Interval overlap

Since the two confidence intervals overlap, a complete separation of the two solutes cannot be achieved in a 100-step countercurrent extraction. The complete distribution of the solutes is shown in Figure A6.4. 

Randomised controlled trials with nondefinitive results (a difference that suggests a clinically significant effect but with confidence intervals overlapping the threshold of this effect)... 

Panel (b) illustrates the application of successive umbrella sampling [35], where each piece of the curve corresponds to several intervals used in the successive sampling. The intervals overlap just by one particle number, which is sufficient if one takes due account of detailed balance at the interval boundaries [36]. Matching the pieces at their boundaries one obtains the distribution across the entire composition range. Of course, both methods yield identical results for the final probability distribution and, provided that one starts with an appropriate modification factor, /, they require comparable amounts of CPU time. 

There are times when a researcher would like to compare multiple regression function lines. One approach is to construct a series of 95% confidence intervals for each of the y values at specific x, values. If the confidence intervals overlap, from regression line A to regression line B, the researcher simply states that no difference exists, and if the confidence intervals do not overlap, the researcher states that the y points are significantly different from each other at a (see Figure 2.37). 

From Figure 1.5 we can see that, for the subset of 25 compounds, the 95% confidence intervals overlap. When the confidence intervals overlap like this, we cannot say that one correlation is superior to the other. For the subset of 50 compounds, there is a very small difference between the upper bound of the 95% confidence interval for Method A (0.91) and the lower bound of the confidence interval for Method B (0.92). While there is some separation, we may still have doubts as to whether one method outperforms the other. The bars on the far right in Figure 1.5 show the correlations for the subset of 100 compounds and the associated... 

The calculated value of Eq. (2) agrees with that of Eq. (1) determined from experiment, with their uncertainty intervals overlapping betw een -139 and -145 , where 6 is an additional uncertainty of the result reported in [8]. Assuming that the actual value of AH lies within the overlap interval, we obtain ... 

It is important to note that shifting the curves vertically does not alter the horizontal overlap between the curves. It therefore does not alter the amount by which the cold composite curve extends beyond the start of the hot composite curve at the hot end of the problem and the amount by which the hot composite curve extends beyond the start of the cold composite curve at the cold end. The shift simply removes the problem of ensuring temperature feasibility within temperature intervals. 

The main cost of this enlianced time resolution compared to fluorescence upconversion, however, is the aforementioned problem of time ordering of the photons that arrive from the pump and probe pulses. Wlien the probe pulse either precedes or trails the arrival of the pump pulse by a time interval that is significantly longer than the pulse duration, the action of the probe and pump pulses on the populations resident in the various resonant states is nnambiguous. When the pump and probe pulses temporally overlap in tlie sample, however, all possible time orderings of field-molecule interactions contribute to the response and complicate the interpretation. Double-sided Feymuan diagrams, which provide a pictorial view of the density matrix s time evolution under the action of the laser pulses, can be used to detenuine the various contributions to the sample response [125]. 

As m increases, At becomes progressively smaller (compare the difference between the square roots of 1 and 2 (= 0.4) with the difference between 100 and 101 (= 0.05). Thus, the difference in arrival times of ions arriving at the detector become increasingly smaller and more difficult to differentiate as mass increases. This inherent problem is a severe restriction even without the second difficulty, which is that not all ions of any one given m/z value reach the same velocity after acceleration nor are they all formed at exactly the same point in the ion source. Therefore, even for any one m/z value, ions at each m/z reach the detector over an interval of time instead of all at one time. Clearly, where separation of flight times is very short, as with TOF instruments, the spread for individual ion m/z values means there will be overlap in arrival times between ions of closely similar m/z values. This effect (Figure 26.2) decreases available (theoretical) resolution, but it can be ameliorated by modifying the instrument to include a reflectron. 

The relaxation at the lowest temperature y relaxation) takes place below - I0O°C. The two polymers with shorter spacers (PDEB and PTEB) show weak relaxations overlapped with the )3 ones due to the low tanS values (0.03 and 0.04, respectively). Notwithstanding this, the y relaxation is clearly distinguished when using loss modulus plots, even in the case of PDEB, that shows the weakest maximum (see Fig. 16). For PTTB, tan6 values in the y relaxation interval are of the order of 0.05. 

In many respects the time-resolved pump-probe technique is similar to the CW counterpart. The use of pulsed laser light permits direct probing of both the magnitude of the PA and its dynamics. The experimental arrangement is practically the same as for the CW version, i.e., both pump and probe beams are focused and overlapped onto same spot on a sample. In addition, the pump and probe pulses are synchronized so that the lime interval t between them is constant and confined to a certain time range (in our case up to 3 ns). 

Figure 4.5. Estimated total analytical cost for one batch of tablets versus the attained confidence interval CI(X). 640 (UV) resp. 336 (HPLC) parameter combinations were investigated (some points overlap on the plot).

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