Research Article The Probability That a Measurement Falls within a Range of n Standard Deviations from an Estimate of the Mean

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1 Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 70806, 8 pages doi:0.540/0/70806 Research Article The Probability That a Measuremet Falls withi a Rage of Stadard Deviatios from a Estimate of the Mea Louis M. Housto The Louisiaa Accelerator Ceter, The Uiversity of Louisiaa at Lafayette, Lafayette, LA , USA Correspodece should be addressed to Louis M. Housto, housto@louisiaa.edu Received 5 November 0; Accepted 5 November 0 Academic Editors: M.-H. Hsu ad X. Liu Copyright q 0 Louis M. Housto. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. We derive a geeral equatio for the probability that a measuremet falls withi a rage of stadard deviatios from a estimate of the mea. So, we provide a format that is compatible with a cofidece iterval cetered about the mea that is aturally idepedet of the sample size. The equatio is derived by iterpolatig theoretical results for extreme sample sizes. The itermediate value of the equatio is cofirmed with a computatioal test.. Itroductio A cofidece iterval is a iterval i which a measuremet or trial falls correspodig to a give probability,. I statistics, cofidece itervals cetered about a estimate of the mea target the mea. Because these cofidece itervals rely o the stadard error of the estimate, they icrease as sample size decreases ad they decrease as sample size icreases. Cosequetly, the cofidece iterval, i statistics, is ofte a margi of error for the estimate of the mea 3. We are iterested i fidig the probability of a cofidece iterval cetered about a estimate of the mea that targets a arbitrary measuremet ad is idepedet of the sample size. I this way, we provide a format that is compatible with a cofidece iterval cetered about the mea that is aturally idepedet of sample size, but has a width of stadard deviatios. We approach this problem by cosiderig the kow case of a cofidece iterval cetered about the mea ad we derive the associated probability. We correlate this result to the case of ifiite sample size. The ext step is to calculate the probability

2 ISRN Applied Mathematics associated with the miimum sample size of oe. This requires that we average the probability over differet possible sample values. Fially, we propose a equatio for probability that aturally iterpolates our results ad we show that the equatio is cosistet with itermediate probability values by comparig the equatio to estimates of probability produced by computer simulatio.. Expected Probability I order to make the ecessary cofidece iterval calculatios, we will have to determie a expected or average probability. This ca be examied by cosiderig the cumulative distributio fuctio 4, 5, F x; μ, σ x e t μ /σ dt,. πσ where x is a measuremet, μ is the mea, ad σ is the stadard deviatio. Let v t μ / σ.so,dv dt/ σ. Therefore,. becomes F x; μ, σ x μ / π σ e v dv 0 e v dv x μ / σ e v dv π π erf x μ σ,. where erf x is the so-called erf fuctio 6, 7. Now, we kow that F x; μ, σ P X x..3 Rather tha specifyig a value of x, we would like to compute the probability averaged over all possible values of x. So we have E F x; μ, σ E P X x..4 This yields E F x; μ, σ πσ e x μ /σ dx x μ πσ erf e x μ /σ dx.5 σ x μ πσ erf e x μ /σ dx..6 σ

3 ISRN Applied Mathematics 3 Let z x μ / σ. So,dz dx/ σ. Therefore,.6 becomes E F x; μ, σ erf z e z dz. π.7 Sice erf z is a odd fuctio, we have E F x; μ, σ..8 This ca be easily cofirmed by a alterative calculatio. The probability desity f x is f x πσ e x μ /σ..9 Therefore, we ca write 8 f df dx..0 The expected value 9 of the cumulative distributio fuctio is thus E F Ff dx.. Note that whe x, F ad whe x, F 0. Therefore, we ca write E F 0. FdF. Cosequetly, as per.4, we ca write the expected probability as E P X x The Probability of a Cofidece Iterval We wat to determie the probability give as P x σ x x σ, 3.

4 4 ISRN Applied Mathematics where x N N x i i 3. is the estimate of the mea, μ. At the maximal value of N,wefidthat N lim x i μ. N N i 3.3 For a ormal distributio, the probability that a measuremet falls withi stadard deviatios σ of the mea μ i.e., withi the iterval μ σ, μ σ is give by P μ σ x μ σ σ π σ π μ σ μ σ μ σ μ e x μ / σ dx e x μ / σ dx. 3.4 Now, let u x μ / σ,sodu dx/ σ. The P μ σ x μ σ σ / σ e u du π / π 0 erf e u du Observe that we ca express 3.7 i terms of the cumulative probability distributio fuctio F. Thus, we ca write P μ σ x μ σ F μ σ; μ, σ F μ σ; μ, σ. 3.8 The miimal value of N is oe, so that x x. I this case, we have P x σ x x σ F x σ; μ, σ F x σ; μ, σ. 3.9 We fid that 3.9 reduces ito P x σ x x σ erf x σ μ σ σ μ erf x σ. 3.0

5 ISRN Applied Mathematics 5 Table : Comparisos of umerical itegratio of / π erf z / e z dz ad erf /. / π erf z / e z dz erf / The probability i 3.0 is writte i terms of the value x. Cosequetly, we ca derive a expected probability by computig E x P σ π σ π [ erf x σ μ σ ] σ μ erf x σ e x μ /σ dx 3. [ erf x μ σ erf x μ ] σ e x μ /σ dx. 3. Let z x μ /σ which implies that dz dx /σ. This reduces 3. ito E x P [ π erf π erf z z e z dz. ] erf z e z dz Equatio 3.4 ca be evaluated umerically to yield the followig: E x P x σ x x σ erf. 3.5 A portio of the umerical results is preseted i Table. Based o 3.7 ad 3.5, we ca propose the equatio P x σ x x σ erf, 3.6 /N i which we are by default referrig to the expected probability. Equatio 3.6 clearly coverges for the extreme estimates of the mea, sice lim N erf lim N erf /N /N erf erf., 3.7

6 6 ISRN Applied Mathematics P N ifiite N = 0 N = N = Figure : A plot of the equatio P x σ x x σ erf / /N for four differet values of N. Aplotof 3.6 for four values of N is show i Figure. Observe that the curve for N is itermediate to the other curves. Cosequetly, we ca have cofidece i 3.6 if we ca show that it is valid for N. 4. Computatioal Test We ca estimate P x σ x x σ computatioally. Simulate the ormal, idepedet radom variables X ad {X i }, for which x X, x i X i. 4. Let the coditio β be β : x σ x x σ. 4. If M β is the umber of trials i which the coditio β is met ad M is the total umber of trials, the a estimate of P is give as P M β M. 4.3 Figure shows a plot of P versus P for N adm 000.

7 ISRN Applied Mathematics P P estimate, N = P equatio, N = Figure : A plot of P versus P for N adm Coclusios We derive a geeral equatio for the probability that a measuremet falls withi a rage of stadard deviatios from a estimate of the mea. So, we provide a format that is compatible with a cofidece iterval cetered about the mea that is aturally idepedet of the sample size. It is cosistet with our equatio that probability reduces with sample size. However, for samples greater tha te, the value of probability begis to coverge. The equatio for probability is derived by cosiderig the miimal ad maximal sample sizes ad producig a equatio which aturally iterpolates the results. Computer simulatio is used to estimate probability for the sample size N that produces itermediate results that are i strog agreemet with the geeral equatio. Ackowledgmet Discussios with Maxwell Lueckehoff are appreciated. Refereces J. F. Keey ad E. S. Keepig, Cofidece limits for the biomial parameter ad Cofidece iterval charts.4 ad.5, i Mathematics of Statistics, part, pp , D. Va Nostrad, Priceto, NJ, USA, 3rd editio, 96. A. Teat ad E. M. Badley, A cofidece iterval approach to ivestigatig o-respose bias ad moitorig respose to postal questioaires, Joural of Epidemiology ad Commuity Health, vol. 45, o., pp. 8 85, D. G. Rees, Essetial Statistics, Chapma & Hall/CRC, 4th editio, D. Zwilliger ad S. Kokoska, CRC stadard probability ad statistics tables ad formulae, Chapma & Hall/CRC, Boca Rato, FL, 000.

8 8 ISRN Applied Mathematics 5 U. Balasooriva, J. Li, ad C. K. Low, O iterpretig ad extractig iformatio from the cumulative distributio fuctio curve: a ew perspective with applicatios, Australia Seior Mathematics Joural, vol. 6, o., 0. 6 J. Spaier ad K. B. Oldham, The error fuctio ad its complemet, i A Atlas of Fuctios,chapter 40, pp , Hemisphere, Washigto, DC, USA, L. M. Housto, G. A. Glass, ad A. D. Dymikov, Sig-bit amplitude recovery i Gaussia oise, Joural of Seismic Exploratio, vol. 9, o. 3, pp. 49 6, N. G. Ushakov, Desity of a probability distributio, i Ecyclopedia of Mathematics, M.Hazewikel, Ed., Spriger, L. M. Housto, G. A. Glass, ad A. D. Dymikov, Sig data derivative recovery, ISRN Applied Mathematics, vol. 0, Article ID 63070, 7 pages, 0.

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