T Distribution Confidence Interval

data

confidence

Statistics

sample size

$$n$$

5

sample mean

$$\bar X = \frac{1}{n} \sum_j X_j$$

0.0

sample variance

$$S^2 = \frac{1}{n-1} \sum_j (X_j - \bar X)^2$$

1.0

degrees of freedom

$$\nu = n - 1$$

0

$$T_\nu = \frac{\bar X - \mu}{\sqrt{S^2 / n}}$$
$$f(z) = \frac{\Gamma \left( \frac{\nu+1}{2} \right)}{\sqrt{\nu \pi} \ \Gamma \left( \frac{\nu}{2} \right)} \ \left( 1 + \frac{z^2}{\nu} \right)^{-\frac{\nu+1}{2}}$$

confidence

$$1 - \alpha$$

0.95

t value

$$t_0: {\cal P} (-t_0 < T_\nu < t_0) = 1-\alpha$$

2.0

lower bound

$$a = \bar X - \sqrt{\frac{S^2}{n}} \ t_0$$

-2.0

upper bound

$$b = \bar X + \sqrt{\frac{S^2}{n}} \ t_0$$

2.0