Pontryagin's minimum principle

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A tolerance interval is a statistical interval within which, with some confidence level, a specified proportion of a sampled population falls. "More specifically, a 100×p%/100×(1−α) tolerance interval provides limits within which at least a certain proportion (p) of the population falls with a given level of confidence (1−α)."[1] "A (p, 1−α) tolerance interval (TI) based on a sample is constructed so that it would include at least a proportion p of the sampled population with confidence 1−α; such a TI is usually referred to as p-content − (1−α) coverage TI."[2] "A (p, 1−α) upper tolerance limit (TL) is simply an 1−α upper confidence limit for the 100 p percentile of the population."[2]

A tolerance interval can be seen as a statistical version of a probability interval. "In the parameters-known case, a 95% tolerance interval and a 95% prediction interval are the same."[3] If we knew a population's exact parameters, we would be able to compute a range within which a certain proportion of the population falls. For example, if we know a population is normally distributed with mean μ and standard deviation σ, then the interval μ±1.96σ includes 95% of the population (1.96 is the z-score for 95% coverage of a normally distributed population).

However, if we have only a sample from the population, we know only the sample mean μ^ and sample standard deviation σ^, which are only estimates of the true parameters. In that case, μ^±1.96σ^ will not necessarily include 95% of the population, due to variance in these estimates. A tolerance interval bounds this variance by introducing a confidence level γ, which is the confidence with which this interval actually includes the specified proportion of the population. For a normally distributed population, a z-score can be transformed into a "k factor" or tolerance factor[4] for a given γ via lookup tables or several approximation formulas.[5] "As the degrees of freedom approach infinity, the prediction and tolerance intervals become equal."[6]

Relation to other intervals

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. The tolerance interval is less widely known than the confidence interval and prediction interval, a situation some educators have lamented, as it can lead to misuse of the other intervals where a tolerance interval is more appropriate.[7][8]

The tolerance interval differs from a confidence interval in that the confidence interval bounds a single-valued population parameter (the mean or the variance, for example) with some confidence, while the tolerance interval bounds the range of data values that includes a specific proportion of the population. Whereas a confidence interval's size is entirely due to sampling error, and will approach a zero-width interval at the true population parameter as sample size increases, a tolerance interval's size is due partly to sampling error and partly to actual variance in the population, and will approach the population's probability interval as sample size increases.[7][8]

The tolerance interval is related to a prediction interval in that both put bounds on variation in future samples. The prediction interval only bounds a single future sample, however, whereas a tolerance interval bounds the entire population (equivalently, an arbitrary sequence of future samples). In other words, a prediction interval covers a specified proportion of a population on average, whereas a tolerance interval covers it with a certain confidence level, making the tolerance interval more appropriate if a single interval is intended to bound multiple future samples.[8][9]

Examples

[7] gives the following example:

So consider once again a proverbial EPA mileage test scenario, in which several nominally identical autos of a particular model are tested to produce mileage figures

y1,y2,...,yn

. 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

μ35

as opposed to being "95% sure" that

yn+135

.) But neither a confidence interval for

μ

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

p=.99

of mileages of such autos.

Another example is given by:[9]

The air lead levels were collected from n=15 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 mu and sigma^2, respectively, denote the population mean and variance for the log-transformed data. If X denotes the corresponding random variable, we thus have X ~ N(mu, sigma^2). We note that exp(mu) is the median air lead level. A confidence interval for mu can be constructed the usual way, based on the t-distribution; this in turn will provide a confidence interval for the median air lead level. If

X¯

and S denote the sample mean and standard deviation of the log-transformed data for a sample of size n, a 95% confidence interval for mu is given by

X¯±tn1,0.975S/(n)

, where

tm,1α

denotes the 1-alpha quantile of a t-distribution with m 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 mu is given by

X¯+tn1,0.95S/n

. Consequently, a 95% upper confidence bound for the median air lead is given by

exp(X¯+tn1,0.95S/n)

. 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

X¯+tn1,0.95S(1+1/n)

. 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

X¯±tn1,0.975S/n

is computed repeatedly from independent samples, 95% of the intervals so computed will include the true value of mu, in the long run. In other words, the interval is meant to provide information concerning the parameter mu 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%.

Further reading

  • "Preliminaries", Chap. 1 in;[9] available at [1]
  • [10]

References

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  1. D. S. Young (2010), Book Reviews: "Statistical Tolerance Regions: Theory, Applications, and Computation", TECHNOMETRICS, FEBRUARY 2010, VOL. 52, NO. 1, pp.143-144.
  2. 2.0 2.1 Krishnamoorthy, K. and Lian, Xiaodong(2011) 'Closed-form approximate tolerance intervals for some general linear models and comparison studies', Journal of Statistical Computation and Simulation,, First published on: 13 June 2011 doi:10.1080/00949655.2010.545061
  3. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  4. Template:Cite web
  5. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  6. Template:Cite doi
  7. 7.0 7.1 7.2 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

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    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

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  8. 8.0 8.1 8.2 Template:Cite web
  9. 9.0 9.1 9.2 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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  10. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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