Student’s t-statistic

The confidence limits for the slope and intercept may be calculated using the Student s t statistic, noting Equations 61-27 through 61-30 below. 

Fig. 2. t probability as a function of Student s t statistics for Enzyme Leach (a best case ) and Bioleach (b worst case )... 

An interlaboratory bias study is a limited form of method performance study used to determine the bias of a standard method or the bias introduced by laboratories that use the standard method. Laboratories are chosen for their competence in performing the method, and the organization is the same as for a method performance study. The number of laboratories in the study is determined by the statistics required. If the bias (6) is calculated as the difference between accepted reference value and mean of n laboratories results, the significance can be tested using the standard deviation of the mean, sR/ /n. The Student s t statistic is calculated as... 

Using Student s t statistics, hypothesis p 0 was tested (i). it was found that the hypothesis P 0 can be rejected at the 0.005 significance level. It therefore appears almost certain that the % carbon conversions to CH4, C H, BTX and Oils can be related to reaction temperature, H2 partial pressure, particle residence time, and gas residence time by the correlations. 

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... 

Student s t statistics (distributions) are widely used in solving statistical problems in chemical analysis involving small numbers of samples (n < 30). For further details, see the texts listed in Section 12.2.2.9. 

These two constraints require that the confidence limits be found when the standard deviation in the population mean is unknown. This is where the Student s t-statistics have a role to play Who was Student Anderson has introduced a little history (9) ... 

Include in your notebook the following data from both modes of calibration for each of the two metals, calculate the confidence limits at 95% probability using the Student s t-statistics for the ICV for both calibration modes. Report on the concentration of Pb in the coded unknown provided to you. Report on the Pb concentration of any unknown drinking water sample that you analyzed. Two statistical computer programs RSD and LSQUARES and written in BASIC are available for your use on the laboratory PCs. These programs are found in Appendix C. To use these programs, first download GWBASIC.exe on the contemporary Windows-based... 

Values of the Student s t-statistic are summarized in Table 2.2. For a sample size of six (five degrees of freedom, n — 1) and a 95% confidence interval a = 95%), the value of the Student s t is 2.571 ( 2.6). It is approximately equal to 2 for a Gaussian distribution. Table 2.3 compares the Student s t with the Gaussian distribution for several common combinations of sample numbers... 

The studentized residual is an indication of how well the calibration model estimates the analyte property in each sample. The studentized residual is similar to the Student s t-statistic the estimation error of each sample is converted to a distance in standard deviations away from zero. An additional term is often added to the calculation to correct for the weight each sample has in determining the calibration model. The studentized residual is increased for samples with a large leverage this is known as the studentized leverage corrected residuals. 

The two tailed student s t statistic for small sample sizes was 5.93 for HK and 5.59 for d. Both of these are greater than the critical t ratio for 17 degrees of freedom at the. 001 probability level (t 3.97). 

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