Editing Interval estimation (section)

==== Tolerance ====
{{main|Tolerance interval}}
Tolerance intervals use collected data set population to obtain an interval, within tolerance limits, containing 100γ% values. Examples typically used to describe tolerance intervals include manufacturing. In this context, a percentage of an existing product set is evaluated to ensure that a percentage of the population is included within tolerance limits. When creating tolerance intervals, the bounds can be written in terms of an upper and lower tolerance limit, utilizing the sample [[mean]], <math>\mu</math>, and the sample [[standard deviation]], s.

:<math>(l_b, u_b) = \mu \pm k_2s</math> for two-sided intervals

for two-sided intervals

And in the case of one-sided intervals where the tolerance is required only above or below a critical value,

:<math>l_{b} = \mu - k_{1}s</math>

:<math>u_{b}=\mu + k_{1} s</math>

<math>k_i</math> varies by distribution and the number of sides, i, in the interval estimate. In a normal distribution, <math>k_2</math> can be expressed as <ref>{{Cite journal |last=Howe |first=W. G. |date=June 1969 |title=Two-Sided Tolerance Limits for Normal Populations, Some Improvements |url=http://dx.doi.org/10.2307/2283644 |journal=Journal of the American Statistical Association |volume=64 |issue=326 |pages=610 |doi=10.2307/2283644 |issn=0162-1459|url-access=subscription }}</ref>

:<math>k_2 = z_{\alpha/2}\sqrt{\frac{\nu(1+\frac{1}{N})}{\chi_{1-\alpha,\nu}^2}}</math>

Where,

:<math>\chi _{1-\alpha,\nu}^2</math> is the critical value of the chi-square distribution utilizing <math>\nu</math> degrees of freedom that is exceeded with probability <math>\alpha</math>.

<math> z_{\alpha/2}</math> is the critical values obtained from the normal distribution.