T distribution table

Since the comparison of rxy with the table values may be considered a somewhat weak test, it is perhaps more meaningful to compare the tr value with values in a t-distribution table for N-2 degrees of freedom (df), as is done for Student s I-tcst. This will give a more exact determination of the degree of statistical correlation between the two groups. 

The new statistic t is usually referred to as student s t-distribution Table C, after W.S. Gosset, who first worked out its distribution. For a normal population ... 

The critical value of the t test can be abbreviated as fa(2),v, where a(2) refers to the two-tailed probability of a and v = n — 1 (degree of freedom). For the two-tailed t test, compare the calculated t value with the critical value from the t distribution table. In general, if t > ta(2),v, then reject the null hypothesis. When comparing the means of replicate determinations, it is desirable that the number of replicates be the same in each case. 

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. 

Roughly, then, x s/y/n is the interval in which the tme population mean jju will be found 68 times out of 100. The 95% confidence interval for the population mean /r is x 2.0s jy/n. However, because s/y/n slightly overestimates the interval, 95 out of 100 times, the true /r will be contained in the interval, x 1.96s/v , given the sample size is large enough to assure a normal distribution. If not, the Student s t distribution (Table B) is used, instead of the Z distribution (Table A). 

A comparison of the calculated test value t with the tabulated value of the t distribution (Table 2.6) at a given risk level a for the degree of freedom f = — al2 /), provides the decision on... 

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. 

Values of t for different numbers of degrees of freedom and upper tail probabilities can be looked up in a t-distribution table such as Table 20.2. More extensive tables can be consulted in books and on the internet. 

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. 

The t (Student s t) distribution is an unbounded distribution where the mean is zero and the variance is v/(v - 2), v being the scale parameter (also called degrees of freedom ). As v -> < , the variance —> 1 (standard normal distribution). A t table such as Table 1-19 is used to find values of the t statistic where... 

The table gives the value of f . — the 100a percentage point of the t-distribution for v degrees of freedom. 

Various substituted styrene-alkyl methacrylate block copolymers and all-acrylic block copolymers have been synthesized in a controlled fashion demonstrating predictable molecular weight and narrow molecular weight distributions. Table I depicts various poly (t-butylstyrene)-b-poly(t-butyl methacrylate) (PTBS-PTBMA) and poly(methyl methacrylate)-b-poly(t-butyl methacrylate) (PMMA-PTBMA) samples. In addition, all-acrylic block copolymers based on poly(2-ethylhexyl methacrylate)-b-poly(t-butyl methacrylate) have been recently synthesized and offer many unique possibilities due to the low glass transition temperature of PEHMA. In most cases, a range of 5-25 wt.% of alkyl methacrylate was incorporated into the block copolymer. This composition not only facilitated solubility during subsequent hydrolysis but also limited the maximum level of derived ionic functionality. 

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. 

CVt = 0.25/1.96 = 0.128. The number 0.128 is the largest true precision for a net error at +25% at the 95% confidence level. The number 1.96 is the appropriate t - statistic from the t distribution at the same confidence level. Since the coefficient of variation of pump error is assumed to be 5%, a method should have a CV analysis <0.102 to meet the CV accuracy standard. Tables IV and V7 shows that the infrared technique meets this requirement. 

Table 2.2. Values of the two-tailed Student s t distribution calculated in Excel by =TINV(0.05,...

Here n is the average refractive index, k is Boltzman s constant, and T is absolute temperature (13). If a polyblend were to form a homogeneous network, the stress would be distributed equally between network chains of different composition. Assuming that the size of the statistical segments of the component polymers remains unaffected by the mixing process, the stress-optical coefficient would simply be additive by composition. Since the stress-optical coefficient of butadiene-styrene copolymers, at constant vinyl content, is a linear function of composition (Figure 9), a homogeneous blend of such polymers would be expected to exhibit the same stress-optical coefficient as a copolymer of the same styrene content. Actually, all blends examined show an elevation of Ka which increases with the breadth of the composition distribution (Table III). Such an elevation can be justified if the blends have a two- or multiphase domain structure in which the phases differ in modulus. If we consider the domains to be coupled either in series or in parallel (the true situation will be intermediate), then it is easily shown that... 

A few values of the t-distribution are given in an accompanying table. We note that t values are considerably higher than corresponding standard normal values for small sample size but as n increases, the t-distribution asymptotically approaches the standard normal distribution. Even at a sample size as small as 30, the deviation from normality is small, so that it is possible to use the standard normal distribution for sample sizes larger than 30 (n>30) and in most cases, for n<30 t-distribution is used. This is equivalent to assuming that Sx is an exact estimate of ox at large sample sizes (n>30). 

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