17.3.3.2 Algorithms (Pair-Sample T-Test)tTest-PairSample-Algorithm
This function is used to test whether or not the difference of the two paired sample means equals to /math-12f3be74f7266e038bdf61eb68a6208c.png?v=0 "\mu_d\,\!") (i.e. to test whether or not their means are equal, you can just test whether or not their difference is 0, /math-590eba6400c9e4df07e0c234f91e4460.png?v=0 "\mu_1-\mu_2=\mu_d=0\,\!") ). And the hypotheses take the form:
/math-f67a81ef315f0c59070ef53cb50b3932.png?v=0 "H_0:\mu_1-\mu_2=\mu_d\,\!") vs /math-5c09300d675833793616502094acaa0f.png?v=0 "H_1:\mu_1-\mu_2 \ne \mu_d") Two Tailed
/math-1fc800dcea3acfe50f9e90c310b1174a.png?v=0 "H_0:\mu_1-\mu_2 \le \mu_d") vs /math-0fbc9ae55928f128a94afcf65c2c2e39.png?v=0 "H_1:\mu_1-\mu_2 > \mu_d") Upper Tailed
/math-8d8812a65564ffde40d299d450d3de7f.png?v=0 "H_0:\mu_1-\mu_2 \ge \mu_d") vs /math-9a968e5033dd7110b7973529fa35c85d.png?v=0 "H_1:\mu_1-\mu_2 < \mu_d") Lower Tailed
Test Statistics
Consider two samples /math-d60f5062564e1ece65993038b62484fa.png?v=0 "x_1\,\!") and/math-9defdb27049cde0b5bdfd4e762a02d6b.png?v=0 "x_2\,\!") which assumed to be drawn from normal populations are equal size, we can define the paired difference as:
/math-fa4b74855447f3d57010bcf1090d05b2.png?v=0 "d_j=x_{1j}-x_{2j},for(j=1,2,...,n)\,\!")
And we have the mean paired difference is:
/math-2239a6e81d4d585208f48542fcff6792.png?v=0 "\bar{d}=\frac{1}{n}\sum_{i=1}^n d_i")
Then we can compute the standard deviation for the difference between paired data points /math-8d0320a6bce227662ff1de19b22c6e25.png?v=0 "s_d\,\!") , with v = n-1 degrees of freedom as:
/math-06d940fc729cf5c8ba1315ebb8ca1ab7.png?v=0 "s_d=\sqrt{\frac{1}{n-1}\sum_{i=1}^n(d_i-\bar{d})}")
And then we can calculate the test statistic by:
/math-a7cc51688366cbdafbc493674976b146.png?v=0 "t=\frac{\bar{d}-\mu_d}{\frac{s_d}{\sqrt{n}}}")
Compare the t value with the critical value and we will reject /math-806277203dedea2ed8321f6cbd465a54.png?v=0 "H_0\,\!") if:
Two tailed test: /math-dc78eff8d710d926c9d712cebdb408d0.png?v=0 "|t| > t_{\sigma/2}\,\!") ;
Upper tailed test: /math-b3c71dc9421063ffaf6de1641540a5d5.png?v=0 "t > t_\sigma\,\!") ;
Lower tailed test: /math-b3c0361ff553afb248a469861134f399.png?v=0 "t < -t_\sigma\,\!") ;
The p-value will also be compared with a user-defined significance level,,/math-3ac6004d77c0cc0055e95c99b9dfd7e0.png?v=0 "\sigma\,\!") , which commonly 0.05 is used. And the null hypothesis will be rejected if /math-d797c315e459e13f929c1778f48760a9.png?v=0 "p < \sigma\,\!") .
Confidence Intervals
The confidence interval for paired sample mean difference /math-32a17f0051e09406b2f689fa5f0bb2c5.png?v=0 "(\mu_1 - \mu_2)\,\!") is:
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Confidence Interval
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![\left[\bar{d} - t_{\alpha/2}\frac{s_d}{\sqrt{n}}, \bar{d} + t_{\alpha/2}\frac{s_d}{\sqrt{n}}\right]](//d2mvzyuse3lwjc.cloudfront.net/doc/en/UserGuide/images/Algorithm_(PairSampletTest)/math-8d10700cb7317b9442a5fcc4d14e79cd.png?v=0 "\left[\bar{d} - t_{\alpha/2}\frac{s_d}{\sqrt{n}}, \bar{d} + t_{\alpha/2}\frac{s_d}{\sqrt{n}}\right]")
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![\left[\bar{d} - t_{\alpha}\frac{s_d}{\sqrt{n}}, \infty\right]](//d2mvzyuse3lwjc.cloudfront.net/doc/en/UserGuide/images/Algorithm_(PairSampletTest)/math-7b92fed8a3f4aabcf278451dad26b32b.png?v=0 "\left[\bar{d} - t_{\alpha}\frac{s_d}{\sqrt{n}}, \infty\right]")
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![\left[-\infty, \bar{d} + t_{\alpha}\frac{s_d}{\sqrt{n}}\right]](//d2mvzyuse3lwjc.cloudfront.net/doc/en/UserGuide/images/Algorithm_(PairSampletTest)/math-5439571d23c3fe70adf8527285fdc5f8.png?v=0 "\left[-\infty, \bar{d} + t_{\alpha}\frac{s_d}{\sqrt{n}}\right]")
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Power Analysis
The power of a two sample t-test is a measurement of its sensitivity. Detail algorithm about calculating power please read the help of Power and Sample Size.
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