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Tolerance interval
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==Examples== <ref name=vardeman/> gives the following example: <blockquote>So consider once again a proverbial [[United States Environmental Protection Agency|EPA]] [[Fuel economy in automobiles|mileage]] test scenario, in which several nominally identical autos of a particular model are tested to produce mileage figures <math>y_1, y_2, ..., y_n</math>. If such data are processed to produce a 95% confidence interval for the mean mileage of the model, it is, for example, possible to use it to project the mean or total gasoline consumption for the manufactured fleet of such autos over their first 5,000 miles of use. Such an interval, would however, not be of much help to a person renting one of these cars and wondering whether the (full) 10-gallon tank of gas will suffice to carry him the 350 miles to his destination. For that job, a prediction interval would be much more useful. (Consider the differing implications of being "95% sure" that <math>\mu \ge 35</math> as opposed to being "95% sure" that <math>y_{n+1} \ge 35</math>.) But neither a confidence interval for <math>\mu</math> nor a prediction interval for a single additional mileage is exactly what is needed by a design engineer charged with determining how large a gas tank the model really needs to guarantee that 99% of the autos produced will have a 400-mile cruising range. What the engineer really needs is a tolerance interval for a fraction <math>p = .99</math> of mileages of such autos.</blockquote> Another example is given by:<ref name=Krishnamoorthy/> <blockquote>The air lead levels were collected from <math>n=15</math> different areas within the facility. It was noted that the log-transformed lead levels fitted a normal distribution well (that is, the data are from a [[lognormal distribution]]. Let <math>\mu</math> and <math>\sigma^2</math>, respectively, denote the population mean and variance for the log-transformed data. If <math>X</math> denotes the corresponding random variable, we thus have <math>X \sim \mathcal{N}(\mu, \sigma^2)</math>. We note that <math>\exp(\mu)</math> is the median air lead level. A confidence interval for <math>\mu</math> can be constructed the usual way, based on the [[Student's t-distribution|''t''-distribution]]; this in turn will provide a confidence interval for the median air lead level. If <math>\bar{X}</math> and <math>S</math> denote the sample mean and standard deviation of the log-transformed data for a sample of size n, a 95% confidence interval for <math>\mu</math> is given by <math>\bar{X} \pm t_{n-1,0.975} S / \sqrt{n}</math>, where <math>t_{m,1-\alpha}</math> denotes the <math>1-\alpha</math> quantile of a [[Student's t-distribution|''t''-distribution]] with <math>m</math> degrees of freedom. It may also be of interest to derive a 95% upper confidence bound for the median air lead level. Such a bound for <math>\mu</math> is given by <math>\bar{X} + t_{n-1,0.95} S / \sqrt{n}</math>. Consequently, a 95% upper confidence bound for the median air lead is given by <math>\exp{\left( \bar{X} + t_{n-1,0.95} S / \sqrt{n} \right)}</math>. Now suppose we want to predict the air lead level at a particular area within the laboratory. A 95% upper prediction limit for the log-transformed lead level is given by <math>\bar{X} + t_{n-1,0.95} S \sqrt{\left( 1 + 1/n \right)}</math>. A two-sided prediction interval can be similarly computed. The meaning and interpretation of these intervals are well known. For example, if the confidence interval <math>\bar{X} \pm t_{n-1,0.975} S / \sqrt{n}</math> is computed repeatedly from independent samples, 95% of the intervals so computed will include the true value of <math>\mu</math>, in the long run. In other words, the interval is meant to provide information concerning the parameter <math>\mu</math> only. A prediction interval has a similar interpretation, and is meant to provide information concerning a single lead level only. Now suppose we want to use the sample to conclude whether or not at least 95% of the population lead levels are below a threshold. The confidence interval and prediction interval cannot answer this question, since the confidence interval is only for the median lead level, and the prediction interval is only for a single lead level. What is required is a tolerance interval; more specifically, an upper tolerance limit. The upper tolerance limit is to be computed subject to the condition that at least 95% of the population lead levels is below the limit, with a certain confidence level, say 99%.</blockquote>
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