Algorithms (One-Sample T-Test)

A one sample t-test can be employed to test whether or not the true mean of a population, \mu\,\!, is equal to or different than a specified test mean, \mu_0\,\!,. The test can either be one- or two-tailed, and the hypotheses take the form:

H_0:\mu=\mu_0 , vs H_1:\mu \ne \mu_0, two tailed

H_0:\mu \le \mu_0 vs H_1:\mu > \mu_0, upper tailed

H_0:\mu \ge \mu_0 vs H_1:\mu < \mu_0, lower tailed

Test Statistics

Let X(x_1,x_2,...,x_n)\,\! be the input dataset, the statistic t-value has a Student's t-distribution with (n-1) degrees of freedom, which computed as:


Where \bar{x}=\frac{1}{n}\sum_{i=1}^n x_i and s=\sqrt{\sum_{i=1}^n \frac{(x_i-\bar{x})^2}{n-1}}

For a given significance level, \alpha\,\!, we will reject the null hypothesis H_0\,\!, when:

|t|>t_{\alpha/2}\,\!, for two tailed test

t>t_{\alpha}\,\!, for upper tailed test

t<-t_{\alpha}\,\!, for lower tailed test

where t_\alpha\,\! is the critical value from t distribution indexed at \alpha\,\! level by (n-1) degrees of freedom. For a better way to express the probability of hypothesis, the P-Value is also reported. And we can reject the null hypothesis H_0\,\! if p < \alpha\,\!. The P-value for t-statistic is associated with the incomplete beta function:


where I_x(\alpha,\beta)=\frac{I'(\alpha+\beta)}{I'(\alpha)\cdot I'(\beta)} \int_0^x t^{\alpha-1}(1-t)^{\beta-1}\,dt

Confidence Intervals

For the specified significance level, the confidence interval for the sample mean is:

Null Hypothesis Confidence Interval
H_0:\mu=\mu_0\,\! \left[\bar{x}-t_{n-1,\alpha/2}\frac{s}{\sqrt{n}},\bar{x}+t_{n-1,\alpha/2}\frac{s}{\sqrt{n}}\right]
H_0:\mu \le \mu_0 \left[\bar{x}-t_{n-1,\alpha}\frac{s}{\sqrt{n}}, \infty\right]
H_0:\mu \ge \mu_0 \left[-\infty, \bar{x}+t_{n-1,\alpha}\frac{s}{\sqrt{n}}\right]

Power Analysis

The power of a one sample t-test is a measurement of its sensitivity. In terms of the null and alternative hypotheses, power is the probability that the test statistic T will be extreme enough to allow the rejection of the null hypothesis when it should in fact be rejected (i.e. given the null hypothesis is not true). For each of the three different null hypotheses, power is mathematically defined below:

Null Hypothesis Power
H_0:\mu=\mu_0 1-P \left\{T \le t_{1-\alpha/2}(n-1)-t\right\}+P\left\{T<t_{\alpha/2}(n-1)-t \right\}
H_0:\mu \le \mu_0 1-P \left\{T \le t_{1-\alpha}(n-1)-t\right\}
H_0:\mu \ge \mu_0 P \left\{T \le t_{\alpha}(n-1)-t\right\}

where T is a random variable having a t-distribution with (n-1) degrees of freedom. The computation for hypothetical power is the same as for actual power except that the test statistic t , the critical value and degree of freedom are recomputed using hypothetical sample sizes instead of the actual sample size.