Student’s t-value

Use Calculate Student s t-values given p and df Student s t is used instead of the normal deviate z when the number of measurements that go into a mean is relatively small and the assumption of p and a being infinitely precise has to be replaced by the assumption of a normally distributed mean and a x -distributed s. ... 

This range is a confidence interval of the average of two values with a Student s t value of 2, which corresponds to a free degree of forty six with which the average coefficient of variation is known. So for benzene the only value of LD50 provided by Sax (3306 mg/kg) would lead to the following range ... 

Thermal mass flowrate Coil heat transfer coefficient Deviation from side average Deviation student s t value... 

Table 4.1 Student s t Values for Various Confidence Levels and Numbers of Measurements...

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. 

Because historical data for the soil is not available, to estimate the number of samples for the stockpile characterization, we collect four preliminary samples and analyze them for lead. The concentrations are 120, 60, 200, and 180 mg/kg, with the average concentration of 140 mg/kg and the standard deviation of 63 mg/kg. We use Equation 11, Appendix 1, for the calculation of the estimated number of samples. Using a one-tailed confidence interval and a probability of 0.05, we determine the Student s t value of 2.353 for 3 degrees of freedom (the number of collected samples less one) from Table 1, Appendix 1. 

Step 7 Obtain Student s t value corresponding to the degree of freedom value determined in Step 5 Look up Table 1, Appendix 1, for the selected t-value t-a. —... 

In this case the student s t value for 95% confidence level requires that (X = 0.05 so that Cf/2 = 0.025 and with 4 degrees of freedom is... 

Sometimes a repetition of a measurement yields a value that differs greatly from the other members of the sample (a discordant value). For example, say that we repeated the measurement of the melting temperature of Ca(N03)2 4H2O in the previous example one more time and obtained a value of 39.75 °C. If we include this eleventh data point, we get a sample mean of 42.43 °C and a sample standard deviation of 0.89 °C. Using the table of Student s t values, we obtain a value for of 0.60 °C at the 95% confidence level. Some people think that the only honest thing to do is to report the melting temperature as 42.4 0.6 °C. 

The Student s t value can be used to test for systematic error (bias) by comparing means of different determinations. The CL equation is rewritten as ... 

Where t = the Student s t value appropriate for a 99% confidence level (one-sided) and a standard deviation estimate with n-1 degrees of freedom and s = the standard deviation of the replicate analyses of standards or samples with low concentration of analyte. Figure 5 depicts the MDL as an error distribution. As originally proposed, the procedure did not allow for blank correction. The latest version of the procedure does allow for blank correction, although the standard deviation of analyte response rather than the standard deviation of the blank Is used to calculate the detection limit. The distribution shown In Figure 5 Is that of the result (presumably after blank correction If necessary). 

These differences are then tested against the experimental error expressed by the standard deviation s multiplied by the Student s t value ... 

For high precision the values of t and of the standard error of variance should be minimized. The value of t will be obtained from Student s t distribution tables in this case and its value will largely be determined by the confidence level required of the estimate. If a 95% confidence level is required then the corresponding Student s t value for 60 samples is 2.00. The 95% confidence level infers that on average one out of twenty estimates of mixture variance fall outside the stated precision limits. If a higher confidence level was required then the value of Student s t would increase and the precision of the estimate would decrease. 

Figure 8.11 Graphical representations of the definition and implications of the EPA definition of an MDL. (a). Assumed normal frequency distribution of measured concentrations of MDL test samples spiked at one to five times the expected MDL concentration, showing the standard deviation s. (b) Assumed standard deviation as a function of analyte concentration, with a region of constant standard deviation at low concentrations, (c) The frequency distribution of the low concentration spike measurements is assumed to be the same as that for replicate blank measurements (analyte not present), (d) The MDL is set at a concentration to provide a false positive rate of no more than 1% (t = Student s t value at the 99 % confidence level), (e) Probability of a false negative when a sample contains the analyte at the EPA MDL concentration. Reproduced with permission from New Reporting Procedures Based on Long-Term Method Detection Levels and Some Considerations for Interpretations of Water-Quality Data Provided by the US Geological Survey NationalWater Quality Laboratory (1999), Open-File Report 99-193.

Confidence interval (95%CI) In most forensic analyses, there will be three or fewer replicates per sample, not enough for standard deviation to be a reliable expression of uncertainty. Even the 10 samples used in the foregoing examples represent a tiny subset of the populabon of measurements that could have been taken. One way to account for a small number of samples is to apply a multiplier called the Student s t-value as follows ... 

Table 2.2 Student s t-Values (Abbreviated) Complete Table See Appendix 11 for...

The obvious solution would be to use SOx directly. The best estimate of the standard deviation at x is a direct measurement, rather than a complicated calculation. Dividing the maximum tolerated deviation of 33% by Student s t-value of 3.3 gives a maximum tolerated cv(%) of 10%. A method can break down as a result of the difEculty of obtaining a cv(%) of 5% instead of 10%. 

Where t = Student s t value for a 95% confidence level and specifies a standard deviation estimate with n - 1 degrees of freedom (f = 3.14 for 7 replicates) and S = the SD of the replicate analyses. 

Method detection limit (MDL) Prepare seven samples at 3-5 times the estimated MDL concentration, as described in the U.S. ERA regulations at 40 CFR, Part 136, Appendix B, for the digestion of samples using the open-vessel technique. Note that the MDL samples should be prepared in the synthetic FGDW matrix. Analyze the MDL standards in triplicate. Calculate the standard deviation of the concentration(s) in pg/L for each analyte. The MDL is calculated as the student s t-value for the degrees of freedom (i.e., 3.143 for 6 degrees of freedom) multiplied by the standard deviation. 

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