This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# PairedTTest

 PairedTTest[data] tests whether the mean of data is zero. PairedTTesttests whether the mean of is zero. PairedTTesttests a location measure against . PairedTTestreturns the value of .
• PairedTTest performs a hypothesis test on data with null hypothesis that the true population mean is some value , and alternative hypothesis that .
• Given and , PairedTTest performs a test on the paired differences of the two datasets.
• By default a probability value or -value is returned.
• A small -value suggests that it is unlikely that is true.
• The data in dspec can be univariate or multivariate .
• If two samples are given, they must be of equal length.
• The argument can be a real number or a real vector with length equal to the dimension of the data.
• PairedTTest returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
• PairedTTest can be used to directly give the value of .
• Properties related to the reporting of test results include:
 "DegreesOfFreedom" the degrees of freedom used in a test "PValue" list of -values "PValueTable" formatted table of -values "ShortTestConclusion" a short description of the conclusion of a test "TestConclusion" a description of the conclusion of a test "TestData" list of pairs of test statistics and -values "TestDataTable" formatted table of -values and test statistics "TestStatistic" list of test statistics "TestStatisticTable" formatted table of test statistics
• The following options can be used:
 AlternativeHypothesis "Unequal" the inequality for the alternative hypothesis SignificanceLevel 0.05 cutoff for diagnostics and reporting VerifyTestAssumptions Automatic what assumptions to verify
• For the PairedTTest, a cutoff is chosen such that is rejected only if . The value of used for the and properties is controlled by the SignificanceLevel option. This value is also used in diagnostic tests of assumptions including tests for normality, equal variance and symmetry. By default is set to .
 "Normality" verify that all data is normally distributed
Test whether the mean of a population is zero:
The full test table:
Test whether the means of two dependent populations differ:
The mean of the differences:
At the level the mean of the differenced data is not significantly different from 0:
Compare the locations of dependent multivariate populations:
The mean of the differences:
At the level the mean of the differenced data is not significantly different from 0:
Test whether the mean of a population is zero:
 Out[2]=
The full test table:
 Out[3]=

Test whether the means of two dependent populations differ:
The mean of the differences:
 Out[2]=
 Out[3]=
At the level the mean of the differenced data is not significantly different from 0:
 Out[4]=

Compare the locations of dependent multivariate populations:
The mean of the differences:
 Out[2]=
 Out[3]=
At the level the mean of the differenced data is not significantly different from 0:
 Out[4]=
 Scope   (13)
Test versus :
The -values are typically large when the mean is close to :
The -values are typically small when the location is far from :
Using Automatic is equivalent to testing for a mean of zero:
Test versus :
The -values are typically large when the mean is close to :
The -values are typically small when the location is far from :
Test whether the mean vector of a multivariate population is the zero vector:
Alternatively test against :
Test whether the mean of differenced datasets is zero:
The -values are generally small when the mean is not zero:
The -values are generally large when the mean is zero:
Test whether the mean of differenced data is 3:
The order of the datasets affects the test results:
Test whether the mean vector of differenced multivariate data is the zero vector:
Alternatively test against :
Create a HypothesisTestData object for repeated property extraction:
The properties available for extraction:
Extract some properties from a HypothesisTestData object:
The -value, test statistic, and degrees of freedom:
Extract any number of properties simultaneously:
The -value, test statistic, and degrees of freedom:
Tabulate the test results:
Retrieve the entries from a test table for customized reporting:
Tabulate -values or test statistics:
The -value from the table:
The test statistic from the table:
 Options   (8)
A two-sided test is performed by default:
Test versus :
Perform a two-sided test or a one-sided alternative:
Test versus :
Test versus :
Test versus :
Perform tests with one-sided alternatives when is given:
Test versus :
Test versus :
Set the significance level for diagnostic tests:
By default, is used:
The significance level is also used for and :
By default normality is tested:
Here normality is assumed:
The result is the same but a warning is issued:
Alternatively, use All:
Bypassing diagnostic tests can save compute time:
It is often useful to bypass diagnostic tests for simulation purposes:
The assumptions of the test hold by design, so a great deal of time can be saved:
The results are identical:
 Applications   (1)
A group of 15 students is convinced that they were more prepared for the mathematics portion of the ACT than that of the SAT. Test this claim assuming that national ACT and SAT scores are normally distributed with means 21.7 and 528.5 and standard deviations 4.1 and 117.2, respectively:
For comparison, the data should be normalized:
Density estimates for the individual and differenced scores:
can be rejected at the 5% level. The students' claim is not refuted:
PairedTTest is equivalent to a TTest for a single dataset:
For two datasets the PairedTTest is equivalent to a TTest of the paired differences:
If the variance of the population is known, the more powerful PairedZTest can be used:
The probability that the PairedZTest will return a -value smaller than the PairedTTest:
If the data can be paired, the PairedTTest is more powerful than the TTest:
The paired test detects a significant difference where the unpaired test does not:
PairedTTest requires that the data be normally distributed: