Student critical values

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. 

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. 

VHien this method is used, Table II shows the results when the regression model is the normal first order linear model. Since the maximum absolute studentized residual (Max ASR) found, 2.29, was less than the critical value relative to this model, 2.78, the conclusion is that there are no inconsistent values. 

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). 

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... 

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. 

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

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. 

The bias is significant if the calculated Student s /-value is larger than the tabulated critical ti-a/2,n - i- The bias should be included in the measurement uncertainty if significant. 

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

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... 

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

Statistics teaches that the deviation of data based on less than 30 measurements is not a normal distribution but Student s t-distribution. So it is suitable to express the binding constant K with 95% confidence interval calculated by applying by Student s t-distribution. Student s t-distribution includes the normal distribution. When the number of measurements is more than 30, Student s t-distribution and the normal distribution are practically the same. The actual function of Student s t-distribution is very complicated so that it is rarely used directly. A conventional way to apply Student s t-distribution is to pick up data from the critical value table of Student s t-distribution under consideration of degree of freedom , level of significance and measured data. It is troublesome to repeat this conventional way many times. Most spreadsheet software even for personal computers has the function of Student s t-distribution. Without any tedious work, namely, picking up data from the table, statistical treatment can be applied to experimental results based on Student s t-distribution with the aid of a computer. In Fig. 2.12, an example is shown. When the measurement data are input into the gray cells, answers can be obtained in the cell D18 and D19 instantaneously. 

The critical value from the studentized range statistic... 

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... 

Upper bounds for externally studentized residuals have not been developed. Externally studentized residuals are distributed as a Student s t-distribution with n—p—1 degrees of freedom. Thus in the case of a single outlier observation, a quick test would be to compare the value of the external studentized residual to the appropriate t-distribution value, although as Cook and Weisberg (1999) point out, because of issues with multiplicity a more appropriate comparison would be Student s t-distribution with a/n critical value and n—p—1 degrees of freedom. In general, however, a yardstick of 2 or 2.5 is usually used as a critical value to flag suspect observations. 

In a table of Student s t we find the critical value for 95% probability and 5 degrees of freedom to be 2.57. It therefore appears that the coefficient h, .22 is statistically significant and the model is inadequate. 

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. 

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