Intercept, confidence interval

The confidence limits for the slope are given by fc where the r-value is taken at the desired confidence level and (A — 2) degrees of freedom. Similarly, the confidence limits for the intercept are given by a ts. The closeness of x to X is answered in terms of a confidence interval for that extends from an upper confidence (UCL) to a lower confidence (LCL) level. Let us choose 95% for the confidence interval. Then, remembering that this is a two-tailed test (UCL and LCL), we obtain from a table of Student s t distribution the critical value of L (U975) the appropriate number of degrees of freedom. 

Example 14 For the best-fit line found in Example 13, express the result in terms of confidence intervals for the slope and intercept. We will choose 95% for the confidence interval. 

These standard deviations can be used to establish confidence intervals for the true slope and the true y-intercept... 

Calculate the 95% confidence intervals for the slope and y-intercept determined in Example 5.10. 

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 ... 

In principle, there is also a test for the intercept. But since the expected value for the intercept depends on the slope, it gets a bit hairy. It also makes the confidence interval so large that the test is nigh on useless - few statisticians recommend it. 

This suggests that a plot of r0 vs pA (Fig. 4) or r0/pA vs pA should be linear. It is very difficult to reject this model on the basis of data curvature, even though it is evident that some curvature could exist in Fig. 4. However, Eq. (16) demands that Fig. 4 also exhibit a zero intercept. In fact, the 99.99% confidence interval on the intercept of a least-square line through the data does not contain zero. Hence the model could be rejected with 99.99% certainty. 

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 ... 

A 95% confidence interval on slope or intercept is obtained by multiplying these standard errors by a t value at the desired probability and n — 2 (n - 1) degrees of freedom ... 

It is usual to test the slope of the regression line to ensure that it is significant using the F ratio, but for most analytical purposes this is an academic exercise. Of more importance is whether the intercept is statistically indistinguishable from zero. In this instance, the 95% confidence interval for the intercept is from — 0.0263 to +0.0266 which indicates that this is the case. 

Then, in the same way that one talks about a certain (percentage) confidence interval for a given series of measurements (see pp 54-57), one can also talk about the (percentage) confidence intervals for the slope and y intercept of a best-fit line. They are related to the standard error of estimate jv/x and, for the desired level of confidence, the t value that corresponds to one less than the number of data pairs, as follows. For the slope,... 

Find the 95% confidence interval of the slope and intercept of the best-fit equation obtained in the problem on p 75 involving thermocouple voltage versus temperature. 

Note Problems 3 through 7 and 10 through 14 are, m part, given at two levels of sophistication and expectation The best" answers involve doing the part marked with an asterisk ( ) instead of the immediately preceding part The parts with an involve the use of the method of least squares, correlation coefficients, and confidence intervals for the slopes and intercepts, the preceding parts do not... 

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