Confidence intervals known standard deviation

When a small number of observations is made, the value of the standard deviation s, does not by itself give a measure of how close the sample mean x might be to the true mean. It is, however, possible to calculate a confidence interval to estimate the range within which the true mean may be found. The limits of this confidence interval, known as the confidence limits, are given by the expression ... 

The calculated values from any sample are considered as point estimates. Any such estimate may be close to the true value v>f the population (/c, a or other) or it may vary substantially from the true value. An indication of the interval around this point estimate, within which the true value is expected to fall with some stated probability, is called a coiifich iHc interval, and the lower and upper boundary values are called the confidence limits. The probability used to set the interval is called the level of eonfidenee. This level is given by (1 - ), where a is the probability as discussed above for rejecting a null hypothesis when it is true. In most circumstances, means are the most important point estimates, and confidence intervals for means are evaluated at some probability / — (I - a) that the true population mean is within the stated confidence limits. This can be expressed for a population with a known standard deviation a as given in Eq. 21. 

The population standard deviation for the amount of aspirin in a batch of analgesic tablets is known to be 7 mg of aspirin. A single tablet is randomly selected, analyzed, and found to contain 245 mg of aspirin. What is the 95% confidence interval for the population mean ... 

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... 

Earlier we introduced the confidence interval as a way to report the most probable value for a population s mean, p, when the population s standard deviation, O, is known. Since is an unbiased estimator of O, it should be possible to construct confidence intervals for samples by replacing O in equations 4.10 and 4.11 with s. Two complications arise, however. The first is that we cannot define for a single member of a population. Consequently, equation 4.10 cannot be extended to situations in which is used as an estimator of O. In other words, when O is unknown, we cannot construct a confidence interval for p, by sampling only a single member of the population. 

If the standard deviation for a method is known, how many results must be Use the confidence interval method obtained to provide a reasonable estimate of the true mean (equation (2.7))... 

Normally the population standard deviation a is not known, and has to be estimated from a sample standard deviation s. This will add an additional uncertainty and therefore will enlarge the confidence interval. This is reflected by using the Student-t-distribution instead of the normal distribution. The t value in the formula can be found in tables for the required confidence limit and n-1 degrees of freedom. 

The reason for this is again a technical one but relates to the uncertainty associated with the use of the sample standard deviation (s) in place of the true population value (a) in the formula for the standard error. When a is knovm, the multiplying constants given earlier apply. When ct is not known (the usual case) we make the confidence intervals slightly wider in order to account for this uncertainty. When n is large of course s will be close to a and so the earlier multiplying constants apply approximately. 

A meta-analysis for continuous data cannot be calculated unless the pertinent standard deviations are known. Unfortunately, clinical reports often give the sample size and mean ratings for the various groups but do not report the standard deviations (or standard error of the mean), which are necessary for effect size calculations. Thus, investigators should always report the indices of variability (e.g., confidence intervals, SDs) for the critical variables related to their primary hypothesis. 

When the repeatability standard deviation is known, it can be used to assess results done in duplicate. If the repeatability is Uj, then the 95% confidence interval of the difference of two results is... 

You purchased a Standard Reference Material (Box 3-1) coal sample certified by the National Institute of Standards and Technology to contain 3.19 wt% sulfur. You are testing a new analytical method to see whether it can reproduce the known value. The measured values are 3.29, 3.22, 3.30, and 3.23 wt% sulfur, giving a mean of 3r = 3.260 and a standard deviation of, v = 0.04,. Does your answer agree with the known answer To find out, compute the 95% confidence interval for your answer and see if that range includes the known answer. If the known answer is not within your 95% confidence interval, then the results do not agree. 

For univariate calibration, the International Union of Pure and Applied Chemistry (IUPAC) defines sensitivity as the slope of the calibration curve when the instrument response is the dependent variable, i.e., y in Equation 5.4, and the independent variable is concentration. This is also known as the calibration sensitivity, contrasted with the analytical sensitivity, which is the calibration sensitivity divided by the standard deviation of an instrumental response at a specified concentration [18], Changing concentration to act as the dependent variable, as in Equation 5.4, shows that the slope of this calibration curve, f, is related to the inverse of the calibration sensitivity. In either case, confidence intervals for concentration estimates are linked to sensitivity [1, 19-22],... 

Usefulness of the normal distribution curve lies in the fact that from two parameters, the true mean p. and the true standard deviation true mean determines the value on which the bell-shaped curve is centered, and most probability concentrated on values near the mean. It is impossible to find the exact value of the true mean from information provided by a sample. But an interval within which the true mean most likely lies can be found with a definite probability, for example, 0.95 or 0.99. The 95 percent confidence level indicates that while the true mean may or may not lie within the specified interval, the odds are 19 to 1 that it does.f Assuming a normal distribution, the 95 percent limits are x 1.96 where a is the true standard deviation of the sample mean. Thus, if a process gave results that were known to fit a normal distribution curve having a mean of 11.0 and a standard deviation of 0.1, it would be clear firm Fig. 17-1 that there is only a 5 percent chance of a result falling outside the range of 10.804 and 11.196. 

In terms of the previously mentioned normal distribution, the probability that a randomly selected observation x from a total population of data will be within so many units of the true mean p can be calculated. However, this leads to an integral which is difficult to evaluate. To overcome this difficulty, tables have been developed in terms of p Ztrue standard deviation a of a particular normal distribution under study is known and assuming that the difference between the sample x and the true mean p is only the result of chance and that the individual observations are normally distributed, then a confidence interval in estimating p can be determined. This measure was referred to previously as the confidence level. 

If the true standard deviation is not known, a corresponding confidence interval can still be determined. However, this estimate must utilize the t-distri-bution instead of the Z-distribution since the t-concept includes the additional variation introduced by the estimate of standard deviation. In this case the rearranged t-equation is used. 

In most instances the population standard deviation is not known and must be estimated from the sample standard deviation (s). Substitution of s for cr in equation (1.1) with M = 1.96 does not result in a 95% confidence interval unless the sample number is infinitely large (in practice >30). When s is used, multipliers, whose values depend on sample number, are chosen from the /-distribution and the denominator in... 

The BEST develops an estimate of the total sample population using a small set of known samples. A point estimate of the center of this known population is also calculated. When a new sample is analyzed, its spectrum is projected into the same hyperspace as the known samples. A vector is then formed in hyperspace to connect the center of the population estimate to the new sample spectral point. A hypercylinder is formed about this vector to contain a number of estimated-population spectral points. The density of these points in both directions along the central axis of the hypercylinder is used to construct an asymmetric nonparametric confidence interval. The use of a central 68% confidence interval produces bootstrap distances analogous to standard deviations. 

Once the laboratory samples have been prepared, the question that remains is how many samples should be taken for the analysis. If we have reduced the measurement uncertainty such that it is less than one third the sampling uncertainty, the latter will limit the precision of the analysis. The number, of course, depends on what confidence interval we want to report for the mean value and the desired relative standard deviation of the method. If the sampling standard deviation cr, is known from previous experience, we can use the values of z from the tables (see Section 7A-1). 

If the standard deviation of a result is known, or if a 95% confidence interval has been calculated, this is a guide to the number of significant figures. 

The average and sample standard deviation are known as estimators of the population mean and standard deviation. We have seen how the estimates improve as the number of data increases. As we have stressed, the use of these statistics requires data that are normally distributed, and for confidence intervals employing the standard deviation of the mean this tends to be so. Real data may be so distributed, but often the distribution will contain data that are seriously flawed, as with the RACI titration competition described in chapter 1. If we can identify such data and remove them from further... 

So far we have assumed that the population standard deviation was known. Yet we only knew a sample estimate. It is true it had been obtained from a quite large sample, but seldom is an experiment replicated 140 times. We shall now see how to break free of this imsatisfactory restriction and obtain confidence intervals without relying on population values or performing an inordinate number of replicate experiments. 

This confidence interval is almost twice the previous example because the standard deviation as well as the estimated mean is only known to df = 3,... 

The concentration of each THM is normally distributed in the drinking water sample. There exists a population mean concentration for each THM. We can never known this mean concentration, p, for each THM from the population we can, however, by measuring L replicates of the sample, calculate a mean concentration, x, and a standard deviation, s. We really do not know the standard deviation for the population, a. We can make L replicate measurements and use t-statistics to define a confidence interval for the population mean using chloroform as our example ... 

It is clear from Equation 2.9 that the length of the confidence interval is linearly proportional to the population standard deviation, and inversely related to the square root of the sample size. If o were known. Equation 2.9 could be used to determine the minimum sample size required to obtain a confidence interval which will contain the unknown mean p, with a (1-cc) probability. An expression for the minimum sample size will, therefore, be... 

Confidence Interval of jt if the Standard Deviation of the Population is Known... 

Since the population standard deviation is known, Z will be used, which in this case is 1.96 (useful number to memorise). Therefore, the confidence interval is... 

If the total population is known and, therefore, also the true mean fi and the standard deviation A, to infer the value that corresponds to a given percent of survival is rather an easy game. Assuming that a normal distribution holds, what shall be done is to evaluate the number k of standard deviation A to subtract to the true mean ji. But when the true mean is not known because the population of data is to large with respect to the sample size (see Eq. 4.1) and the mean available x is just the sample mean, the question arises as to how close or far we really are from the true one. The question can be answered only in terms of confidence interval C. In general terms, if a population parameter is not known, for instance the true mean it can always be estimated using observed sample data. Estimated actually means that its value will never be exactly determined, but it may be included in a range of values whose size depends on the confidence we want to know it. As the... 

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