Normal 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

$$Z \approx \frac{\bar X - \mu}{\sqrt{S^2 / n}}$$
$$f(z) = \frac{1}{\sqrt{2 \pi}} \ \exp \left( -\frac{z^2}{2} \right)$$

confidence

$$1 - \alpha$$

0.95

z value

$$z_0: {\cal P} (-z_0 < Z < z_0) = 1-\alpha$$

2.0

lower bound

$$a = \bar X - \sqrt{\frac{S^2}{n}} \ z_0$$

-2.0

upper bound

$$b = \bar X + \sqrt{\frac{S^2}{n}} \ z_0$$

2.0