Tolerance Factors for Normal Distributions

Sumber:
CRC Standard Probability and Statistics Tables and Formule, pp. 175-177, Daniel Zwillinger and Stephen Kokoska.

Suppose $X_1, X_2, \ldots, X_n$ is a random sample of size $n$ from a normal distribution with mean $\mu$ and standar deviation $\sigma.$ Using the summary statistics $\bar{x}$ and $s,$ a tolerance interval $[L,U]$ may be constructed to capture $100P\%$ of the population with probability $1-\alpha.$ The following procedures may be used.

1. Two-sided tolerance interval: A $100(1-\alpha)\%$ tolerance interval that captures $100P\%$ of the population has as endpoints

$[L,U] = \bar{x}\pm K_{\alpha,n,P}\cdot s$

2. One-sided tolerance interval, upper tailed: A $100(1-\alpha)\%$ tolerance interval bounded below has

$L = \bar{x} - k_{\alpha,n,P}\cdot s \qquad U = \infty$

3. One-sided tolerance interval, lower tailed: A $100(1-\alpha)\%$ tolerance interval bounded above has

$L = -\infty \qquad U = \bar{x} + k_{\alpha,n,P} \cdot s$

where $K_{\alpha,n,P}$ is the tolerance factor given in section 7.3.1 and $k_{\alpha,n,P}$ is computed using the formula below.

Values of $K_{\alpha,n,P}$ are given in section 7.3.1 for $P = 0.75, 0.90, 0.95, 0.99, 0.999, \alpha = 0.75, 0.90, 0.95, 0.99,$ and various values of $n.$

The value of $k_{\alpha,n,P}$ is given by

$k_{\alpha,n,P} = \frac{z_{1-P} + \sqrt{(z_{1-P})^2-ab}}{a}$

$a = 1 - \frac{(z_\alpha)^2}{2(n-1)}$

$b= z_{1-P}^2 - \frac{z_\alpha^2}{n}$

where $z_{1-P}$ and $z_\alpha$ are critical values for a standard normal random variable (see page 175).

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