></a> <FONT color="green">111</FONT> * <p><a name="line.111"></a> <FONT color="green">112</FONT> * Returns <code>true</code> iff the null hypothesis can be rejected with<a name="line.112"></a> <FONT color="green">113</FONT> * confidence <code>1 - alpha</code>. To perform a 1-sided test, use<a name="line.113"></a> <FONT color="green">114</FONT> * <code>alpha * 2</code></p><a name="line.114"></a> <FONT color="green">115</FONT> * <p><a name="line.115"></a> <FONT color="green">116</FONT> * <strong>Usage Note:</strong><br><a name="line.116"></a> <FONT color="green">117</FONT> * The validity of the test depends on the assumptions of the parametric<a name="line.117"></a> <FONT color="green">118</FONT> * t-test procedure, as discussed<a name="line.118"></a> <FONT color="green">119</FONT> * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"><a name="line.119"></a> <FONT color="green">120</FONT> * here</a></p><a name="line.120"></a> <FONT color="green">121</FONT> * <p><a name="line.121"></a> <FONT color="green">122</FONT> * <strong>Preconditions</strong>: <ul><a name="line.122"></a> <FONT color="green">123</FONT> * <li>The input array lengths must be the same and their common length<a name="line.123"></a> <FONT color="green">124</FONT> * must be at least 2.<a name="line.124"></a> <FONT color="green">125</FONT> * </li><a name="line.125"></a> <FONT color="green">126</FONT> * <li> <code> 0 < alpha < 0.5 </code><a name="line.126"></a> <FONT color="green">127</FONT> * </li></ul></p><a name="line.127"></a> <FONT color="green">128</FONT> *<a name="line.128"></a> <FONT color="green">129</FONT> * @param sample1 array of sample data values<a name="line.129"></a> <FONT color="green">130</FONT> * @param sample2 array of sample data values<a name="line.130"></a> <FONT color="green">131</FONT> * @param alpha significance level of the test<a name="line.131"></a> <FONT color="green">132</FONT> * @return true if the null hypothesis can be rejected with<a name="line.132"></a> <FONT color="green">133</FONT> * confidence 1 - alpha<a name="line.133"></a> <FONT color="green">134</FONT> * @throws IllegalArgumentException if the preconditions are not met<a name="line.134"></a> <FONT color="green">135</FONT> * @throws MathException if an error occurs performing the test<a name="line.135"></a> <FONT color="green">136</FONT> */<a name="line.136"></a> <FONT color="green">137</FONT> boolean pairedTTest(<a name="line.137"></a> <FONT color="green">138</FONT> double[] sample1,<a name="line.138"></a> <FONT color="green">139</FONT> double[] sample2,<a name="line.139"></a> <FONT color="green">140</FONT> double alpha)<a name="line.140"></a> <FONT color="green">141</FONT> throws IllegalArgumentException, MathException;<a name="line.141"></a> <FONT color="green">142</FONT> /**<a name="line.142"></a> <FONT color="green">143</FONT> * Computes a <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm#formula"><a name="line.143"></a> <FONT color="green">144</FONT> * t statistic </a> given observed values and a comparison constant.<a name="line.144"></a> <FONT color="green">145</FONT> * <p><a name="line.145"></a> <FONT color="green">146</FONT> * This statistic can be used to perform a one sample t-test for the mean.<a name="line.146"></a> <FONT color="green">147</FONT> * </p><p><a name="line.147"></a> <FONT color="green">148</FONT> * <strong>Preconditions</strong>: <ul><a name="line.148"></a> <FONT color="green">149</FONT> * <li>The observed array length must be at least 2.<a name="line.149"></a> <FONT color="green">150</FONT> * </li></ul></p><a name="line.150"></a> <FONT color="green">151</FONT> *<a name="line.151"></a> <FONT color="green">152</FONT> * @param mu comparison constant<a name="line.152"></a> <FONT color="green">153</FONT> * @param observed array of values<a name="line.153"></a> <FONT color="green">154</FONT> * @return t statistic<a name="line.154"></a> <FONT color="green">155</FONT> * @throws IllegalArgumentException if input array length is less than 2<a name="line.155"></a> <FONT color="green">156</FONT> */<a name="line.156"></a> <FONT color="green">157</FONT> double t(double mu, double[] observed)<a name="line.157"></a> <FONT color="green">158</FONT> throws IllegalArgumentException;<a name="line.158"></a> <FONT color="green">159</FONT> /**<a name="line.159"></a> <FONT color="green">160</FONT> * Computes a <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm#formula"><a name="line.160"></a> <FONT color="green">161</FONT> * t statistic </a> to use in comparing the mean of the dataset described by<a name="line.161"></a> <FONT color="green">162</FONT> * <code>sampleStats</code> to <code>mu</code>.<a name="line.162"></a> <FONT color="green">163</FONT> * <p><a name="line.163"></a> <FONT color="green">164</FONT> * This statistic can be used to perform a one sample t-test for the mean.<a name="line.164"></a> <FONT color="green">165</FONT> * </p><p><a name="line.165"></a> <FONT color="green">166</FONT> * <strong>Preconditions</strong>: <ul><a name="line.166"></a> <FONT color="green">167</FONT> * <li><code>observed.getN() > = 2</code>.<a name="line.167"></a> <FONT color="green">168</FONT> * </li></ul></p><a name="line.168"></a> <FONT color="green">169</FONT> *<a name="line.169"></a> <FONT color="green">170</FONT> * @param mu comparison constant<a name="line.170"></a> <FONT color="green">171</FONT> * @param sampleStats DescriptiveStatistics holding sample summary statitstics<a name="line.171"></a> <FONT color="green">172</FONT> * @return t statistic<a name="line.172"></a> <FONT color="green">173</FONT> * @throws IllegalArgumentException if the precondition is not met<a name="line.173"></a> <FONT color="green">174</FONT> */<a name="line.174"></a> <FONT color="green">175</FONT> double t(double mu, StatisticalSummary sampleStats)<a name="line.175"></a> <FONT color="green">176</FONT> throws IllegalArgumentException;<a name="line.176"></a> <FONT color="green">177</FONT> /**<a name="line.177"></a> <FONT color="green">178</FONT> * Computes a 2-sample t statistic, under the hypothesis of equal<a name="line.178"></a> <FONT color="green">179</FONT> * subpopulation variances. To compute a t-statistic without the<a name="line.179"></a> <FONT color="green">180</FONT> * equal variances hypothesis, use {@link #t(double[], double[])}.<a name="line.180"></a> <FONT color="green">181</FONT> * <p><a name="line.181"></a> <FONT color="green">182</FONT> * This statistic can be used to perform a (homoscedastic) two-sample<a name="line.182"></a> <FONT color="green">183</FONT> * t-test to compare sample means.</p><a name="line.183"></a> <FONT color="green">184</FONT> * <p><a name="line.184"></a> <FONT color="green">185</FONT> * The t-statisitc is</p><a name="line.185"></a> <FONT color="green">186</FONT> * <p><a name="line.186"></a> <FONT color="green">187</FONT> * &nbsp;&nbsp;<code> t = (m1 - m2) / (sqrt(1/n1 +1/n2) sqrt(var))</code><a name="line.187"></a> <FONT color="green">188</FONT> * </p><p><a name="line.188"></a> <FONT color="green">189</FONT> * where <strong><code>n1</code></strong> is the size of first sample;<a name="line.189"></a> <FONT color="green">190</FONT> * <strong><code> n2</code></strong> is the size of second sample;<a name="line.190"></a> <FONT color="green">191</FONT> * <strong><code> m1</code></strong> is the mean of first sample;<a name="line.191"></a> <FONT color="green">192</FONT> * <strong><code> m2</code></strong> is the mean of second sample</li><a name="line.192"></a> <FONT color="green">193</FONT> * </ul><a name="line.193"></a> <FONT color="green">194</FONT> * and <strong><code>var</code></strong> is the pooled variance estimate:<a name="line.194"></a> <FONT color="green">195</FONT> * </p><p><a name="line.195"></a> <FONT color="green">196</FONT> * <code>var = sqrt(((n1 - 1)var1 + (n2 - 1)var2) / ((n1-1) + (n2-1)))</code><a name="line.196"></a> <FONT color="green">197</FONT> * </p><p><a name="line.197"></a> <FONT color="green">198</FONT> * with <strong><code>var1<code></strong> the variance of the first sample and<a name="line.198"></a> <FONT color="green">199</FONT> * <strong><code>var2</code></strong> the variance of the second sample.<a name="line.199"></a> <FONT color="green">200</FONT> * </p><p><a name="line.200"></a> <FONT color="green">201</FONT> * <strong>Preconditions</strong>: <ul><a name="line.201"></a> <FONT color="green">202</FONT> * <li>The observed array lengths must both be at least 2.<a name="line.202"></a> <FONT color="green">203</FONT> * </li></ul></p><a name="line.203"></a> <FONT color="green">204</FONT> *<a name="line.204"></a> <FONT color="green">205</FONT> * @param sample1 array of sample data values<a name="line.205"></a> <FONT color="green">206</FONT> * @param sample2 array of sample data values<a name="line.206"></a> <FONT color="green">207</FONT> * @return t statistic<a name="line.207"></a> <FONT color="green">208</FONT> * @throws IllegalArgumentException if the precondition is not met<a name="line.208"></a> <FONT color="green">209</FONT> */<a name="line.209"></a> <FONT color="green">210</FONT> double homoscedasticT(double[] sample1, double[] sample2)<a name="line.210"></a> <FONT color="green">211</FONT> throws IllegalArgumentException;<a name="line.211"></a> <FONT color="green">212</FONT> /**<a name="line.212"></a> <FONT color="green">213</FONT> * Computes a 2-sample t statistic, without the hypothesis of equal<a name="line.213"></a> <FONT color="green">214</FONT> * subpopulation variances. To compute a t-statistic assuming equal<a name="line.214"></a> <FONT color="green">215</FONT> * variances, use {@link #homoscedasticT(double[], double[])}.<a name="line.215"></a> <FONT color="green">216</FONT> * <p><a name="line.216"></a> <FONT color="green">217</FONT> * This statistic can be used to perform a two-sample t-test to compare<a name="line.217"></a> <FONT color="green">218</FONT> * sample means.</p><a name="line.218"></a> <FONT color="green">219</FONT> * <p><a name="line.219"></a> <FONT color="green">220</FONT> * The t-statisitc is</p><a name="line.220"></a> <FONT color="green">221</FONT> * <p><a name="line.221"></a> <FONT color="green">222</FONT> * &nbsp;&nbsp; <code> t = (m1 - m2) / sqrt(var1/n1 + var2/n2)</code><a name="line.222"></a> <FONT color="green">223</FONT> * </p><p><a name="line.223"></a> <FONT color="green">224</FONT> * where <strong><code>n1</code></strong> is the size of the first sample<a name="line.224"></a> <FONT color="green">225</FONT> * <strong><code> n2</code></strong> is the size of the second sample;<a name="line.225"></a> <FONT color="green">226</FONT> * <strong><code> m1</code></strong> is the mean of the first sample;<a name="line.226"></a> <FONT color="green">227</FONT> * <strong><code> m2</code></strong> is the mean of the second sample;<a name="line.227"></a> <FONT color="green">228</FONT> * <strong><code> var1</code></strong> is the variance of the first sample;<a name="line.228"></a> <FONT color="green">229</FONT> * <strong><code> var2</code></strong> is the variance of the second sample;<a name="line.229"></a> <FONT color="green">230</FONT> * </p><p><a name="line.230"></a> <FONT color="green">231</FONT> * <strong>Preconditions</strong>: <ul><a name="line.231"></a> <FONT color="green">232</FONT> * <li>The observed array lengths must both be at least 2.<a name="line.232"></a> <FONT color="green">233</FONT> * </li></ul></p><a name="line.233"></a> <FONT color="green">234</FONT> *<a name="line.234"></a> <FONT color="green">235</FONT> * @param sample1 array of sample data values<a name="line.235"></a> <FONT color="green">236</FONT> * @param sample2 array of sample data values<a name="line.236"></a> <FONT color="green">237</FONT> * @return t statistic<a name="line.237"></a> <FONT color="green">238</FONT> * @throws IllegalArgumentException if the precondition is not met<a name="line.238"></a> <FONT color="green">239</FONT> */<a name="line.239"></a> <FONT color="green">240</FONT> double t(double[] sample1, double[] sample2)<a name="line.240"></a> <FONT color="green">241</FONT> throws IllegalArgumentException;<a name="line.241"></a> <FONT color="green">242</FONT> /**<a name="line.242"></a> <FONT color="green">243</FONT> * Computes a 2-sample t statistic </a>, comparing the means of the datasets<a name="line.243"></a> <FONT color="green">244</FONT> * described by two {@link StatisticalSummary} instances, without the<a name="line.244"></a> <FONT color="green">245</FONT> * assumption of equal subpopulation variances. Use<a name="line.245"></a> <FONT color="green">246</FONT> * {@link #homoscedasticT(StatisticalSummary, StatisticalSummary)} to<a name="line.246"></a> <FONT color="green">247</FONT> * compute a t-statistic under the equal variances assumption.<a name="line.247"></a> <FONT color="green">248</FONT> * <p><a name="line.248"></a> <FONT color="green">249</FONT> * This statistic can be used to perform a two-sample t-test to compare<a name="line.249"></a> <FONT color="green">250</FONT> * sample means.</p><a name="line.250"></a> <FONT color="green">251</FONT> * <p><a name="line.251"></a> <FONT color="green">252</FONT> * The returned t-statisitc is</p><a name="line.252"></a> <FONT color="green">253</FONT> * <p><a name="line.253"></a> <FONT color="green">254</FONT> * &nbsp;&nbsp; <code> t = (m1 - m2) / sqrt(var1/n1 + var2/n2)</code><a name="line.254"></a> <FONT color="green">255</FONT> * </p><p><a name="line.255"></a> <FONT color="green">256</FONT> * where <strong><code>n1</code></strong> is the size of the first sample;<a name="line.256"></a> <FONT color="green">257</FONT> * <strong><code> n2</code></strong> is the size of the second sample;<a name="line.257"></a> <FONT color="green">258</FONT> * <strong><code> m1</code></strong> is the mean of the first sample;<a name="line.258"></a> <FONT color="green">259</FONT> * <strong><code> m2</code></strong> is the mean of the second sample<a name="line.259"></a> <FONT color="green">260</FONT> * <strong><code> var1</code></strong> is the variance of the first sample;<a name="line.260"></a> <FONT color="green">261</FONT> * <strong><code> var2</code></strong> is the variance of the second sample<a name="line.261"></a> <FONT color="green">262</FONT> * </p><p><a name="line.262"></a> <FONT color="green">263</FONT> * <strong>Preconditions</strong>: <ul><a name="line.263"></a> <FONT color="green">264</FONT> * <li>The datasets described by the two Univariates must each contain<a name="line.264"></a> <FONT color="green">265</FONT> * at least 2 observations.<a name="line.265"></a> <FONT color="green">266</FONT> * </li></ul></p><a name="line.266"></a> <FONT color="green">267</FONT> *<a name="line.267"></a> <FONT color="green">268</FONT> * @param sampleStats1 StatisticalSummary describing data from the first sample<a name="line.268"></a> <FONT color="green">269</FONT> * @param sampleStats2 StatisticalSummary describing data from the second sample<a name="line.269"></a> <FONT color="green">270</FONT> * @return t statistic<a name="line.270"></a> <FONT color="green">271</FONT> * @throws IllegalArgumentException if the precondition is not met<a name="line.271"></a> <FONT color="green">272</FONT> */<a name="line.272"></a> <FONT color="green">273</FONT> double t(<a name="line.273"></a> <FONT color="green">274</FONT> StatisticalSummary sampleStats1,<a name="line.274"></a> <FONT color="green">275</FONT> StatisticalSummary sampleStats2)<a name="line.275"></a> <FONT color="green">276</FONT> throws IllegalArgumentException;<a name="line.276"></a> <FONT color="green">277</FONT> /**<a name="line.277"></a> <FONT color="green">278</FONT> * Computes a 2-sample t statistic, comparing the means of the datasets<a name="line.278"></a> <FONT color="green">279</FONT> * described by two {@link StatisticalSummary} instances, under the<a name="line.279"></a> <FONT color="green">280</FONT> * assumption of equal subpopulation variances. To compute a t-statistic<a name="line.280"></a> <FONT color="green">281</FONT> * without the equal variances assumption, use<a name="line.281"></a> <FONT color="green">282</FONT> * {@link #t(StatisticalSummary, StatisticalSummary)}.<a name="line.282"></a> <FONT color="green">283</FONT> * <p><a name="line.283"></a> <FONT color="green">284</FONT> * This statistic can be used to perform a (homoscedastic) two-sample<a name="line.284"></a> <FONT color="green">285</FONT> * t-test to compare sample means.</p><a name="line.285"></a> <FONT color="green">286</FONT> * <p><a name="line.286"></a> <FONT color="green">287</FONT> * The t-statisitc returned is</p><a name="line.287"></a> <FONT color="green">288</FONT> * <p><a name="line.288"></a> <FONT color="green">289</FONT> * &nbsp;&nbsp;<code> t = (m1 - m2) / (sqrt(1/n1 +1/n2) sqrt(var))</code><a name="line.289"></a> <FONT color="green">290</FONT> * </p><p><a name="line.290"></a> <FONT color="green">291</FONT> * where <strong><code>n1</code></strong> is the size of first sample;<a name="line.291"></a> <FONT color="green">292</FONT> * <strong><code> n2</code></strong> is the size of second sample;<a name="line.292"></a> <FONT color="green">293</FONT> * <strong><code> m1</code></strong> is the mean of first sample;<a name="line.293"></a> <FONT color="green">294</FONT> * <strong><code> m2</code></strong> is the mean of second sample<a name="line.294"></a> <FONT color="green">295</FONT> * and <strong><code>var</code></strong> is the pooled variance estimate:<a name="line.295"></a> <FONT color="green">296</FONT> * </p><p><a name="line.296"></a> <FONT color="green">297</FONT> * <code>var = sqrt(((n1 - 1)var1 + (n2 - 1)var2) / ((n1-1) + (n2-1)))</code><a name="line.297"></a> <FONT color="green">298</FONT> * </p><p><a name="line.298"></a> <FONT color="green">299</FONT> * with <strong><code>var1<code></strong> the variance of the first sample and<a name="line.299"></a> <FONT color="green">300</FONT> * <strong><code>var2</code></strong> the variance of the second sample.<a name="line.300"></a> <FONT color="green">301</FONT> * </p><p><a name="line.301"></a> <FONT color="green">302</FONT> * <strong>Preconditions</strong>: <ul><a name="line.302"></a> <FONT color="green">303</FONT> * <li>The datasets described by the two Univariates must each contain<a name="line.303"></a> <FONT color="green">304</FONT> * at least 2 observations.<a name="line.304"></a> <FONT color="green">305</FONT> * </li></ul></p><a name="line.305"></a> <FONT color="green">306</FONT> *<a name="line.306"></a> <FONT color="green">307</FONT> * @param sampleStats1 StatisticalSummary describing data from the first sample<a name="line.307"></a> <FONT color="green">308</FONT> * @param sampleStats2 StatisticalSummary describing data from the second sample<a name="line.308"></a> <FONT color="green">309</FONT> * @return t statistic<a name="line.309"></a> <FONT color="green">310</FONT> * @throws IllegalArgumentException if the precondition is not met<a name="line.310"></a> <FONT color="green">311</FONT> */<a name="line.311"></a> <FONT color="green">312</FONT> double homoscedasticT(<a name="line.312"></a> <FONT color="green">313</FONT> StatisticalSummary sampleStats1,<a name="line.313"></a> <FONT color="green">314</FONT> StatisticalSummary sampleStats2)<a name="line.314"></a> <FONT color="green">315</FONT> throws IllegalArgumentException;<a name="line.315"></a> <FONT color="green">316</FONT> /**<a name="line.316"></a> <FONT color="green">317</FONT> * Returns the <i>observed significance level</i>, or<a name="line.317"></a> <FONT color="green">318</FONT> * <i>p-value</i>, associated with a one-sample, two-tailed t-test<a name="line.318"></a> <FONT color="green">319</FONT> * comparing the mean of the input array with the constant <code>mu</code>.<a name="line.319"></a> <FONT color="green">320</FONT> * <p><a name="line.320"></a> <FONT color="green">321</FONT> * The number returned is the smallest significance level<a name="line.321"></a> <FONT color="green">322</FONT> * at which one can reject the null hypothesis that the mean equals<a name="line.322"></a> <FONT color="green">323</FONT> * <code>mu</code> in favor of the two-sided alternative that the mean<a name="line.323"></a> <FONT color="green">324</FONT> * is different from <code>mu</code>. For a one-sided test, divide the<a name="line.324"></a> <FONT color="green">325</FONT> * returned value by 2.</p><a name="line.325"></a> <FONT color="green">326</FONT> * <p><a name="line.326"></a> <FONT color="green">327</FONT> * <strong>Usage Note:</strong><br><a name="line.327"></a> <FONT color="green">328</FONT> * The validity of the test depends on the assumptions of the parametric<a name="line.328"></a> <FONT color="green">329</FONT> * t-test procedure, as discussed<a name="line.329"></a> <FONT color="green">330</FONT> * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html">here</a><a name="line.330"></a> <FONT color="green">331</FONT> * </p><p><a name="line.331"></a> <FONT color="green">332</FONT> * <strong>Preconditions</strong>: <ul><a name="line.332"></a> <FONT color="green">333</FONT> * <li>The observed array length must be at least 2.<a name="line.333"></a> <FONT color="green">334</FONT> * </li></ul></p><a name="line.334"></a> <FONT color="green">335</FONT> *<a name="line.335"></a> <FONT color="green">336</FONT> * @param mu constant value to compare sample mean against<a name="line.336"></a> <FONT color="green">337</FONT> * @param sample array of sample data values<a name="line.337"></a> <FONT color="green">338</FONT> * @return p-value<a name="line.338"></a> <FONT color="green">339</FONT> * @throws IllegalArgumentException if the precondition is not met<a name="line.339"></a> <FONT color="green">340</FONT> * @throws MathException if an error occurs computing the p-value<a name="line.340"></a> <FONT color="green">341</FONT> */<a name="line.341"></a> <FONT color="green">342</FONT> double tTest(double mu, double[] sample)<a name="line.342"></a> <FONT color="green">343</FONT> throws IllegalArgumentException, MathException;<a name="line.343"></a> <FONT color="green">344</FONT> /**<a name="line.344"></a> <FONT color="green">345</FONT> * Performs a <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm"><a name="line.345"></a> <FONT color="green">346</FONT> * two-sided t-test</a> evaluating the null hypothesis that the mean of the population from<a name="line.346"></a> <FONT color="green">347</FONT> * which <code>sample</code> is drawn equals <code>mu</code>.<a name="line.347"></a> <FONT color="green">348</FONT> * <p><a name="line.348"></a> <FONT color="green">349</FONT> * Returns <code>true</code> iff the null hypothesis can be<a name="line.349"></a> <FONT color="green">350</FONT> * rejected with confidence <code>1 - alpha</code>. To<a name="line.350"></a> <FONT color="green">351</FONT> * perform a 1-sided test, use <code>alpha * 2</code></p><a name="line.351"></a> <FONT color="green">352</FONT> * <p><a name="line.352"></a> <FONT color="green">353</FONT> * <strong>Examples:</strong><br><ol><a name="line.353"></a> <FONT color="green">354</FONT> * <li>To test the (2-sided) hypothesis <code>sample mean = mu </code> at<a name="line.354"></a> <FONT color="green">355</FONT> * the 95% level, use <br><code>tTest(mu, sample, 0.05) </code><a name="line.355"></a> <FONT color="green">356</FONT> * </li><a name="line.356"></a> <FONT color="green">357</FONT> * <li>To test the (one-sided) hypothesis <code> sample mean < mu </code><a name="line.357"></a> <FONT color="green">358</FONT> * at the 99% level, first verify that the measured sample mean is less<a name="line.358"></a> <FONT color="green">359</FONT> * than <code>mu</code> and then use<a name="line.359"></a> <FONT color="green">360</FONT> * <br><code>tTest(mu, sample, 0.02) </code><a name="line.360"></a> <FONT color="green">361</FONT> * </li></ol></p><a name="line.361"></a> <FONT color="green">362</FONT> * <p><a name="line.362"></a> <FONT color="green">363</FONT> * <strong>Usage Note:</strong><br><a name="line.363"></a> <FONT color="green">364</FONT> * The validity of the test depends on the assumptions of the one-sample<a name="line.364"></a> <FONT color="green">365</FONT> * parametric t-test procedure, as discussed<a name="line.365"></a> <FONT color="green">366</FONT> * <a href="http://www.basic.nwu.edu/statguidefiles/sg_glos.html#one-sample">here</a><a name="line.366"></a> <FONT color="green">367</FONT> * </p><p><a name="line.367"></a> <FONT color="green">368</FONT> * <strong>Preconditions</strong>: <ul><a name="line.368"></a> <FONT color="green">369</FONT> * <li>The observed array length must be at least 2.<a name="line.369"></a> <FONT color="green">370</FONT> * </li></ul></p><a name="line.370"></a> <FONT color="green">371</FONT> *<a name="line.371"></a> <FONT color="green">372</FONT> * @param mu constant value to compare sample mean against<a name="line.372"></a> <FONT color="green">373</FONT> * @param sample array of sample data values<a name="line.373"></a> <FONT color="green">374</FONT> * @param alpha significance level of the test<a name="line.374"></a> <FONT color="green">375</FONT> * @return p-value<a name="line.375"></a> <FONT color="green">376</FONT> * @throws IllegalArgumentException if the precondition is not met<a name="line.376"></a> <FONT color="green">377</FONT> * @throws MathException if an error computing the p-value<a name="line.377"></a> <FONT color="green">378</FONT> */<a name="line.378"></a> <FONT color="green">379</FONT> boolean tTest(double mu, double[] sample, double alpha)<a name="line.379"></a> <FONT color="green">380</FONT> throws IllegalArgumentException, MathException;<a name="line.380"></a> <FONT color="green">381</FONT> /**<a name="line.381"></a> <FONT color="green">382</FONT> * Returns the <i>observed significance level</i>, or<a name="line.382"></a> <FONT color="green">383</FONT> * <i>p-value</i>, associated with a one-sample, two-tailed t-test<a name="line.383"></a> <FONT color="green">384</FONT> * comparing the mean of the dataset described by <code>sampleStats</code><a name="line.384"></a> <FONT color="green">385</FONT> * with the constant <code>mu</code>.<a name="line.385"></a> <FONT color="green">386</FONT> * <p><a name="line.386"></a> <FONT color="green">387</FONT> * The number returned is the smallest significance level<a name="line.387"></a> <FONT color="green">388</FONT> * at which one can reject the null hypothesis that the mean equals<a name="line.388"></a> <FONT color="green">389</FONT> * <code>mu</code> in favor of the two-sided alternative that the mean<a name="line.389"></a> <FONT color="green">390</FONT> * is different from <code>mu</code>. For a one-sided test, divide the<a name="line.390"></a> <FONT color="green">391</FONT> * returned value by 2.</p><a name="line.391"></a> <FONT color="green">392</FONT> * <p><a name="line.392"></a> <FONT color="green">393</FONT> * <strong>Usage Note:</strong><br><a name="line.393"></a> <FONT color="green">394</FONT> * The validity of the test depends on the assumptions of the parametric<a name="line.394"></a> <FONT color="green">395</FONT> * t-test procedure, as discussed<a name="line.395"></a> <FONT color="green">396</FONT> * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"><a name="line.396"></a> <FONT color="green">397</FONT> * here</a></p><a name="line.397"></a> <FONT color="green">398</FONT> * <p><a name="line.398"></a> <FONT color="green">399</FONT> * <strong>Preconditions</strong>: <ul><a name="line.399"></a> <FONT color="green">400</FONT> * <li>The sample must contain at least 2 observations.<a name="line.400"></a> <FONT color="green">401</FONT> * </li></ul></p><a name="line.401"></a> <FONT color="green">402</FONT> *<a name="line.402"></a> <FONT color="green">403</FONT> * @param mu constant value to compare sample mean against<a name="line.403"></a> <FONT color="green">404</FONT> * @param sampleStats StatisticalSummary describing sample data<a name="line.404"></a> <FONT color="green">405</FONT> * @return p-value<a name="line.405"></a> <FONT color="green">406</FONT> * @throws IllegalArgumentException if the precondition is not met<a name="line.406"></a> <FONT color="green">407</FONT> * @throws MathException if an error occurs computing the p-value<a name="line.407"></a> <FONT color="green">408</FONT> */<a name="line.408"></a> <FONT color="green">409</FONT> double tTest(double mu, StatisticalSummary sampleStats)<a name="line.409"></a> <FONT color="green">410</FONT> throws IllegalArgumentException, MathException;<a name="line.410"></a> <FONT color="green">411</FONT> /**<a name="line.411"></a> <FONT color="green">412</FONT> * Performs a <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm"><a name="line.412"></a> <FONT color="green">413</FONT> * two-sided t-test</a> evaluating the null hypothesis that the mean of the<a name="line.413"></a> <FONT color="green">414</FONT> * population from which the dataset described by <code>stats</code> is<a name="line.414"></a> <FONT color="green">415</FONT> * drawn equals <code>mu</code>.<a name="line.415"></a> <FONT color="green">416</FONT> * <p><a name="line.416"></a> <FONT color="green">417</FONT> * Returns <code>true</code> iff the null hypothesis can be rejected with<a name="line.417"></a> <FONT color="green">418</FONT> * confidence <code>1 - alpha</code>. To perform a 1-sided test, use<a name="line.418"></a> <FONT color="green">419</FONT> * <code>alpha * 2.</code></p><a name="line.419"></a> <FONT color="green">420</FONT> * <p><a name="line.420"></a> <FONT color="green">421</FONT> * <strong>Examples:</strong><br><ol><a name="line.421"></a> <FONT color="green">422</FONT> * <li>To test the (2-sided) hypothesis <code>sample mean = mu </code> at<a name="line.422"></a> <FONT color="green">423</FONT> * the 95% level, use <br><code>tTest(mu, sampleStats, 0.05) </code><a name="line.423"></a> <FONT color="green">424</FONT> * </li><a name="line.424"></a> <FONT color="green">425</FONT> * <li>To test the (one-sided) hypothesis <code> sample mean < mu </code><a name="line.425"></a> <FONT color="green">426</FONT> * at the 99% level, first verify that the measured sample mean is less<a name="line.426"></a> <FONT color="green">427</FONT> * than <code>mu</code> and then use<a name="line.427"></a> <FONT color="green">428</FONT> * <br><code>tTest(mu, sampleStats, 0.02) </code><a name="line.428"></a> <FONT color="green">429</FONT> * </li></ol></p><a name="line.429"></a> <FONT color="green">430</FONT> * <p><a name="line.430"></a> <FONT color="green">431</FONT> * <strong>Usage Note:</strong><br><a name="line.431"></a> <FONT color="green">432</FONT> * The validity of the test depends on the assumptions of the one-sample<a name="line.432"></a> <FONT color="green">433</FONT> * parametric t-test procedure, as discussed<a name="line.433"></a> <FONT color="green">434</FONT> * <a href="http://www.basic.nwu.edu/statguidefiles/sg_glos.html#one-sample">here</a><a name="line.434"></a> <FONT color="green">435</FONT> * </p><p><a name="line.435"></a> <FONT color="green">436</FONT> * <strong>Preconditions</strong>: <ul><a name="line.436"></a> <FONT color="green">437</FONT> * <li>The sample must include at least 2 observations.<a name="line.437"></a> <FONT color="green">438</FONT> * </li></ul></p><a name="line.438"></a> <FONT color="green">439</FONT> *<a name="line.439"></a> <FONT color="green">440</FONT> * @param mu constant value to compare sample mean against<a name="line.440"></a> <FONT color="green">441</FONT> * @param sampleStats StatisticalSummary describing sample data values<a name="line.441"></a> <FONT color="green">442</FONT> * @param alpha significance level of the test<a name="line.442"></a> <FONT color="green">443</FONT> * @return p-value<a name="line.443"></a> <FONT color="green">444</FONT> * @throws IllegalArgumentException if the precondition is not met<a name="line.444"></a> <FONT color="green">445</FONT> * @throws MathException if an error occurs computing the p-value<a name="line.445"></a> <FONT color="green">446</FONT> */<a name="line.446"></a> <FONT color="green">447</FONT> boolean tTest(<a name="line.447"></a> <FONT color="green">448</FONT> double mu,<a name="line.448"></a> <FONT color="green">449</FONT> StatisticalSummary sampleStats,<a name="line.449"></a> <FONT color="green">450</FONT> double alpha)<a name="line.450"></a> <FONT color="green">451</FONT> throws IllegalArgumentException, MathException;<a name="line.451"></a> <FONT color="green">452</FONT> /**<a name="line.452"></a> <FONT color="green">453</FONT> * Returns the <i>observed significance level</i>, or<a name="line.453"></a> <FONT color="green">454</FONT> * <i>p-value</i>, associated with a two-sample, two-tailed t-test<a name="line.454"></a> <FONT color="green">455</FONT> * comparing the means of the input arrays.<a name="line.455"></a> <FONT color="green">456</FONT> * <p><a name="line.456"></a> <FONT color="green">457</FONT> * The number returned is the smallest significance level<a name="line.457"></a> <FONT color="green">458</FONT> * at which one can reject the null hypothesis that the two means are<a name="line.458"></a> <FONT color="green">459</FONT> * equal in favor of the two-sided alternative that they are different.<a name="line.459"></a> <FONT color="green">460</FONT> * For a one-sided test, divide the returned value by 2.</p><a name="line.460"></a> <FONT color="green">461</FONT> * <p><a name="line.461"></a> <FONT color="green">462</FONT> * The test does not assume that the underlying popuation variances are<a name="line.462"></a> <FONT color="green">463</FONT> * equal and it uses approximated degrees of freedom computed from the<a name="line.463"></a> <FONT color="green">464</FONT> * sample data to compute the p-value. The t-statistic used is as defined in<a name="line.464"></a> <FONT color="green">465</FONT> * {@link #t(double[], double[])} and the Welch-Satterthwaite approximation<a name="line.465"></a> <FONT color="green">466</FONT> * to the degrees of freedom is used,<a name="line.466"></a> <FONT color="green">467</FONT> * as described<a name="line.467"></a> <FONT color="green">468</FONT> * <a href="http://www.itl.nist.gov/div898/handbook/prc/section3/prc31.htm"><a name="line.468"></a> <FONT color="green">469</FONT> * here.</a> To perform the test under the assumption of equal subpopulation<a name="line.469"></a> <FONT color="green">470</FONT> * variances, use {@link #homoscedasticTTest(double[], double[])}.</p><a name="line.470"></a> <FONT color="green">471</FONT> * <p><a name="line.471"></a> <FONT color="green">472</FONT> * <strong>Usage Note:</strong><br><a name="line.472"></a> <FONT color="green">473</FONT> * The validity of the p-value depends on the assumptions of the parametric<a name="line.473"></a> <FONT color="green">474</FONT> * t-test procedure, as discussed<a name="line.474"></a> <FONT color="green">475</FONT> * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"><a name="line.475"></a> <FONT color="green">476</FONT> * here</a></p><a name="line.476"></a> <FONT color="green">477</FONT> * <p><a name="line.477"></a> <FONT color="green">478</FONT> * <strong>Preconditions</strong>: <ul><a name="line.478"></a> <FONT color="green">479</FONT> * <li>The observed array lengths must both be at least 2.<a name="line.479"></a> <FONT color="green">480</FONT> * </li></ul></p><a name="line.480"></a> <FONT color="green">481</FONT> *<a name="line.481"></a> <FONT color="green">482</FONT> * @param sample1 array of sample data values<a name="line.482"></a> <FONT color="green">483</FONT> * @param sample2 array of sample data values<a name="line.483"></a> <FONT color="green">484</FONT> * @return p-value for t-test<a name="line.484"></a> <FONT color="green">485</FONT> * @throws IllegalArgumentException if the precondition is not met<a name="line.485"></a> <FONT color="green">486</FONT> * @throws MathException if an error occurs computing the p-value<a name="line.486"></a> <FONT color="green">487</FONT> */<a name="line.487"></a> <FONT color="green">488</FONT> double tTest(double[] sample1, double[] sample2)<a name="line.488"></a> <FONT color="green">489</FONT> throws IllegalArgumentException, MathException;<a name="line.489"></a> <FONT color="green">490</FONT> /**<a name="line.490"></a> <FONT color="green">491</FONT> * Returns the <i>observed significance level</i>, or<a name="line.491"></a> <FONT color="green">492</FONT> * <i>p-value</i>, associated with a two-sample, two-tailed t-test<a name="line.492"></a> <FONT color="green">493</FONT> * comparing the means of the input arrays, under the assumption that<a name="line.493"></a> <FONT color="green">494</FONT> * the two samples are drawn from subpopulations with equal variances.<a name="line.494"></a> <FONT color="green">495</FONT> * To perform the test without the equal variances assumption, use<a name="line.495"></a> <FONT color="green">496</FONT> * {@link #tTest(double[], double[])}.</p><a name="line.496"></a> <FONT color="green">497</FONT> * <p><a name="line.497"></a> <FONT color="green">498</FONT> * The number returned is the smallest significance level<a name="line.498"></a> <FONT color="green">499</FONT> * at which one can reject the null hypothesis that the two means are<a name="line.499"></a> <FONT color="green">500</FONT> * equal in favor of the two-sided alternative that they are different.<a name="line.500"></a> <FONT color="green">501</FONT> * For a one-sided test, divide the returned value by 2.</p><a name="line.501"></a> <FONT color="green">502</FONT> * <p><a name="line.502"></a> <FONT color="green">503</FONT> * A pooled variance estimate is used to compute the t-statistic. See<a name="line.503"></a> <FONT color="green">504</FONT> * {@link #homoscedasticT(double[], double[])}. The sum of the sample sizes<a name="line.504"></a> <FONT color="green">505</FONT> * minus 2 is used as the degrees of freedom.</p><a name="line.505"></a> <FONT color="green">506</FONT> * <p><a name="line.506"></a> <FONT color="green">507</FONT> * <strong>Usage Note:</strong><br><a name="line.507"></a> <FONT color="green">508</FONT> * The validity of the p-value depends on the assumptions of the parametric<a name="line.508"></a> <FONT color="green">509</FONT> * t-test procedure, as discussed<a name="line.509"></a> <FONT color="green">510</FONT> * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"><a name="line.510"></a> <FONT color="green">511</FONT> * here</a></p><a name="line.511"></a> <FONT color="green">512</FONT> * <p><a name="line.512"></a> <FONT color="green">513</FONT> * <strong>Preconditions</strong>: <ul><a name="line.513"></a> <FONT color="green">514</FONT> * <li>The observed array lengths must both be at least 2.<a name="line.514"></a> <FONT color="green">515</FONT> * </li></ul></p><a name="line.515"></a> <FONT color="green">516</FONT> *<a name="line.516"></a> <FONT color="green">517</FONT> * @param sample1 array of sample data values<a name="line.517"></a> <FONT color="green">518</FONT> * @param sample2 array of sample data values<a name="line.518"></a> <FONT color="green">519</FONT> * @return p-value for t-test<a name="line.519"></a> <FONT color="green">520</FONT> * @throws IllegalArgumentException if the precondition is not met<a name="line.520"></a> <FONT color="green">521</FONT> * @throws MathException if an error occurs computing the p-value<a name="line.521"></a> <FONT color="green">522</FONT> */<a name="line.522"></a> <FONT color="green">523</FONT> double homoscedasticTTest(<a name="line.523"></a> <FONT color="green">524</FONT> double[] sample1,<a name="line.524"></a> <FONT color="green">525</FONT> double[] sample2)<a name="line.525"></a> <FONT color="green">526</FONT> throws IllegalArgumentException, MathException;<a name="line.526"></a> <FONT color="green">527</FONT> /**<a name="line.527"></a> <FONT color="green">528</FONT> * Performs a<a name="line.528"></a> <FONT color="green">529</FONT> * <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm"><a name="line.529"></a> <FONT color="green">530</FONT> * two-sided t-test</a> evaluating the null hypothesis that <code>sample1</code><a name="line.530"></a> <FONT color="green">531</FONT> * and <code>sample2</code> are drawn from populations with the same mean,<a name="line.531"></a> <FONT color="green">532</FONT> * with significance level <code>alpha</code>. This test does not assume<a name="line.532"></a> <FONT color="green">533</FONT> * that the subpopulation variances are equal. To perform the test assuming<a name="line.533"></a> <FONT color="green">534</FONT> * equal variances, use<a name="line.534"></a> <FONT color="green">535</FONT> * {@link #homoscedasticTTest(double[], double[], double)}.<a name="line.535"></a> <FONT color="green">536</FONT> * <p><a name="line.536"></a> <FONT color="green">537</FONT> * Returns <code>true</code> iff the null hypothesis that the means are<a name="line.537"></a> <FONT color="green">538</FONT> * equal can be rejected with confidence <code>1 - alpha</code>. To<a name="line.538"></a> <FONT color="green">539</FONT> * perform a 1-sided test, use <code>alpha * 2</code></p><a name="line.539"></a> <FONT color="green">540</FONT> * <p><a name="line.540"></a> <FONT color="green">541</FONT> * See {@link #t(double[], double[])} for the formula used to compute the<a name="line.541"></a> <FONT color="green">542</FONT> * t-statistic. Degrees of freedom are approximated using the<a name="line.542"></a> <FONT color="green">543</FONT> * <a href="http://www.itl.nist.gov/div898/handbook/prc/section3/prc31.htm"><a name="line.543"></a> <FONT color="green">544</FONT> * Welch-Satterthwaite approximation.</a></p><a name="line.544"></a> <FONT color="green">545</FONT> * <p><a name="line.545"></a> <FONT color="green">546</FONT> * <strong>Examples:</strong><br><ol><a name="line.546"></a> <FONT color="green">547</FONT> * <li>To test the (2-sided) hypothesis <code>mean 1 = mean 2 </code> at<a name="line.547"></a> <FONT color="green">548</FONT> * the 95% level, use<a name="line.548"></a> <FONT color="green">549</FONT> * <br><code>tTest(sample1, sample2, 0.05). </code><a name="line.549"></a> <FONT color="green">550</FONT> * </li><a name="line.550"></a> <FONT color="green">551</FONT> * <li>To test the (one-sided) hypothesis <code> mean 1 < mean 2 </code>,<a name="line.551"></a> <FONT color="green">552</FONT> * at the 99% level, first verify that the measured mean of <code>sample 1</code><a name="line.552"></a> <FONT color="green">553</FONT> * is less than the mean of <code>sample 2</code> and then use<a name="line.553"></a> <FONT color="green">554</FONT> * <br><code>tTest(sample1, sample2, 0.02) </code><a name="line.554"></a> <FONT color="green">555</FONT> * </li></ol></p><a name="line.555"></a> <FONT color="green">556</FONT> * <p><a name="line.556"></a> <FONT color="green">557</FONT> * <strong>Usage Note:</strong><br><a name="line.557"></a> <FONT color="green">558</FONT> * The validity of the test depends on the assumptions of the parametric<a name="line.558"></a> <FONT color="green">559</FONT> * t-test procedure, as discussed<a name="line.559"></a> <FONT color="green">560</FONT> * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"><a name="line.560"></a> <FONT color="green">561</FONT> * here</a></p><a name="line.561"></a> <FONT color="green">562</FONT> * <p><a name="line.562"></a> <FONT color="green">563</FONT> * <strong>Preconditions</strong>: <ul><a name="line.563"></a> <FONT color="green">564</FONT> * <li>The observed array lengths must both be at least 2.<a name="line.564"></a> <FONT color="green">565</FONT> * </li><a name="line.565"></a> <FONT color="green">566</FONT> * <li> <code> 0 < alpha < 0.5 </code><a name="line.566"></a> <FONT color="green">567</FONT> * </li></ul></p><a name="line.567"></a> <FONT color="green">568</FONT> *<a name="line.568"></a> <FONT color="green">569</FONT> * @param sample1 array of sample data values<a name="line.569"></a> <FONT color="green">570</FONT> * @param sample2 array of sample data values<a name="line.570"></a> <FONT color="green">571</FONT> * @param alpha significance level of the test<a name="line.571"></a> <FONT color="green">572</FONT> * @return true if the null hypothesis can be rejected with<a name="line.572"></a> <FONT color="green">573</FONT> * confidence 1 - alpha<a name="line.573"></a> <FONT color="green">574</FONT> * @throws IllegalArgumentException if the preconditions are not met<a name="line.574"></a> <FONT color="green">575</FONT> * @throws MathException if an error occurs performing the test<a name="line.575"></a> <FONT color="green">576</FONT> */<a name="line.576"></a> <FONT color="green">577</FONT> boolean tTest(<a name="line.577"></a> <FONT color="green">578</FONT> double[] sample1,<a name="line.578"></a> <FONT color="green">579</FONT> double[] sample2,<a name="line.579"></a> <FONT color="green">580</FONT> double alpha)<a name="line.580"></a> <FONT color="green">581</FONT> throws IllegalArgumentException, MathException;<a name="line.581"></a> <FONT color="green">582</FONT> /**<a name="line.582"></a> <FONT color="green">583</FONT> * Performs a<a name="line.583"></a> <FONT color="green">584</FONT> * <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm"><a name="line.584"></a> <FONT color="green">585</FONT> * two-sided t-test</a> evaluating the null hypothesis that <code>sample1</code><a name="line.585"></a> <FONT color="green">586</FONT> * and <code>sample2</code> are drawn from populations with the same mean,<a name="line.586"></a> <FONT color="green">587</FONT> * with significance level <code>alpha</code>, assuming that the<a name="line.587"></a> <FONT color="green">588</FONT> * subpopulation variances are equal. Use<a name="line.588"></a> <FONT color="green">589</FONT> * {@link #tTest(double[], double[], double)} to perform the test without<a name="line.589"></a> <FONT color="green">590</FONT> * the assumption of equal variances.<a name="line.590"></a> <FONT color="green">591</FONT> * <p><a name="line.591"></a> <FONT color="green">592</FONT> * Returns <code>true</code> iff the null hypothesis that the means are<a name="line.592"></a> <FONT color="green">593</FONT> * equal can be rejected with confidence <code>1 - alpha</code>. To<a name="line.593"></a> <FONT color="green">594</FONT> * perform a 1-sided test, use <code>alpha * 2.</code> To perform the test<a name="line.594"></a> <FONT color="green">595</FONT> * without the assumption of equal subpopulation variances, use<a name="line.595"></a> <FONT color="green">596</FONT> * {@link #tTest(double[], double[], double)}.</p><a name="line.596"></a> <FONT color="green">597</FONT> * <p><a name="line.597"></a> <FONT color="green">598</FONT> * A pooled variance estimate is used to compute the t-statistic. See<a name="line.598"></a> <FONT color="green">599</FONT> * {@link #t(double[], double[])} for the formula. The sum of the sample<a name="line.599"></a> <FONT color="green">600</FONT> * sizes minus 2 is used as the degrees of freedom.</p><a name="line.600"></a> <FONT color="green">601</FONT> * <p><a name="line.601"></a> <FONT color="green">602</FONT> * <strong>Examples:</strong><br><ol><a name="line.602"></a> <FONT color="green">603</FONT> * <li>To test the (2-sided) hypothesis <code>mean 1 = mean 2 </code> at<a name="line.603"></a> <FONT color="green">604</FONT> * the 95% level, use <br><code>tTest(sample1, sample2, 0.05). </code><a name="line.604"></a> <FONT color="green">605</FONT> * </li><a name="line.605"></a> <FONT color="green">606</FONT> * <li>To test the (one-sided) hypothesis <code> mean 1 < mean 2, </code><a name="line.606"></a> <FONT color="green">607</FONT> * at the 99% level, first verify that the measured mean of<a name="line.607"></a> <FONT color="green">608</FONT> * <code>sample 1</code> is less than the mean of <code>sample 2</code><a name="line.608"></a> <FONT color="green">609</FONT> * and then use<a name="line.609"></a> <FONT color="green">610</FONT> * <br><code>tTest(sample1, sample2, 0.02) </code><a name="line.610"></a> <FONT color="green">611</FONT> * </li></ol></p><a name="line.611"></a> <FONT color="green">612</FONT> * <p><a name="line.612"></a> <FONT color="green">613</FONT> * <strong>Usage Note:</strong><br><a name="line.613"></a> <FONT color="green">614</FONT> * The validity of the test depends on the assumptions of the parametric<a name="line.614"></a> <FONT color="green">615</FONT> * t-test procedure, as discussed<a name="line.615"></a> <FONT color="green">616</FONT> * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"><a name="line.616"></a> <FONT color="green">617</FONT> * here</a></p><a name="line.617"></a> <FONT color="green">618</FONT> * <p><a name="line.618"></a> <FONT color="green">619</FONT> * <strong>Preconditions</strong>: <ul><a name="line.619"></a> <FONT color="green">620</FONT> * <li>The observed array lengths must both be at least 2.<a name="line.620"></a> <FONT color="green">621</FONT> * </li><a name="line.621"></a> <FONT color="green">622</FONT> * <li> <code> 0 < alpha < 0.5 </code><a name="line.622"></a> <FONT color="green">623</FONT> * </li></ul></p><a name="line.623"></a> <FONT color="green">624</FONT> *<a name="line.624"></a> <FONT color="green">625</FONT> * @param sample1 array of sample data values<a name="line.625"></a> <FONT color="green">626</FONT> * @param sample2 array of sample data values<a name="line.626"></a> <FONT color="green">627</FONT> * @param alpha significance level of the test<a name="line.627"></a> <FONT color="green">628</FONT> * @return true if the null hypothesis can be rejected with<a name="line.628"></a> <FONT color="green">629</FONT> * confidence 1 - alpha<a name="line.629"></a> <FONT color="green">630</FONT> * @throws IllegalArgumentException if the preconditions are not met<a name="line.630"></a> <FONT color="green">631</FONT> * @throws MathException if an error occurs performing the test<a name="line.631"></a> <FONT color="green">632</FONT> */<a name="line.632"></a> <FONT color="green">633</FONT> boolean homoscedasticTTest(<a name="line.633"></a> <FONT color="green">634</FONT> double[] sample1,<a name="line.634"></a> <FONT color="green">635</FONT> double[] sample2,<a name="line.635"></a> <FONT color="green">636</FONT> double alpha)<a name="line.636"></a> <FONT color="green">637</FONT> throws IllegalArgumentException, MathException;<a name="line.637"></a> <FONT color="green">638</FONT> /**<a name="line.638"></a> <FONT color="green">639</FONT> * Returns the <i>observed significance level</i>, or<a name="line.639"></a> <FONT color="green">640</FONT> * <i>p-value</i>, associated with a two-sample, two-tailed t-test<a name="line.640"></a> <FONT color="green">641</FONT> * comparing the means of the datasets described by two StatisticalSummary<a name="line.641"></a> <FONT color="green">642</FONT> * instances.<a name="line.642"></a> <FONT color="green">643</FONT> * <p><a name="line.643"></a> <FONT color="green">644</FONT> * The number returned is the smallest significance level<a name="line.644"></a> <FONT color="green">645</FONT> * at which one can reject the null hypothesis that the two means are<a name="line.645"></a> <FONT color="green">646</FONT> * equal in favor of the two-sided alternative that they are different.<a name="line.646"></a> <FONT color="green">647</FONT> * For a one-sided test, divide the returned value by 2.</p><a name="line.647"></a> <FONT color="green">648</FONT> * <p><a name="line.648"></a> <FONT color="green">649</FONT> * The test does not assume that the underlying popuation variances are<a name="line.649"></a> <FONT color="green">650</FONT> * equal and it uses approximated degrees of freedom computed from the<a name="line.650"></a> <FONT color="green">651</FONT> * sample data to compute the p-value. To perform the test assuming<a name="line.651"></a> <FONT color="green">652</FONT> * equal variances, use<a name="line.652"></a> <FONT color="green">653</FONT> * {@link #homoscedasticTTest(StatisticalSummary, StatisticalSummary)}.</p><a name="line.653"></a> <FONT color="green">654</FONT> * <p><a name="line.654"></a> <FONT color="green">655</FONT> * <strong>Usage Note:</strong><br><a name="line.655"></a> <FONT color="green">656</FONT> * The validity of the p-value depends on the assumptions of the parametric<a name="line.656"></a> <FONT color="green">657</FONT> * t-test procedure, as discussed<a name="line.657"></a> <FONT color="green">658</FONT> * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"><a name="line.658"></a> <FONT color="green">659</FONT> * here</a></p><a name="line.659"></a> <FONT color="green">660</FONT> * <p><a name="line.660"></a> <FONT color="green">661</FONT> * <strong>Preconditions</strong>: <ul><a name="line.661"></a> <FONT color="green">662</FONT> * <li>The datasets described by the two Univariates must each contain<a name="line.662"></a> <FONT color="green">663</FONT> * at least 2 observations.<a name="line.663"></a> <FONT color="green">664</FONT> * </li></ul></p><a name="line.664"></a> <FONT color="green">665</FONT> *<a name="line.665"></a> <FONT color="green">666</FONT> * @param sampleStats1 StatisticalSummary describing data from the first sample<a name="line.666"></a> <FONT color="green">667</FONT> * @param sampleStats2 StatisticalSummary describing data from the second sample<a name="line.667"></a> <FONT color="green">668</FONT> * @return p-value for t-test<a name="line.668"></a> <FONT color="green">669</FONT> * @throws IllegalArgumentException if the precondition is not met<a name="line.669"></a> <FONT color="green">670</FONT> * @throws MathException if an error occurs computing the p-value<a name="line.670"></a> <FONT color="green">671</FONT> */<a name="line.671"></a> <FONT color="green">672</FONT> double tTest(<a name="line.672"></a> <FONT color="green">673</FONT> StatisticalSummary sampleStats1,<a name="line.673"></a> <FONT color="green">674</FONT> StatisticalSummary sampleStats2)<a name="line.674"></a> <FONT color="green">675</FONT> throws IllegalArgumentException, MathException;<a name="line.675"></a> <FONT color="green">676</FONT> /**<a name="line.676"></a> <FONT color="green">677</FONT> * Returns the <i>observed significance level</i>, or<a name="line.677"></a> <FONT color="green">678</FONT> * <i>p-value</i>, associated with a two-sample, two-tailed t-test<a name="line.678"></a> <FONT color="green">679</FONT> * comparing the means of the datasets described by two StatisticalSummary<a name="line.679"></a> <FONT color="green">680</FONT> * instances, under the hypothesis of equal subpopulation variances. To<a name="line.680"></a> <FONT color="green">681</FONT> * perform a test without the equal variances assumption, use<a name="line.681"></a> <FONT color="green">682</FONT> * {@link #tTest(StatisticalSummary, StatisticalSummary)}.<a name="line.682"></a> <FONT color="green">683</FONT> * <p><a name="line.683"></a> <FONT color="green">684</FONT> * The number returned is the smallest significance level<a name="line.684"></a> <FONT color="green">685</FONT> * at which one can reject the null hypothesis that the two means are<a name="line.685"></a> <FONT color="green">686</FONT> * equal in favor of the two-sided alternative that they are different.<a name="line.686"></a> <FONT color="green">687</FONT> * For a one-sided test, divide the returned value by 2.</p><a name="line.687"></a> <FONT color="green">688</FONT> * <p><a name="line.688"></a> <FONT color="green">689</FONT> * See {@link #homoscedasticT(double[], double[])} for the formula used to<a name="line.689"></a> <FONT color="green">690</FONT> * compute the t-statistic. The sum of the sample sizes minus 2 is used as<a name="line.690"></a> <FONT color="green">691</FONT> * the degrees of freedom.</p><a name="line.691"></a> <FONT color="green">692</FONT> * <p><a name="line.692"></a> <FONT color="green">693</FONT> * <strong>Usage Note:</strong><br><a name="line.693"></a> <FONT color="green">694</FONT> * The validity of the p-value depends on the assumptions of the parametric<a name="line.694"></a> <FONT color="green">695</FONT> * t-test procedure, as discussed<a name="line.695"></a> <FONT color="green">696</FONT> * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html">here</a><a name="line.696"></a> <FONT color="green">697</FONT> * </p><p><a name="line.697"></a> <FONT color="green">698</FONT> * <strong>Preconditions</strong>: <ul><a name="line.698"></a> <FONT color="green">699</FONT> * <li>The datasets described by the two Univariates must each contain<a name="line.699"></a> <FONT color="green">700</FONT> * at least 2 observations.<a name="line.700"></a> <FONT color="green">701</FONT> * </li></ul></p><a name="line.701"></a> <FONT color="green">702</FONT> *<a name="line.702"></a> <FONT color="green">703</FONT> * @param sampleStats1 StatisticalSummary describing data from the first sample<a name="line.703"></a> <FONT color="green">704</FONT> * @param sampleStats2 StatisticalSummary describing data from the second sample<a name="line.704"></a> <FONT color="green">705</FONT> * @return p-value for t-test<a name="line.705"></a> <FONT color="green">706</FONT> * @throws IllegalArgumentException if the precondition is not met<a name="line.706"></a> <FONT color="green">707</FONT> * @throws MathException if an error occurs computing the p-value<a name="line.707"></a> <FONT color="green">708</FONT> */<a name="line.708"></a> <FONT color="green">709</FONT> double homoscedasticTTest(<a name="line.709"></a> <FONT color="green">710</FONT> StatisticalSummary sampleStats1,<a name="line.710"></a> <FONT color="green">711</FONT> StatisticalSummary sampleStats2)<a name="line.711"></a> <FONT color="green">712</FONT> throws IllegalArgumentException, MathException;<a name="line.712"></a> <FONT color="green">713</FONT> /**<a name="line.713"></a> <FONT color="green">714</FONT> * Performs a<a name="line.714"></a> <FONT color="green">715</FONT> * <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm"><a name="line.715"></a> <FONT color="green">716</FONT> * two-sided t-test</a> evaluating the null hypothesis that<a name="line.716"></a> <FONT color="green">717</FONT> * <code>sampleStats1</code> and <code>sampleStats2</code> describe<a name="line.717"></a> <FONT color="green">718</FONT> * datasets drawn from populations with the same mean, with significance<a name="line.718"></a> <FONT color="green">719</FONT> * level <code>alpha</code>. This test does not assume that the<a name="line.719"></a> <FONT color="green">720</FONT> * subpopulation variances are equal. To perform the test under the equal<a name="line.720"></a> <FONT color="green">721</FONT> * variances assumption, use<a name="line.721"></a> <FONT color="green">722</FONT> * {@link #homoscedasticTTest(StatisticalSummary, StatisticalSummary)}.<a name="line.722"></a> <FONT color="green">723</FONT> * <p><a name="line.723"></a> <FONT color="green">724</FONT> * Returns <code>true</code> iff the null hypothesis that the means are<a name="line.724"></a> <FONT color="green">725</FONT> * equal can be rejected with confidence <code>1 - alpha</code>. To<a name="line.725"></a> <FONT color="green">726</FONT> * perform a 1-sided test, use <code>alpha * 2</code></p><a name="line.726"></a> <FONT color="green">727</FONT> * <p><a name="line.727"></a> <FONT color="green">728</FONT> * See {@link #t(double[], double[])} for the formula used to compute the<a name="line.728"></a> <FONT color="green">729</FONT> * t-statistic. Degrees of freedom are approximated using the<a name="line.729"></a> <FONT color="green">730</FONT> * <a href="http://www.itl.nist.gov/div898/handbook/prc/section3/prc31.htm"><a name="line.730"></a> <FONT color="green">731</FONT> * Welch-Satterthwaite approximation.</a></p><a name="line.731"></a> <FONT color="green">732</FONT> * <p><a name="line.732"></a> <FONT color="green">733</FONT> * <strong>Examples:</strong><br><ol><a name="line.733"></a> <FONT color="green">734</FONT> * <li>To test the (2-sided) hypothesis <code>mean 1 = mean 2 </code> at<a name="line.734"></a> <FONT color="green">735</FONT> * the 95%, use<a name="line.735"></a> <FONT color="green">736</FONT> * <br><code>tTest(sampleStats1, sampleStats2, 0.05) </code><a name="line.736"></a> <FONT color="green">737</FONT> * </li><a name="line.737"></a> <FONT color="green">738</FONT> * <li>To test the (one-sided) hypothesis <code> mean 1 < mean 2 </code><a name="line.738"></a> <FONT color="green">739</FONT> * at the 99% level, first verify that the measured mean of<a name="line.739"></a> <FONT color="green">740</FONT> * <code>sample 1</code> is less than the mean of <code>sample 2</code><a name="line.740"></a> <FONT color="green">741</FONT> * and then use<a name="line.741"></a> <FONT color="green">742</FONT> * <br><code>tTest(sampleStats1, sampleStats2, 0.02) </code><a name="line.742"></a> <FONT color="green">743</FONT> * </li></ol></p><a name="line.743"></a> <FONT color="green">744</FONT> * <p><a name="line.744"></a> <FONT color="green">745</FONT> * <strong>Usage Note:</strong><br><a name="line.745"></a> <FONT color="green">746</FONT> * The validity of the test depends on the assumptions of the parametric<a name="line.746"></a> <FONT color="green">747</FONT> * t-test procedure, as discussed<a name="line.747"></a> <FONT color="green">748</FONT> * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"><a name="line.748"></a> <FONT color="green">749</FONT> * here</a></p><a name="line.749"></a> <FONT color="green">750</FONT> * <p><a name="line.750"></a> <FONT color="green">751</FONT> * <strong>Preconditions</strong>: <ul><a name="line.751"></a> <FONT color="green">752</FONT> * <li>The datasets described by the two Univariates must each contain<a name="line.752"></a> <FONT color="green">753</FONT> * at least 2 observations.<a name="line.753"></a> <FONT color="green">754</FONT> * </li><a name="line.754"></a> <FONT color="green">755</FONT> * <li> <code> 0 < alpha < 0.5 </code><a name="line.755"></a> <FONT color="green">756</FONT> * </li></ul></p><a name="line.756"></a> <FONT color="green">757</FONT> *<a name="line.757"></a> <FONT color="green">758</FONT> * @param sampleStats1 StatisticalSummary describing sample data values<a name="line.758"></a> <FONT color="green">759</FONT> * @param sampleStats2 StatisticalSummary describing sample data values<a name="line.759"></a> <FONT color="green">760</FONT> * @param alpha significance level of the test<a name="line.760"></a> <FONT color="green">761</FONT> * @return true if the null hypothesis can be rejected with<a name="line.761"></a> <FONT color="green">762</FONT> * confidence 1 - alpha<a name="line.762"></a> <FONT color="green">763</FONT> * @throws IllegalArgumentException if the preconditions are not met<a name="line.763"></a> <FONT color="green">764</FONT> * @throws MathException if an error occurs performing the test<a name="line.764"></a> <FONT color="green">765</FONT> */<a name="line.765"></a> <FONT color="green">766</FONT> boolean tTest(<a name="line.766"></a> <FONT color="green">767</FONT> StatisticalSummary sampleStats1,<a name="line.767"></a> <FONT color="green">768</FONT> StatisticalSummary sampleStats2,<a name="line.768"></a> <FONT color="green">769</FONT> double alpha)<a name="line.769"></a> <FONT color="green">770</FONT> throws IllegalArgumentException, MathException;<a name="line.770"></a> <FONT color="green">771</FONT> }<a name="line.771"></a> </PRE> </BODY> </HTML>