Population values

The confidence interval for a given sample mean indicates the range of values within which the true population value can be expected to be found and the probability that this will occur. For example, the 95% confidence limits for a given mean are given by... 

Less information is available on PCP concentrations in serum. PCP concentrations lower than 0.5 mg/L have been observed in non-exposed populations. Values ranging from 0.2 to 1.8 mg/L have been measured in occupants of PCP-treated houses, and concentrations from 0.4 to 13 mg/L have been measured in occupationally exposed workers. PCP concentrations higher than 40 mg/L have been observed in fatal poisonings. 

This value for the standard deviation was accepted as the best available approximation to the population value for a. The next step was to take several different aliquots from a large sample (a different sample than used previously) and collect multiple readings from each of them. Six aliquots were placed in each of six flasks, and six repeat measurements were made on each of these six flasks. Each aliquot consisted of 10 g of test sample/lOOml water. The results are shown in Table 9-2. 

The first of these assumptions is the use of the Normal distribution. When we perform an experiment using a sequential design, we are implicitly using the experimentally determined value of s, the sample standard deviation, against which to compare the difference between the data and the hypothesis. As we have discussed previously, the use of the experimental value of s for the standard deviation, rather than the population value of a, means that we must use the f-distribution as the basis of our comparisons, rather than the Normal distribution. This, of course, causes a change in the critical value we must consider, especially at small values of n (which is where we want to be working, after all). 

LD50 Amount of liquid or solid material required to kill half of the exposed population. Values are for ingestion (Ing), percutaneous (Per) exposures, and subcutaneous injection (Sub). These values are expressed as total grams per individual. 

Most of the observations are likely to be within the range x 2 s, and practically all within the range x 3. . In common with the sample mean, the value of the standard deviation calculated from a set of observations is only an estimate of the true or population value of the standard deviation s becomes a better estimate of [Pg.275]

We therefore make the assumption that the sample data gathered in vector y are only our best estimates of the real (population) values which justifies the bar on the symbol as representing measured values. This notation contradicts the standard usage, but is consistent with the basic definitions of Chapter 4. Indeed, for an unbiased estimate, we can still write that... 

In some textbooks, a confidence interval is described as the interval within which there is a certain probability of finding the true value of the estimated quantity. Does the term true used in this sense indicate the statistical population value (e.g., p if one is estimating a mean) or the bias-free value (e.g., 6.21% iron in a mineral) Could these two interpretations of true value be a source of misunderstanding in conversations between a statistician and a geologist ... 

Random error is the divergence, due to chance alone, of an observation on a sample from the true population value, leading to lack of precision in the measurement of an association. There are three major sources of random error individual/biological variation, sampling error, and measurement error. Random error can be minimized but can never be completely eliminated since only a sample of the population can be studied, individual variation always occurs, and no measurement is perfectly accurate. 

It is usual to replace the n in the denominator by (n — 1) so that the resulting estimate of the population value is unbiased. 

The estimation of a parameter alone is not sufficient since a single estimate tells us nothing about how accurate the estimate is. The main purpose of confidence intervals is to indicate the precision, or imprecision, of the estimated statistic as representing the population values. The confidence interval will give us a range of values within which we can have a chosen confidence of it containing the population value. The degree of confidence usually presented is 95%. 

When the population does indeed follow this distribution then the standard deviation, a, has a more specific interpretation. If we move a units below the mean, to x — a and a units above the mean, to x -F ct, then that interval ( x — a, jjL -F a) will capture 68.3 per cent of the population values. This is true whatever we are considering diastolic blood pressure, fall in diastolic blood pressure over a six-month period, cholesterol level, FEVj etc. and whatever the values of x and a in all cases 68.3 per cent of the patients will have data values in the range jx — a to x -F a providing the data are normally distributed. 

More generally, whatever statistic we are interested in, there is always a formula that allows us to calculate its standard error. The formulas change but their interpretation always remains the same a small standard error is indicative of high precision, high reliability. Conversely a large standard error means that the observed value of the statistic is an unreliable estimate of the true (population) value. It is also always the case that the standard error is an estimate of the standard deviation of the list of repeat values of the statistic that we would get were we to repeat the sampling process, a measure of the inherent sampling variability. 

As discussed in the previous section the standard error simply provides indirect information about reliability, it is not something we can use in any specific way, as yet, to tell us where the truth lies. We also have no way of saying what is large and what is small in standard error terms. We will, however, in the next chapter cover the concept of the confidence interval and we will see how this provides a methodology for making use of the standard error to enable us to make statements about where we think the true (population) value lies. 

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. 

In summary, disease-specific instruments are the most sensitive to changes in the patient s experience, whereas findings collected using generic instruments are more easily compared with population values for other therapies. 

Predict volume of distribution (Vd) and clearance (CL) based on standard population values (eg, Table 3-1) with adjustments for factors such as weight and renal function. 

Conventional strain energies (CSE) in kcal mol1 from HF/6-31G(d) calculations. Hybridization ratios n and m, overlap S, and interbond population values from a configuration analysis47. 

Suppose we could use b0 and sbo from only one set of experiments to construct a confidence interval about b(l such that there is a given probability that the interval contains the population value of /30 (see Section 3.4). The interpretation of such a confidence interval is this if we find that the interval includes the value zero, then (with our data) we cannot disprove the null hypothesis that fi0 = 0 that is, on the basis of the estimates b0 and sho, it is not improbable that the true value of y30 could be zero. Suppose, however, we find that the confidence interval does not contain the value zero because we know that if /30 were really equal to zero this lack of overlap... 

Extrapolation of the D value from large microbial population values to fractional (e.g., 10 ) values predicts the number of log reductions a given exposure period will produce. 

A range of values estimated by a sample within which the true population value is expected to fall. For example, if an LC50 and its 95% confidence intervals are estimated from a toxicity test, the true population LC50 is expected to fall within the interval 95% of the time. Volume 1(10), Volume 2(5). 

Measurement of the range of values within which the true population value probably lies. 

Table 23.2 shows the net charge of the center lithium ion and the bond overlap population between the center lithium ion and neighboring halide ions obtained for model II. The net charge values decrease with increasing atomic number of halide ion. On the other hand, the bond overlap population values are almost identical. This is probably due to increased electronic charge around the lithium ion. 

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