Critical values of Student

The other piece of information (in addition to bg and required to establish a confidence interval for a parameter estimate was not available until 1908 when W. S. Gosset, an English chemist who used the pseudonym Student (1908), provided a solution to the statistical problem [J. Box (1981)]. The resulting values are known as critical values of Student s t and may be obtained from so-called /-tables (see Appendix B for values at the 95% level of confidence). 

The data in this example represent an investigation of ammonia production in the presence of a particular catalyst. The measured yield in percent is shown in the far right column of Table 8.10. Suppose we wish to achieve a prediction error less than fi = 1.5% in an example, where the standard deviation (measurement error) is y.2 = 1.05 estimated with 11 measurements, i.e., with degrees of freedom v= 10. The critical value of Student s /-statistic is found to be t10A95 = 2.228. At IV = 16 experiments, we check to see if the desired level of accuracy is achieved and obtain Kj-a Jd = 2.228 0.438 x 1.05 =1.51 > 1.5. At N= 16 experiments, weobtain tvl aJd =1.48 < 1.5 therefore, we can stop at N = 17 and be assured that, 95% of the time, we will achieve a prediction error not worse than +1.5%, which is considerably smaller than the range of the variation in the response value. 

Table 41.2 Critical values of Student s t statistic (for two-tailed tests). Reject the null hypothesis at probability P if your calculated t value exceeds the value shown for the appropriate degrees of freedom = (ni — 1) + (02 1)...

On many occasions, sample statistics are used to provide an estimate of the population parameters. It is extremely useful to indicate the reliability of such estimates. This can be done by putting a confidence limit on the sample statistic. The most common application is to place confidence limits on the mean of a sample from a normally distributed population. This is done by working out the limits as F— ( />[ i] x SE) and F-I- (rr>[ - ij x SE) where //>[ ij is the tabulated critical value of Student s t statistic for a two-tailed test with n — 1 degrees of freedom and SE is the standard error of the mean (p. 268). A 95% confidence limit (i.e. P = 0.05) tells you that on average, 95 times out of 100, this limit will contain the population... 

Note the value of 2.26 was obtained from tables of critical values of Student s t statistics at the 95% confidence interval for n — l. k This infers that in taking 10 samples, an error of 5.14 ppm was tolerated, and that the concentration of lead in the sample should be expressed as 93 5.14 ppm. 

Note the value of 1.96 was obtained from tables of critical values of Student s t statistics at the 95% confidence interval for n = oo. f... 

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. 

Table 1 Tabulated values of Student s t critical values...

Table 2.10 Critical Values of x (from CRC Handbook) for the Student t Distribution...

Table 5 Critical values of the Chi-square and student s t distributions ...

Certified materials were used to evaluate the accuracy of the technique. These included coal, coal fly ash, soil, and crude oil. Table 3 shows that measured concentrations of mercury in the SRMs agree well with certified values. With only one exception, the Relative Percent Difference (RPD) is within 5 % of the certified values. The one exception is for a value that is less than half of the claimed detection limit for the published method. Six repetitions were made for each material. The expanded uncertainty of each measurement is calculated by multiplying the value for of one experimentally derived standard deviation of the data times the critical value. The critical value of 2.57 is determined from a Student s t-distiibution [12] with 5 degrees of fi-eedom and a confidence level of 95 %. 

Three analyses were made of each spike sample. The uncertainty was determined in a similar manner as before. The critical value of 4.3 is determined from a Student s t-distribution [12] with 2 degrees of freedom and a confidence level of 95 %. With just one exception, the recoveries were within 5 % of the expected concentrations. The fact that dimethyl mercury was the most volatile of the analyte species used in the study may have contributed to the slightly lower recovery of its spike sample. 

Our laboratory has tested over 80 crude oil and condensate samples to date. Three external laboratories also tested some of the samples as part of a cooperative effort to develop standard procedures for handling and analysis. The laboratories used techniques other than combustion-CVAAS. Details of the outside labs testing statistics are not known. Table 5 shows a representative subset of these results for comparison. Each analysis by our laboratory was done three times and averaged. The same statistics apply as those discussed in the preceding paragraph. The critical value of 4.3 with 2 degrees of freedom from the Student s t distribution table [12] exaggerates the small differences that were actually observed between the three mercury determinations. 

The probability value (p-value) of a statistical hypothesis test is closely related to the significance level a the p-value is the critical value of the statistical parameter (z or m in the Gaussian case in Section 8.2.3, or the Student-t introduced in Section 8.2.5) for which H, would be only just rejected for a given value of a. That is, the p-value is the threshold value for false positives (type I... 

Estimated DL values for the 24 PAH analytes were obtained as follows. Six (empty bottle) procedural blank samples were analyzed in a parallel study, yielding concentration values for an equivalent 0.5 L water sample. The SE values were multiplied by the one-sided critical value of the Student s t-distribution (t.01[5j) statistic to obtain an estimate of the upper 99 percent confidence limit for the actual concentration. This upper limit value was accepted as the estimated detection limit of the method (Table 1). However, we elected not to quantify concentrations below 0.1 ng/L (procedural blank values <0.1 ng/L were not used to obtain net concentrations). 

Table 1.3. Critical Student s t-Factors for the One- and Two-Sided Cases for Three Values of the Error Probability p and 7 Degrees of Freedom f...

Critical ( -values for p - 0.05 are available. " - In lieu of using these tables, the calculated -values can be divided by the appropriate Student s t(f, 0.05) and V2 and compared to the reduced critical -vdues (see Table 1.12), and data file QRED TBL.dat. A reduced -value that is smaller than the appropriate critical value signals that the tested means belong to the same population. A fully worked example is found in Chapter 4, Process Validation. Data file MOISTURE.dat used with program MULTI gives a good idea of how this concept is applied. MULTI uses Table 1.12 to interpolate the cutoff point for p = 0.05. With little risk of error, this table can also be used fo = 0.025 and 0.1 (divide q by t(/, 0.025) /2 respectively t f, 0.1) V 2, as appropriate. 

QRED-TBL.dat Table 1.7 Reduced critical -values (division of -values for p = 0.05 by the appropriate Student s f-factor and SQR(2)), as used with the multiple range test in MULTI. 

We calculate the confidence limits as shown above and translate onr Nnll hypothesis into a mathematical formnla resulting in a formnla for an observed Student-t-factor tobseived- We can now compare this observed valne with the critical value for 95% confidence and the degrees of freedom for onr nnmber... 

The majority of extreme data were received from the universities. It was evident that the switch-on, inject, switch-off approach was applied (likely by the students) without critical evaluation of data produced by the computers software. The least scattered M values were obtained in the industrial laboratories, in which evidently the skilled operators performed the measurements. Better data accuracy was obtained for polyamides [154] and for oligomeric polyepoxides [155] than for the unproblematic poly(dimethyl siloxane)s and even for the most simple polymer, polystyrene likely because only experts measured the latter difficult samples. 

The t-statistic follows what is known as the Student s t-distribution, after the statistician William Sealy Gosset (1876-1937) who published under the pseudonym StudenT. The shape of the t-distribution is similar to that of the normal distribution, but forms a family of curves distinguished by a parameter known as the degrees of freedom. The 5% critical point in the t-distribution always exceeds the normal value of 1.96, but is nevertheless close to 2.0 for all but quite small values of degrees of freedom. 

For common statistics, such as the Student s t value, chi-square, and Fisher F, Excel has functions that return the critical value at a given probability and degrees of freedom (e.g., =TINV (0.05,10) for the two-tailed Lvalue at a probability of 95% and 10 degrees of freedom), or which accept a calculated statistic and give the associated probability (e.g., =TDIST( t, 10, 2 ) for 10 degrees of freedom and two tails). Table 2.3 gives common statistics calculated in the course of laboratory quality control. 

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