CONFIDENCE INTERVAL ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWN in .NET Creation QR Code in .NET CONFIDENCE INTERVAL ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWN 8-2 CONFIDENCE INTERVAL ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWNEncode qr with .netgenerate, create qr code none on .net projectsCon dence and tolerance intervals bound unknown elements of a distribution. In this chapter you will learn to appreciate the value of these intervals. A prediction interval provides bounds on one (or more) future observations from the population. For example, a prediction interval could be used to bound a single, new measurement of viscosity another useful interval. With a large sample size, the prediction interval for normally distributed data tends to the tolerance interval in Equation 8-1, but for more modest sample sizes the prediction and tolerance intervals are different. Keep the purpose of the three types of interval estimates clear: A con dence interval bounds population or distribution parameters (such as the mean viscosity). A tolerance interval bounds a selected proportion of a distribution. A prediction interval bounds future observations from the population or distribution.Visual Studio .NET qr-codes recognizer in .netUsing Barcode scanner for VS .NET Control to read, scan read, scan image in VS .NET applications.8-2 CONFIDENCE INTERVAL ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWN Barcode creator in .netgenerate, create bar code none for .net projectsThe basic ideas of a con dence interval (CI) are most easily understood by initially considering a simple situation. Suppose that we have a normal population with unknown mean and known variance 2. This is a somewhat unrealistic scenario because typically we know the distribution mean before we know the variance. However, in subsequent sections we will present con dence intervals for more general situations.Get bar code with .netgenerate, create barcode none on .net projects8-2.1Control qr code jis x 0510 data for visual c#to generate qr and qr code 2d barcode data, size, image with visual c#.net barcode sdkDevelopment of the Con dence Interval and its Basic Properties QR Code generator with .netusing barcode implementation for asp.net webform control to generate, create qr-code image in asp.net webform applications.Suppose that X1, X2, p , Xn is a random sample from a normal distribution with unknown mean and known variance 2. From the results of 5 we know that the sample mean X is normally distributed with mean and variance 2 n. We may standardize X by subtracting the mean and dividing by the standard deviation, which results in the variable Z X 1n (8-3)Control qr bidimensional barcode image for visual basic.netusing barcode encoding for .net vs 2010 control to generate, create quick response code image in .net vs 2010 applications.Now Z has a standard normal distribution. A con dence interval estimate for is an interval of the form l u, where the endpoints l and u are computed from the sample data. Because different samples will produce different values of l and u, these end-points are values of random variables L and U, respectively. Suppose that we can determine values of L and U such that the following probability statement is true: P 5L U6 1 (8-4)Barcode 3 Of 9 writer on .netusing visual .net tocompose code 39 full ascii on asp.net web,windows applicationwhere 0 1. There is a probability of 1 of selecting a sample for which the CI will contain the true value of . Once we have selected the sample, so that X1 x1, X2 x2, p , Xn xn, and computed l and u, the resulting con dence interval for is l u (8-5)Linear barcode library for .netuse .net linear barcode generating torender linear barcode in .netCHAPTER 8 STATISTICAL INTERVALS FOR A SINGLE SAMPLE Barcode barcode library in .netuse vs .net crystal bar code encoder toencode bar code for .netThe end-points or bounds l and u are called the lower- and upper-con dence limits, respectively, and 1 is called the con dence coef cient. 2 1 1n2 has a standard normal distribuIn our problem situation, because Z 1X tion, we may write Pe z X2d Matrix Barcode barcode library with .netgenerate, create matrix barcode none with .net projects1 1n.net Framework Crystal leitcode integration with .netgenerate, create leitcode none on .net projectsNow manipulate the quantities inside the brackets by (1) multiplying through by subtracting X from each term, and (3) multiplying through by 1. This results in P eX z 1n X z f 1Incoporate barcode on javausing ireport touse barcode in asp.net web,windows application1n, (2)Code 3 Of 9 printer in office exceluse office excel barcode 39 development toincoporate code39 on office excel(8-6)Visual Studio .NET 3 of 9 barcode decoder on .netUsing Barcode scanner for visual .net Control to read, scan read, scan image in visual .net applications.From consideration of Equation 8-4, the lower and upper limits of the inequalities in Equation 8-6 are the lower- and upper-con dence limits L and U, respectively. This leads to the following de nition. De nition 1n 1nPdf417 2d Barcode implementation in vb.netusing .net vs 2010 todevelop barcode pdf417 for asp.net web,windows applicationIf x is the sample mean of a random sample of size n from a normal population with known variance 2, a 100(1 )% CI on is given by x where zControl upc a data with vb.netto use upca and universal product code version a data, size, image with visual basic barcode sdk(8-7)Generate gs1 - 13 with .netusing barcode maker for asp.net web control to generate, create european article number 13 image in asp.net web applications.is the upper 100 Control uss-128 size on .netto receive gs1 barcode and ucc.ean - 128 data, size, image with .net barcode sdk2 percentage point of the standard normal distribution. ANSI/AIM Code 128 development on .netgenerate, create code128b none on .net projectsEXAMPLE 8-1ASTM Standard E23 de nes standard test methods for notched bar impact testing of metallic materials. The Charpy V-notch (CVN) technique measures impact energy and is often used to determine whether or not a material experiences a ductile-to-brittle transition with decreasing temperature. Ten measurements of impact energy (J) on specimens of A238 steel cut at 60 C are as follows: 64.1, 64.7, 64.5, 64.6, 64.5, 64.3, 64.6, 64.8, 64.2, and 64.3. Assume that impact energy is normally distributed with 1J. We want to nd a 95% CI for , the mean 10, 1, and impact energy. The required quantities are z 2 z0.025 1.96, n x 64.46. The resulting 95% CI is found from Equation 8-7 as follows: x 64.46 1n 1 1.96 110 63.84 z64.46 65.081 110That is, based on the sample data, a range of highly plausible vaules for mean impact energy for A238 steel at 60 C is 63.84J 65.08J. Interpreting a Con dence Interval How does one interpret a con dence interval In the impact energy estimation problem in Example 8-1 the 95% CI is 63.84 65.08, so it is tempting to conclude that is within