Straight confidence intervals

Fig. 7.8. Schematic three-dimensional representation of a calibration straight line of the form y = a + bx with the limits of its two-sided confidence interval and three probability density function (pdf) p(y) of measured values y belonging to the analytical values (contents, concentrations) X(A) = 0 (A), x = x(B) (B) and X(q = ld (C) yc is the critical value of the measurement quantity a the intercept of the calibration function yBL the blank x(B) the analytical value belonging to the critical value yc (which corresponds approximately to Kaiser s a3cr-limit ) xLD limit of detection... 

The limit of detection can also be estimated by means of data of the calibration function, namely the intercept a which is taken as an estimate of the blank, a ylu, and the confidence interval of the calibration straight fine ... 

The denominator n 2 is used here because two parameters are necessary for a fitted straight line, and this makes s2 an unbiased estimator for a2. The estimated residual variance is necessary for constructing confidence intervals and tests. Here the above model assumptions are required, and confidence intervals for intercept, b0, and slope, b, can be derived as follows ... 

In this figure, each individual confidence interval has been drawn as a vertical straight line joining the lower and upper limits. The horizontal line is positioned at the value 4.055 mmol/L- the population mean. This gives us 40 sample means that are not equal to one another, so they on their own like the original measurement show random... 

Figure 6 RBE variation of neutron beams as a function of energy. The p(65) + Be beam of Louvain-la-Neuve is taken as reference (RBE = 1). The closed squares and circles correspond to six different visited neutron facilities. The open squares and circles correspond to beams produced at the variable-energy cyclotron of Louvain-la-Neuve. On the abscissa, the effective energy of the neutron beams is expressed by their half-value thickness (HVT 5/15) measured in specified conditions. Intestinal crypt regeneration in mice, after a single fraction irradiation, is taken as the biological system. The 95% confidence intervals are shown. A straight line is fitted through the points (squares) corresponding to neutron beams produced by the (p + Be) reaction. For comparison, the neutron beams produced by the (d + Be) reaction are represented by circles. (From Ref. 22.)...

Fig. 13. Plasma concentration vs. time profiles of XYZ1234 following a single oral 100 mg dose given to poor CYP 2C19 metabolizing male and female subjects (straight lines), to extensive CYP 2C19 metabolizing females (dashed lines), and to extensive CYP 2C19 metabolizing males (dotted lines) respectively Mean values (bold) and 90% Confidence Interval...

Fig. 15. Straight line calibration with confidence interval of calibration defined in y direction a and confidence intervals of the analytical result defined in x-direction b applying variing amounts of repeated measurements in the analytical procedure S8>...

The general straight-line model is often used as a starting point for model development. The confidence interval often widens at the ends (see Section 3.5). More levels of the factor act to reduce the confidence interval. Equally spaced levels are often tested. The number of levels depends on the complexity of the model or on the number of additional terms to be considered. Enough levels should be provided for left-out terms. The number of levels should exceed the number of parameters expected. Even if a straight line is likely, more than three levels should be considered, plus some replication. 

Fig. 3 Characteristic strength versus (effective) volume in a double logarithmic plot. Shown are test results on specimens of different size. The straight line shows the Weibull extrapolation based on the four point bending test results. The dashed lines are the 90 % confidence intervals of the prediction.

For the confidence intervals of the predicted y values, the approaches given for the straight-line model hold (Eqs. (6.25) - (6.27)). Instead of the parameter vector b, the matrix of the parameter estimations B is now used. For prediction of a single y value according to Eq. (6.25), we get... 

The coefficients p, and p2 refer to the intercept and slope of the straight line fitting the data points (see Fig. 4 a). Being based on the random variables X and y, the coefficients p, and p2 are outcomes of random variables themselves. The true coefficients can be covered by the following 95 % confidence intervals ... 

The characteristic quantities obtained for the straight lines must be tested for their statistical significance and should be quoted with a confidence interval. Their standard deviation is a function of the residual standard deviation, Srcs of the regression model. This is defined as ... 

A more compact comparison between measured and calculated response is achieved by comparing the standard deviation of the angular response. The results in Figure 11 provide as comparison as a function of the mean heading direction. The straight horizontal lines correspond to the measured response with the middle line corresponding to the mean value and the upper and lower lines represent the 95 percent confidence interval. 

Use the simplest (straight-line) model to fity = f(x). Find the regressed parameters, their 95% confidence interval, Adj-R, SSE, and RMSE. Plot the residuals and see if the errors are randomly distributed around their zero-value mean. 

For an upper dwell temperature of 1020°C and upper dwell time of 16 h, the predicted time to spall is 95.1 h with a 95% confidence interval of 61 to 129 h. The experimentally determined values for Alloy 800 at 1000°C and 1050°C were 96 and 32 for 8 h upper dwell time and 160 and 40 for 20 h upper dwell time for comparison. Figure 18.11 shows the observed times to spall plotted against the times predicted by the model. Figure 18.11 shows a good fit between the observed values and the model because the points fall approximately on a straight line. The points for trials 4 and 9 are slightly off the line but there is no obvious connection between trials 4 and 9 to suggest a trend. 

In many applications, the Weibull analysis is applied to predict the part reliability or unreliability based on limited data with the help of modern computer technology. The limited data will inevitably introduce some statistical uncertainty to the results. Figure 6.11 shows the unreliability as a function of time in a Weibull plot. The six data points reasonably fit on the straight line and verify that the data are a Weibull distribution. The hourglass curves plotted on each side of the Weibull line represent the bounds of 90% confidence intervals for fhis analysis. The width of the intervals depends on the sample size it narrows when more samples are analyzed. 

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