Standard errors confidence intervals

We used a two-tailed test. Upon rereading the problem, we realize that this was pure FeO whose iron content was 77.60% so that p = 77.60 and the confidence interval does not include the known value. Since the FeO was a standard, a one-tailed test should have been used since only random values would be expected to exceed 77.60%. Now the Student t value of 2.13 (for —to05) should have been used, and now the confidence interval becomes 77.11 0.23. A systematic error is presumed to exist. 

The F statistic, along with the z, t, and statistics, constitute the group that are thought of as fundamental statistics. Collectively they describe all the relationships that can exist between means and standard deviations. To perform an F test, we must first verify the randomness and independence of the errors. If erf = cr, then s ls2 will be distributed properly as the F statistic. If the calculated F is outside the confidence interval chosen for that statistic, then this is evidence that a F 2. 

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... 

Determine the density at least five times, (a) Report the mean, the standard deviation, and the 95% confidence interval for your results, (b) Eind the accepted value for the density of your metal, and determine the absolute and relative error for your experimentally determined density, (c) Use the propagation of uncertainty to determine the uncertainty for your chosen method. Are the results of this calculation consistent with your experimental results ff not, suggest some possible reasons for this disagreement. 

The true standard deviation Ox is expected inside the confidence interval CI(5 , ) = /Vi. .. /V with a total error probability 2 p (in connection with F and x P taken to be one-sided). 

The only drawback in using this method is that any numerical errors introduced in the estimation of the time derivatives of the state variables have a direct effect on the estimated parameter values. Furthermore, by this approach we can not readily calculate confidence intervals for the unknown parameters. This method is the standard procedure used by the General Algebraic Modeling System (GAMS) for the estimation of parameters in ODE models when all state variables are observed. 

If on the other hand we wish to compute the(l-a) 100% confidence interval of the response of y, at t=to, we must include the error term (e0) in the calculation of the standard error, namely we have... 

Additional measurements were made with the 17-)im sizing screen to obtain more information on the variability of our measurement techniques. Eight lint samples from a single source of cotton were analyzed by the procedures outlined previously. The dust levels obtained in this test were 11.7, 12.1, 13.5, 11.8, 10.8, 11.2, 10.9, and 9.7 mg, respectively, per 20 g of lint. The mean and standard deviation of these measurements were 11.5 and 1.1, respectively. The estimated standard error of the mean was 0.42, and the interval from 10.5 to 12.5 represented a 95% confidence interval for the lot mean. 

The difference between these two concepts, the population standard deviation and the standard error of the mean, is important and we will return to it when considering confidence intervals. 

One question that is often asked of statisticians is in what sense can we be 95% confident that the population mean lies within the limits 3.84 and 4.13 To answer the question we can again conduct a sampling experiment as follows. Suppose that the 40 blood glucose measurements in Figure 8.3 comprised the total population of values. For random sample of size 10 from the populations of blood glucose values determine the sample mean, standard error and the corresponding 95% confidence interval. Repeat the process 100 times. The results of such an experiment are shown in Figure 8.6. 

From the formula for a confidence interval, its width is determined by three parameters the sample size, population variability and the degree of confidence. Plainly, if the sample size is increased then we have seen the standard error will be reduced and hence the width of the interval will also be reduced. If we can reduce the variability of the characteristic being studied then... 

In this notation, N d is the number of independent samples contained in the trajectory, and fsim the length of the trajectory. The standard error can be used to approximate confidence intervals, with a rule of thumb being that + 2SE represents roughly a 95% confidence interval [26]. The actual interval depends on the underlying distribution and the sampling quality as embodied in Nfd fSimA/ see ref. 25 for a more careful discussion. 

The classical, frequentist approach in statistics requires the concept of the sampling distribution of an estimator. In classical statistics, a data set is commonly treated as a random sample from a population. Of course, in some situations the data actually have been collected according to a probability-sampling scheme. Whether that is the case or not, processes generating the data will be snbject to stochastic-ity and variation, which is a sonrce of uncertainty in nse of the data. Therefore, sampling concepts may be invoked in order to provide a model that accounts for the random processes, and that will lead to confidence intervals or standard errors. The population may or may not be conceived as a finite set of individnals. In some situations, such as when forecasting a fnture value, a continuous probability distribution plays the role of the popnlation. 

An approach that is sometimes helpful, particularly for recent pesticide risk assessments, is to use the parameter values that result in best fit (in the sense of LS), comparing the fitted cdf to the cdf of the empirical distribution. In some cases, such as when fitting a log-normal distribution, formulae from linear regression can be used after transformations are applied to linearize the cdf. In other cases, the residual SS is minimized using numerical optimization, i.e., one uses nonlinear regression. This approach seems reasonable for point estimation. However, the statistical assumptions that would often be invoked to justify LS regression will not be met in this application. Therefore the use of any additional regression results (beyond the point estimates) is questionable. If there is a need to provide standard errors or confidence intervals for the estimates, bootstrap procedures are recommended. 

It is not possible at this stage to say precisely what we mean by small and large in this context, we need the concept of the confidence interval to be able to say more in this regard and we will cover this topic in the next chapter. For the moment just look upon the standard error as an informal measure of precision high values mean low precision and vice versa. Further if the standard error is small, it is likely that our estimate x is close to the true mean, p,. If the standard error is large, however, there is no guarantee that we will be close to the true mean. 

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. 

Note the role played by the standard error in the formula for the confidence interval. We have previously seen that the standard error of the mean provides an indirect measure of the precision with which we have calculated the mean. The confidence interval has now translated the numerical value for the standard error into something useful in terms of being able to make a statement about where jl lies. A large standard error will lead to a wide confidence interval reflecting the imprecision and resulting poor information about the value of jjl. In contrast a... 

Finally, returning again to the formula for the standard error, sl.y/n, we can, at least in principle, see how we could make the standard error smaller increase the sample size n and reduce the patient-to-patient variability. These actions will translate into narrower confidence intervals. 

At the end of the previous chapter we saw how to extend the idea of a standard error for a single mean to a standard error for the difference between two means. The extension of the confidence interval is similarly straightforward. Consider the placebo controlled trial in cholesterol lowering described in Example 2.3 in Chapter 2. We had an observed difference in the sample means 3cj — 3c2 of 1.4 mmol/1 and a standard error of 0.29. The formula for the 95 per cent confidence interval for the difference between two means — p.2) i -... 

In Section 2.5.2 we set down the formulas for the standard errors for both individual rates and the difference between two rates. These lead naturally to expressions for the confidence intervals. 

With a ratio it is not possible to obtain a standard error formula directly however it is possible to obtain standard errors for log ratios. (Taking logs converts a ratio into a difference with log A/B = log A — log B.) So we first of all calculate confidence intervals on the log scale. It does, in fact, not make any difference what base we use for the logs but by convention we usually use natural logarithms, denoted fn . 

Previously when we had calculated a confidence interval, for example for a difference in rates or for a difference in means, then the confidence interval was symmetric around the estimated difference in other words the estimated difference sat squarely in the middle of the interval and the endpoints were obtained by adding and subtracting the same amount (2 x standard error). When we calculate a confidence interval for the odds ratio, that interval is symmetric only on the log scale. Once we convert back to the odds ratio scale by taking anti-logs that symmetry is lost. This is not a problem, but it is something that you will notice. Also, it is a property of all standard confidence intervals calculated for ratios. 

Produces improvements in efficiency (smaller standard errors, narrower confidence intervals, increased power). 

---