Confidence mean value

Confidence intervals also can be reported using the mean for a sample of size n, drawn from a population of known O. The standard deviation for the mean value. Ox, which also is known as the standard error of the mean, is... 

Unpaired Data Consider two samples, A and B, for which mean values, Xa and Ab, and standard deviations, sa and sb, have been measured. Confidence intervals for Pa and Pb can be written for both samples... 

Determine the %CO for each sample, and report the mean value and the 95% confidence interval. 

From standard Student tables, the value for t l/fT= 1.049 at the 95% confidence level. Thus, mean value = 56.3 13.0 (95% confidence level) and one could be confident that 95% of measured values would fall in the range 69.3 - 43.3. This is a large range and is not very precise. 

If a heat exchanger is sized usiag the mean values of the design parameters, then the probabiUty, or the confidence level, of the exchanger to meet its design thermal duty is only 50%. Therefore, in order to increase the confidence level of the design, a proper uncertainty analysis must be performed for all principal design parameters. 

Internal accidents, alone, have 267 sequences with a mean value (2.3E-4) slightly higher than the point estimate. The 5% and 95% confidence values are 1.7E-5 and l.OE-3/ reactor-year, respectively. 

Confidence interval That portion of a distribution which is expected to contain the mean value a certain percentage of time. 

Population confidence interval The limits on either side of a mean value of a group of observations which will, in a stated fraction or percent of the cases, include the... 

Tab. 1.2 Limits, ranges and mean values 99.9% confidence limits of the molecular properties of acetylcholine conformers generated during MD simulations.

One of the most common complaints from the inexperienced user is that the result obtained in the routine laboratory does not fall in the confidence interval. Pau-wels (1999) makes considerable reference to this problem, which he calls the Jor-hem Paradox . Even though Pauwels goes on to explain this paradox, in doing so he highlights the problem when he states two results (the certified value and the subsequent laboratory determination) which both claim to contain the most probable mean value of the material with a probability of 95 % do, effectively, not overlap . 

So if we take 385 measurements we conclude with a 95% confidence that the true analyte value (mean value) will be between the average of the 385 results (x) 0.1. 

Given the same underlying spread of data (standard deviation, s), as more data are gathered, we become more confident of the mean value, x, being an accurate representation of the population mean, x. 

The critical value for r tiom the table below with eight degree oi liccdum. aid ai ilie, 5 conti-dence level is 2.11. Because the calculated value for f does not exceed the critical value lor V, it can be concluded w nh l)5 < confidence that no significant difference c i t bci ween ihe mean value for the two sets of dui.i and the two methods have similar decrees of aiciuuev. Vs w nh Procedure 1.2. this information would be verv much more reliable if more replicate value had been available. 

The critical value for 7 with four degrees of freedom is 2.78 and as the calculated value for t does not exceed this value it can be concluded with confidence that there is no significant difference between the mean values for the two sets of data arid the two methods show similar degrees of accuracy. 

A major difficulty in drawing graphs lies in the nature of the relationship between the two variables and a minimum of five points is essential in order to be able to draw a particular line or curve confidently. In the preparation of a calibration graph, the standard or calibrator solutions are usually analysed either in duplicate or in a limited number of replicates and the mean value is used in the preparation of the curve. Because the mean value from a limited number of replicates is unlikely to be the true mean, it might help in drawing the graph if the mean is plotted and a bar is drawn to indicate the highest and lowest values obtained in the replicates. 

The Bandwidth is essentially a normalized half confidence band. The confidence interval bandwidths for 9 data sets using inverse transformed data are given in Table X. The bandwidths are approximately the vertical widths of response from the line to either band. The best band was found for chlorpyrifos, 1.5%, at the minimum width (located at the mean value of the response) and 4.9% at the minimum or lowest point on the graph. Values for fenvalerate and chlorothalonil were slightly higher, 2.1-2.2% at the mean level. The width at the lowest amount for the former was smaller due to a lower scatter of its points. The same reason explains the difference between fenvalerate and Dataset B. Similarly, the lack of points in Dataset A produced a band that was twice as wide when compared to Dataset B. Dataset C gave a much wider band when compared to Dataset B. 

The t value is the number of standard deviations that the single value differs from the mean value. This t value is then compared to the critical t value obtained from a t-table, given a desired statistical confidence (i.e., 90%, 95%, or 99% confidence) and the number of degrees of freedom (typically iV-1), to assess whether the value is statistically different from the other values in the series. In chemometrics, the t test can be useful for evaluating outliers in data sets. 

The simple approach is just using the mean value of several determinations of blank samples plus a multiple (factor k) of the standard deviation of these measnrements. With the choice of k we define the level of confidence. 

Most of the 95 per cent confidence intervals do contain the true mean of 80 mmHg, but not all. Sample number 4 gave a mean value 3c = 81.58 mmHg with a 95 per cent confidence interval (80.33, 82.83), which has missed the true mean at the lower end. Similarly samples 35, 46, 66, 98 and 99 have given confidence intervals that do not contain p = 80 mmHg. So we have a method that seems to work most of the time, but not all of the time. For this simulation we have a 94 per cent (94/100) success rate. If we were to extend the simulation and take many thousands of samples from this population, constructing 95 per cent confidence intervals each time, we would in fact see a success rate of 95 per cent exactly 95 per cent of those intervals would contain the true (population) mean value. This provides us with the interpretation of a 95 per cent confidence interval in... 

Using a spreadsheet function or analysis tool, perform a Smdent s f-test to compare the mean values of each of the first four wear rates with that of gold (the last value in the column) at an agreed-upon confidence level—for example, 95% confidence level. If you have not yet learned how to do Student s f-test, check out any book on elementary statistics, and learn how to do a f-test, or use the Help menus in your spreadsheet package. The goal is to determine if the mean wear rates of the four ceramic materials are statistically different from the wear rate for gold. If done correctly, you will have performed four separate f-tests in this step. 

UNODC has also started to conduct yield surveys in some countries, measuring the yield of test fields, and to develop methodologies to extrapolate the yields from proxy variables, such as the volume of poppy capsules or the number of plants per plot. This approach is used in South-East Asia as well as in Afghanistan. All of this is intended to improve yield estimates, aiming at information that is independent from farmers reports. The accuracy of the calculated yields depends on a number of factors, including the number of sites investigated. In the case of Afghanistan the confidence interval for the mean yield results in the 2006 survey was, for instance, +/- 3% of the mean value (tt= 0.1). 

There are two different ways of carrying out this test. The first one involves taking a single sample and analysing it by both methods a number of times. The usual procedure is to undertake a number of analyses (preferably not less than 6) for the chosen sample with both methods and calculate the value of the t-statistic. This is then compared with the tabular value for the appropriate degrees of freedom at the selected confidence level. If the calculated value is less than the tabulated t value then the mean values, and hence the methods, are accounted equivalent. This method has the advantage that the number of replicates undertaken for each method does not have to be equal. However, it is not always recognised that for this test to be valid the precision of the two methods should be equal. The method used to compare the precisions of methods is the F-ratio test and is carried out as part of the procedure. 

In a penalty test, a property cf the system is modified to reduce the probability of the desired result. For example, to predict safety, a particular expl train interface may be tested with a standard donor and a more sensitive acceptor conversely, to predict reliability, a less sensitive acceptor material is used. If this probability is reduced sufficiently, it is possible to obtain mixed responses (that is, some fires and some no-fires) with samples of reasonable size, and to develop data from which the mean value of the penalty and its standard deviation (as well as confidence limits) can be established. These estimates can be used iri statistical extrapolation to estimate safety or reliability under the original design conditions. The term VARICOMP (VARIation of explosive COMPosition) was coined by J.N. Ayres for a method developed at the Naval Ordnance Lab, White Oak, in the 1950 s and early 1960 s (Ref 1)... 

Figure 4-5 illustrates the meaning of confidence intervals. A computer chose numbers at random from a Gaussian population with a population mean (p.) of 10 000 and a population standard deviation (o) of 1 000 in Equation 4-3. In trial 1, four numbers were chosen, and their mean and standard deviation were calculated with Equations 4-1 and 4-2. The 50% confidence interval was then calculated with Equation 4-6, using t = 0.765 from Table 4-2 (50% confidence, 3 degrees of freedom). This trial is plotted as the first point at the left in Figure 4-5a the square is centered at the mean value of 9 526, and the error bar extends from the lower limit to the upper limit of the 50% confidence interval ( 290). The experiment was repeated 100 times to produce the points in Figure 4-5a. 

Enter the measured mean value of y for replicate measurements of the unknown in cell B18. In cell B19, enter the number of replicate measurements of the unknown. Cell B20 computes the value of x corresponding to the measured mean value of v. Cell B21 uses Equation 4-27 to find the uncertainty (the standard deviation) in the value of x for the unknown. If you want a confidence interval for x, multiply sv times Student s t from Table 4-2 for n — 2 degrees of freedom and the desired confidence level. 

After you select a confidence level. Student s t is used to find confidence intervals (p, = x ts/Vn) and to compare mean values measured by different methods. The F test is used to decide whether two standard deviations are significantly different from each other. The Q test helps you to decide whether or not a questionable datum should be discarded. It is best to repeal the measurement several times to increase the probability that your decision is correct. 

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