Student s f distribution

In the case of finite sample size in analytical practice, the quantiles of Student s f-distribution are used as realistic limits. 

Table 6.3. Values of the two-tailed Student s f distribution for 90%, 95%, and 99% confidence calculated using the Excel function...

Thus, SE(ai) = and SE(af) = Now returning to Equation (B.3.1), T is the experimental deviation over the standard error and if this value is larger than the value in a Student s F-distribution table (see any text on statistics for this table) for a given degree of confidence, for example, 95 percent (7 xp = expected deviation/standard error), then the hypothesis is rejected, that is, the y-intercept is significantly different than zero. If < t xp then the hypothesis is accepted and dij can be reported as ... 

Figure 1.15. Student s f-distributions for 1 (bottom), 2, 5, and 100 (top) degrees of freedom /. The hatehed area between the innermost marks is in all cases 80% of the total area under the respective curve. The other marks designate the points at which the area reaches 90, resp. 95% of the total area. This shows how the f-factor varies with/. The f-distribution for/ = 100 already very closely matches the normal distribution. The normal distribution, which corresponds to t( f = ==), does not depend on /.

Statistical theory teaches that under the assumption that the population means of the two groups are the same (i.e. if Hq is true), the distribution of variable T depends only on the sample size but not on the value of the common mean or on the measurements population variance and thus can be tabulated independently of the particulars of any given experiment. This is the so-called Student s f-distribution. Using tables of the f-distribution, we can calculate the probability that a variable T calculated as above assumes a value greater or equal to 4.7, the value obtained in our example, given that H0 is true. This probability is <0.0001. Thus, if H0 is true, the result obtained in our experiment is extremely unlikely, although not impossible. We are forced to choose between two possible explanations to this. One is that a very unlikely event occurred. The second is that the result of our experiment is not a fluke, rather, the difference Mb — Ma is a positive number, sufficiently large to make the probability of this outcome a likely event. We elect the latter explanation and reject H0 in favor of the alternative hypothesis Hx. 

For smaller sample sizes (n < 30), the use of the Studentized approach is recommended, as it follows the Student s f-distribution with n — k — df. The Studentized residual Sr,) is computed as... 

The jackknife procedure reflects an expectation si iV(0, (r ), which is the basis for the Student s f-distribution at a/2 and n k 2 degrees of freedom. The jackknife residual, however, must be adjusted, because there are, in fact, n tests performed, one for each observation. If n = 20, a = 0.05, and a two-tail test is conducted, then the adjustment factor is... 

Table 2.6 Quantile of the one-sided Student s f distribution for three significance levels a and different degrees of freedom f. Note how the distribution approaches the Gaussian distribution if the degrees of freedom tend to infinity (cf. Table 2.5).

Table A.3 Two- and one-sided Student s f-distribution for different risk levels a and the degrees of freedom from f = to f = 20.

Two-sided Student s f-distribution One-sided Student s f-distribution ... 

Only —1% of measurements for a blank are expected to exceed the detection limit. However, 50% of measurements for a sample containing analyte at the detection limit will be below the detection limit. There is a 1% chance of concluding that a blank has analyte above the detection limit. If a sample contains analyte at the detection limit, there is a 50% chance of concluding that analyte is absent because its signal is below the detection limit. Curves in this figure are Student s f distributions, which are broader than the Gaussian distribution. 

Figure 1.9 shows examples of how these kinds of problems can appear on a probabiUty plot. Figure 1.9a shows a normal probability distribution with mean 0 and variance 1 with 2 outliers (circled). Notice how the outliers can cause some of the adjacent points to also be skewed from the ideal location. Figure 1.9b shows the case where the tails of the distribution do not match. In this case, a 2-degree-of-freedom Student s f-distribution was compared against the normal distribution. The f-distribution has larger tails than the normal distribution. This can clearly be seen by the deviations on both sides from the central line. Figure 1.9c shows the case... 

The Student s t-distribution, denoted as t(v) or more commonly as f, is a statistical distribution that is used for dealing with the estimation of the mean of a normal distribution when the sample size is small and the population standard deviation is unknown. It approaches the normal distribution as the number of degrees of freedom, v, approaches infinity. In general, the Student s f-distribution has larger tails than the normal distribution. Useful properties of the Student s f-distribution are summarised in Table 2.2, while Fig. 2.3 compares the Student s f-distribution with the normal distribution. 

A conventional way to apply Student s t-distribution is to pick up data from the critical value table of Student s f-distribution and considering degrees of freedom, levels of significance, and measured data. It is troublesome to repeat this conventional method many times. Most spreadsheet software even for personal computers includes a Student s f-distribution function. Without any tedious work, namely, picking up data from the table, statistical treatment can be applied to experimental results based on Student s f-distribution with the aid of a computer. Figure 15.12 displays an... 

FIGURE 15.12 Spreadsheet for statistical data-treatment based on Student s f-distribution. 

To continue the calculations one needs a value of Student s f-distribution coefficient for the given number of degrees of freedom. Special tables are usually used for this however, they are not required when working with Mathcad as the suite has a built-in function qt. For the most used confidence level of 95%, equals ... 

In the previous section, the problem of estimating the population mean has been addressed when dealing with a sample mean x, but knowing the population standard deviation A. Student s f-distribution or simply the f-distribution, is used when not only the population mean fi is unknown because the sample size is small, but also the population standard deviation A is unknown and has to be estimated from the data available that provide a sample standard deviation s. Its theoretical basis was published for the hrst time in i908 by Gosset [3]. Gosset, however, was employed at a brewery that forbade the publication of researches and studies by its staff members. To circumvent this restriction, Gosset, that was a student, decided... 

The shape of the Student s f-distribution (4.17) resembles the bell shape of the Gaussian distribution with mean /r = 0 and variance = 1, except that it is a little lower and wider. As the number of degrees of freedom grows, the t-distribution approaches the normal distribution with mean ix — 0 and variance = 1. Figure 4.10 is a sketch of the Student s distribution. Its main feature is... 

Fig. 4.10 Sketch of Student s f-distribution with mean /i = 0 and variance...

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