/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:
The two sample t-test calculates a Student's t statistic and the associated probability to test whether or not the difference of the two sample means equals to (i.e. to test whether or not their means are equal, you can just test whether or not their difference is 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
Consider two independent samples/math-d60f5062564e1ece65993038b62484fa.png?v=0 "x_1\,\!")
and/math-9defdb27049cde0b5bdfd4e762a02d6b.png?v=0 "x_2\,\!")
, of size /math-8d88a43872d6db6535d8672a15f09ce2.png?v=0 "n_1\,\!")
and /math-2f6fac59fce80e76d698bcf5ea77bab6.png?v=0 "n_2\,\!")
drawn from two normal population with means /math-4ed789d5f6cd3fea302e306da621cb83.png?v=0 "\mu_1\,\!")
and /math-917bb8defee48f460766703f23fc0822.png?v=0 "\mu_2\,\!")
, and variances /math-0c153418ac24896b413d1c3c8ad3e95c.png?v=0 "\sigma_1^2\,\!")
and /math-f94e0c3605e0ebe3aa890455f27416e5.png?v=0 "\sigma_2^2\,\!")
respectively, we have:
/math-6dc73b5670604b65621d1ad70209c678.png?v=0 "\bar{x}_1=\frac{1}{n_1}\sum_{j=1}^{n_1}x_{1j}")
, /math-06a309ba82fba50c69561e78b4ffdf7f.png?v=0 "\bar{x}_2=\frac{1}{n_2}\sum_{j=1}^{n_2}x_{2j}")
,
/math-06931d63b90e0e6a6d7d81d64e149e48.png?v=0 "s_1^2=\frac{1}{n_1-1}\sum_{j=1}^{n_1}{(x_{1j}-\bar{x}_1)^2}")
, /math-1ea85773f946cf6770cf4f76022e133d.png?v=0 "s_2^2=\frac{1}{n_2-1}\sum_{j=1}^{n_2}{(x_{2j}-\bar{x}_2)^2}")
where /math-875d0b783f9b167155a8f54a2c3f9e6a.png?v=0 "\bar{x}_1\,\!")
and/math-68dc9b353bc272e9d32196f5e30bff4e.png?v=0 "\bar{x}_2\,\!")
are sample means and /math-f963d3c66aa9a89e64a6db3d0dd63351.png?v=0 "s_1^2\,\!")
and /math-8a65733320d52c6da4efeeb5888dba00.png?v=0 "s_2^2\,\!")
are sample variances. Then we can compute the t test statistic by:
For equal variance is assumed, that is /math-3728dc0e9ed8d4ea6c77089a7c1bb2fb.png?v=0 "\sigma_1^2=\sigma_2^2\,\!")
:
In this case the test statistic t:
/math-8992dd1682d9d05a300c39478ada35f5.png?v=0 "t=\frac{(\bar{x}_1-\bar{x}_2)-\mu_d}{s_p\sqrt{(1/n_1+1/n_2)}}")
has a t-distribution with /math-6350a1d09d381cb5b9c2420a49cec064.png?v=0 "(v = n_1+n_2-2)")
degrees of freedom and
/math-49682ab10edeb0f5ed248d7195b85d78.png?v=0 "s_p=\sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}}")
is the pooled variance of the two samples.
For equal variance is not assumed:
In this case the usual two sample t-statistic no longer has a t-distribution and an approximate test statistic, t'is used:
/math-2f3931fdb28db4a39f9b7140cc965c81.png?v=0 "t'=\frac{(\bar{x}_1-\bar{x}_2)-\mu_d}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}")
And a t-distribution with v degrees of freedom is used to approximate the distribution of t'where
/math-fdd079fbba82fc1bb066171f062bab57.png?v=0 "v=\frac{(s_1^2/n_1+s_2^2/n_2)^2}{\frac{(s_1^2/n_1)^2}{n_1-1}+\frac{(s_2^2/n_2)^2}{n_2-1}}")
Then 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-350fad5bdd3c28b92d7b6a076336a463.png?v=0 "p < \mu\,\!")
.
The upper and lower /math-45828455265148a331fe94e5c5175f93.png?v=0 "(1-\sigma )\times 100\%")
confident limits for mean difference /math-32a17f0051e09406b2f689fa5f0bb2c5.png?v=0 "(\mu_1 - \mu_2)\,\!")
are calculated as:
For equal variance is assumed:
| Null Hypothesis | Confidence Interval |
|---|---|
|
|
![\left[(\bar{x}_1-\bar{x}_2)- t_{\alpha/2}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}, (\bar{x}_1-\bar{x}_2)+ t_{\alpha/2}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\right]](/origin-help/en/images/Algorithm_(TwoSampletTest)/math-35e508ffa325356798bfeb78b55c5162.png?v=0 "\left[(\bar{x}_1-\bar{x}_2)- t_{\alpha/2}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}, (\bar{x}_1-\bar{x}_2)+ t_{\alpha/2}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\right]")
|
|
|
![\left[(\bar{x}_1-\bar{x}_2)- t_{\alpha}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}, \infty\right]](/origin-help/en/images/Algorithm_(TwoSampletTest)/math-438038a816d1733a3f4947f61e539c61.png?v=0 "\left[(\bar{x}_1-\bar{x}_2)- t_{\alpha}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}, \infty\right]")
|
|
|
![\left[-\infty, (\bar{x}_1-\bar{x}_2)+ t_{\alpha}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\right]](/origin-help/en/images/Algorithm_(TwoSampletTest)/math-043e91f03e23cf4cb47273b881ee6f02.png?v=0 "\left[-\infty, (\bar{x}_1-\bar{x}_2)+ t_{\alpha}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\right]")
|
For equal variance is not assumed:
| Null Hypothesis | Confidence Interval |
|---|---|
|
|
![\left[(\bar{x}_1-\bar{x}_2)- t_{\alpha/2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}, (\bar{x}_1-\bar{x}_2)+ t_{\alpha/2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\right]](/origin-help/en/images/Algorithm_(TwoSampletTest)/math-1dc3a8412c3e70c868d0def30834cdfd.png?v=0 "\left[(\bar{x}_1-\bar{x}_2)- t_{\alpha/2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}, (\bar{x}_1-\bar{x}_2)+ t_{\alpha/2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\right]")
|
|
|
![\left[(\bar{x}_1-\bar{x}_2)- t_{\alpha}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}, \infty\right]](/origin-help/en/images/Algorithm_(TwoSampletTest)/math-6317c80637d203c8b436ce4d361f825c.png?v=0 "\left[(\bar{x}_1-\bar{x}_2)- t_{\alpha}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}, \infty\right]")
|
|
|
![\left[-\infty, (\bar{x}_1-\bar{x}_2)+ t_{\alpha}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\right]](/origin-help/en/images/Algorithm_(TwoSampletTest)/math-39eb7689ff8eaf4fc97f74a82869d201.png?v=0 "\left[-\infty, (\bar{x}_1-\bar{x}_2)+ t_{\alpha}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\right]")
|
where /math-f1004b741c7a8310ad2a3b7c0d726855.png?v=0 "t_{\sigma/2}\,\!")
is the critical value of the t-distribution with v degrees of freedom.
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.
The two-sample t-test is implemented with a Nag function, nag_2_sample_t_test (g07cac). Please refer to the corresponding Nag document for more details on the algorithm.