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Statistics

 TwoSamplePairedTTest
 apply the paired t-test for population means

 Calling Sequence TwoSamplePairedTTest(X1, X2, beta, options)

Parameters

 X1 - first data sample X2 - second data sample beta - realcons; the test value for the difference between the two means options - (optional) equation(s) of the form option=value where option is one of alternative, confidence, ignore, output, summarize, weights1 or weights2; specify options for the TwoSamplePairedTTest function

Description

 • The TwoSamplePairedTTest function computes the paired t-test on datasets X1 and X2 (which have been constructed by paired sampling).  This calculation is used to determine the significance of the difference between sample means and an assumed difference in population means when the standard deviation of the population is unknown.
 • The first parameter X1 is the first data sample to use in the analysis.
 • The second parameter X2 is the second data sample to use in the analysis.
 • The third parameter beta is the assumed difference in population means (assumed population mean of X1 minus the assumed population mean of X2), specified as a real constant.

Options

 The options argument can contain one or more of the options shown below.
 • alternative='twotailed', 'lowertail', or 'uppertail'
 This option is used to specify the type or interval used in the analysis, or similarly, the alternative hypothesis to consider when performing the analysis.
 • confidence=float
 This option is used to specify the confidence level of the interval and must be a floating-point value between 0 and 1.  By default this is set to 0.95.
 • ignore=truefalse
 This option is used to specify how to handle non-numeric data. If ignore is set to true all non-numeric items in the data samples will be ignored.
 • output='report', 'statistic', 'pvalue', 'confidenceinterval', 'distribution', 'hypothesis', or list('statistic', 'pvalue', 'confidenceinterval', 'distribution', 'hypothesis')
 This option is used to specify the desired format of the output from the function.  If 'report' is specified then a module containing all output from this test is returned.  If a single parameter name is specified other than 'report' then that quantity alone is returned.  If a list of parameter names is specified then a list containing those quantities in the specified order will be returned.
 • summarize= 'true', 'false', 'embed'
 This option controls the display of a printed or embedded summary for the hypothesis test. Unlike the output option, the displayed summary is not assignable output.
 • weights=rtable
 Vector of weights (one-dimensional rtable). If these weights are given, the TwoSamplePairedTTest function will scale each data point in X1 and X2 to have given weight. Note that the weights provided must have type realcons and the results are floating-point, even if the problem is specified with exact values. Both the data array and the weights array must have the same number of elements.

Notes

 • This test generates a complete report of all calculations in the form of a userinfo message.  In order to access this report, specify infolevel[Statistics] := 1 or use the summarize option.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$

Specify the data sample.

 > $X≔\mathrm{Array}\left(\left[9,10,8,4,8,3,0,10,15,9\right]\right):$
 > $Z≔\mathrm{Array}\left(\left[10,11,7,3,10,5,2,12,14,10\right]\right):$
 > $\mathrm{Mean}\left(X\right)-\mathrm{Mean}\left(Z\right)$
 ${-0.800000000000001}$ (1)

Calculate the paired t-test on an array of values.

 > $\mathrm{TwoSamplePairedTTest}\left(X,Z,1,\mathrm{confidence}=0.95,\mathrm{summarize}=\mathrm{embed}\right):$

Null Hypothesis:

Sample drawn from populations with difference of means equal to 1

Alternative Hypothesis:

Sample drawn from population with difference of means not equal to 1

 Sample Size Difference in  Means Difference Standard Deviation Distribution Computed Statistic Computed p-value Confidence Interval ${10.}$ ${-0.800000}$ ${1.31656}$ ${\mathrm{StudentT}}{}\left({9}\right)$ ${-4.32346}$ ${0.00192341}$ ${-1.74181}{..}{0.141811}$

Result:

Rejected: This statistical test provides evidence that the null hypothesis is false.

Calculate the lower tail paired t-test.

 > $\mathrm{TwoSamplePairedTTest}\left(X,Z,1,\mathrm{confidence}=0.95,\mathrm{alternative}='\mathrm{lowertail}',\mathrm{summarize}=\mathrm{true}\right)$
 Standard T-Test with Paired Samples ----------------------------------- Null Hypothesis: Sample drawn from populations with difference of means greater than 1 Alt. Hypothesis: Sample drawn from population with difference of means less than 1   Sample Size:             10 Difference in Means:     -0.8 Difference Std. Dev.:    1.31656 Distribution:            StudentT(9) Computed Statistic:      -4.32346015250714 Computed p-value:        .000961705861738372 Confidence Interval:     -infinity .. -.0368142919477125                          (difference of population means)   Result: [Rejected] This statistical test provides evidence that the null hypothesis is false.
 ${\mathrm{hypothesis}}{=}{\mathrm{false}}{,}{\mathrm{confidenceinterval}}{=}{-}{\mathrm{\infty }}{..}{-0.0368142919477125}{,}{\mathrm{distribution}}{=}{\mathrm{StudentT}}{}\left({9}\right){,}{\mathrm{pvalue}}{=}{0.000961705861738372}{,}{\mathrm{statistic}}{=}{-4.32346015250714}$ (2)

As an alternative to using the summarize option, setting infolevel[Statistics] := 1 also returns the printed summary.

 > ${\mathrm{infolevel}}_{\mathrm{Statistics}}≔1:$

Calculate the upper tail paired t-test.

 > $\mathrm{TwoSamplePairedTTest}\left(X,Z,1,\mathrm{confidence}=0.95,\mathrm{alternative}='\mathrm{uppertail}'\right)$
 Standard T-Test with Paired Samples ----------------------------------- Null Hypothesis: Sample drawn from populations with difference of means less than 1 Alt. Hypothesis: Sample drawn from population with difference of means greater than 1   Sample Size:             10 Difference in Means:     -0.8 Difference Std. Dev.:    1.31656 Distribution:            StudentT(9) Computed Statistic:      -4.32346015250714 Computed p-value:        .999038294138262 Confidence Interval:     -1.56318570805229 .. infinity                          (difference of population means)   Result: [Accepted] This statistical test does not provide enough evidence to conclude that the null hypothesis is false.
 ${\mathrm{hypothesis}}{=}{\mathrm{true}}{,}{\mathrm{confidenceinterval}}{=}{-1.56318570805229}{..}{\mathrm{\infty }}{,}{\mathrm{distribution}}{=}{\mathrm{StudentT}}{}\left({9}\right){,}{\mathrm{pvalue}}{=}{0.999038294138262}{,}{\mathrm{statistic}}{=}{-4.32346015250714}$ (3)
 > 

References

 Kanji, Gopal K. 100 Statistical Tests. London: SAGE Publications Ltd., 1994.
 Sheskin, David J. Handbook of Parametric and Nonparametric Statistical Procedures. London: CRC Press, 1997.

Compatibility

 • The Statistics[TwoSamplePairedTTest] command was updated in Maple 2016.
 • The summarize option was introduced in Maple 2016.