T values, Student

We used a two-tailed test. Upon rereading the problem, we realize that this was pure FeO whose iron content was 77.60% so that p = 77.60 and the confidence interval does not include the known value. Since the FeO was a standard, a one-tailed test should have been used since only random values would be expected to exceed 77.60%. Now the Student t value of 2.13 (for —to05) should have been used, and now the confidence interval becomes 77.11 0.23. A systematic error is presumed to exist. 

Upper case letters are used for a quantity, for example Y may be the current at a glucose electrode. Small letters denote a particular quantity, for example y= 10nA. Example a correct statement of a /-test is that p(T>t) = 0.05 which reads the probability of finding a Student t value (T) equal to or greater than the t calculated from the data is 0.05. (Section 5.3)... 

Calculate a two-tailed Student t-value for a 95% confidence limit =TINV(0.05, df) df = degrees of freedom... 

Calculate the probability of a Student t-value =TDIST(f, df, tails) df = degrees of freedom tails = 1 (one-tailed) or 2 (two-tailed)... 

Table A.2 One-tailed Student t-values. As TINY only gives two-tailed values, we must multiply a by 2 to calculate the correct value, i.e., =TINV(2 x a, df)... 

In addition to the calibration coefficients, most multiple regression programs calculate a number of auxiliary statistics, which have the purpose of helping the operator decide how well the calibration fits the data, and how well the calibration can be expected to predict future samples. The most important of these statistics are the SEE, the multiple correlation coefficient (R), the statistic F for regression, and the Student t values for the regression coefficients. 

The standard deviation of the distribution of means equals cr/N. Since cr is not usually known, its approximation for a finite number of measurements is overcome by the Student t test. It is a measure of error between p and x. The Student t takes into account both the possible variation of the value of x from p on the basis of the expected variance and the reliability of using 5- in... 

Figure 1.18. The Student s ( resp. t/Vn for various confidence levels are plotted the curves for p = 0.05 are enhanced. The other curves are for p = 0.5 (bottom), 0.2, 0.1, 0.02, 0.01, 0.002, 0.001, and 0.0001 (top). By plotting a horizontal, the number of measurements necessary to obtain the same confidence intervals for different confidence levels can be estimated. While it takes n - 9 measurements (/ = 8) for a t-value of 7.12 and p = 0.0001, just n = 3 f - 2) will give the same limits on the population for p = 0.02 (line A - C). For the CL on the mean, in order to obtain the same t/ /n for p = 0.02 as for p = 0.0001, it will take n = 4 measurements (line B ) note the difference between points D and ...

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 ... 

Figure 2.3 shows the results obtained in a comparision of these methods on a range of deep-sea and offshore samples. The line of gradient 1.0 on each diagram shows the result which would have occurred had agreement been obtained. The Student t-test showed that in both exercises the colorimetric method with iodine water treatment yielded higher values than that without... 

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

The mean x of these measurements is 0.710259, while their standard deviation s is 0.0000104 (we take one more digit to keep a reasonable precision on ratios). Let us form the variable t which is meant to represent a specific value taken by the Student-t variable and such that... 

Statistical Methods. Means of treatment groups for plasma retention of BSP, plasma osmolality, total plasma protein concentration and urine flow rates were compared by students t test for independent sample means (17). Plasma enzyme activity data were converted to a quantal form and analyzed by the Fischer Exact Probability Test (18). Values greater than 2 standard deviations (P < 0.05) from the control value were chosen to indicate a positive response in treated fish. 

The value for is obtained from a table of student s values (see, for example. Table T-5 in Ref. 13, or Table A-4 in Ref. 26) for the desired confidence level and number of samples... 

Construction of an Approximate Confidence Interval. An approxi-mate confidence interval can be constructed for an assumed class of distributions, if one is willing to neglect the bias introduced by the spline approximation. This is accomplished by estimation of the standard deviation in the transformed domain of y-values from the replicates. The degrees of freedom for this procedure is then diminished by one accounting for the empirical search for the proper transformation. If one accepts that the distribution of data can be approximated by a normal distribution the Student t-distribution gives... 

Figure 8. Concentration of markers of adhesion (A) and differentiation (B) in rat aortic smooth muscle cells in cultures on polyethylene (PE) modified by irradiation with ions (energy 30 keV, doses from lO to lO ions/cm ). Measured by enzymatic immunosorbent assay (ELISA) per mg of protein, absorbances expressed in % of the values obtained on pristine non-modified PE. Mean + SEM from 4 experiments. Student t-test for unpaired data, p<0.05 p<0.01 compared to the values on pristine PE.

Normally the population standard deviation a is not known, and has to be estimated from a sample standard deviation s. This will add an additional uncertainty and therefore will enlarge the confidence interval. This is reflected by using the Student-t-distribution instead of the normal distribution. The t value in the formula can be found in tables for the required confidence limit and n-1 degrees of freedom. 

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... 

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

The variables 17, Ua, and are the corresponding uncertainty values for each parameter. They are computed to the 67% confidence interval by taking the standard error of each parameter in the regressions (i.e., ai, U2 and (73, and multiplying by their Student f-score ts (i.e., =ts SEo ), where ts is the Student t-score at the confidence level of interest and SEai is the corresponding standard error for the parameter ai. The period can be chosen based on the maximum value or another statistical parameter. The results of four experiments are given in Figure 9.8. 

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