F test

The F statistic, along with the z, t, and statistics, constitute the group that are thought of as fundamental statistics. Collectively they describe all the relationships that can exist between means and standard deviations. To perform an F test, we must first verify the randomness and independence of the errors. If erf = cr, then s ls2 will be distributed properly as the F statistic. If the calculated F is outside the confidence interval chosen for that statistic, then this is evidence that a F 2. 

It is possible to compare the means of two relatively small sets of observations when the variances within the sets can be regarded as the same, as indicated by the F test. One can consider the distribution involving estimates of the true variance. With sj determined from a group of observations and S2 from a second group of N2 observations, the distribution of the ratio of the sample variances is given by the F statistic ... 

The larger variance is placed in the numerator. For example, the F test allows judgment regarding the existence of a significant difference in the precision between two sets of data or between two analysts. The hypothesis assumed is that both variances are indeed alike and a measure of the same a. 

As applied in Example 12, the F test was one-tailed. The F test may also be applied as a two-tailed test in which the alternative to the null hypothesis is erj A cr. This doubles the probability that the null hypothesis is invalid and has the effect of changing the confidence level, in the above example, from 95% to 90%. 

A typical application of this significance test, which is known as a f-test of A to p, is outlined in the following example. 

The test statistic for evaluating the null hypothesis is called an f-test, and is given as either... 

The mean values obtained by the two analysts are compared using a two-tailed f-test. The null and alternative hypotheses are... 

The value of fexp is then compared with a critical value, f(a, v), which is determined by the chosen significance level, a, the degrees of freedom for the sample, V, and whether the significance test is one-tailed or two-tailed. For paired data, the degrees of freedom is - 1. If fexp is greater than f(a, v), then the null hypothesis is rejected and the alternative hypothesis is accepted. If fexp is less than or equal to f(a, v), then the null hypothesis is retained, and a significant difference has not been demonstrated at the stated significance level. This is known as the paired f-test. 

This is an example of a paired data set since the acquisition of samples over an extended period introduces a substantial time-dependent change in the concentration of monensin. The comparison of the two methods must be done with the paired f-test, using the following null and two-tailed alternative hypotheses... 

A statistical analysis allows us to determine whether our results are significantly different from known values, or from values obtained by other analysts, by other methods of analysis, or for other samples. A f-test is used to compare mean values, and an F-test to compare precisions. Comparisons between two sets of data require an initial evaluation of whether the data... 

In this experiment students standardize a solution of HGl by titration using several different indicators to signal the titration s end point. A statistical analysis of the data using f-tests and F-tests allows students to compare results obtained using the same indicator, with results obtained using different indicators. The results of this experiment can be used later when discussing the selection of appropriate indicators. 

In this experiment students measure the length of a pestle using a wooden meter stick, a stainless-steel ruler, and a vernier caliper. The data collected in this experiment provide an opportunity to discuss significant figures and sources of error. Statistical analysis includes the Q-test, f-test, and F-test. 

Students use a commercial diluter to prepare five sets of dilutions of a stock dye solution (each set contains ten replicates) using two different diluters. Results are compared using f-tests and F-tests. 

The stretching properties of polymers are investigated by examining the effect of polymer orientation, polymer chain length, stretching rate, and temperature. Homogeneity of polymer films and consistency between lots of polymer films also are investigated. Statistical analysis of data includes Q-tests and f-tests. 

The absorbance of solutions of food dyes is used to explore the treatment of outliers and the application of the f-test for comparing means. 

If systematic errors due to the analysts are significantly larger than random errors, then St should be larger than sd. This can be tested statistically using a one-tailed F-test... 

Once a significant difference has been demonstrated by an analysis of variance, a modified version of the f-test, known as Fisher s least significant difference, can be used to determine which analyst or analysts are responsible for the difference. The test statistic for comparing the mean values Xj and X2 is the f-test described in Chapter 4, except that Spool is replaced by the square root of the within-sample variance obtained from an analysis of variance. 

Fisher s least significant difference a modified form of the f-test for comparing several sets of data. (p. 696) flame ionization detector a nearly universal GC detector in which the solutes are combusted in an H2/air flame, producing a measurable current, (p. 570)... 

F-test statistical test for comparing two variances to see if their difference is too large to be explained by indeterminate error, (p. 87)... 

Textile motors Crane motors Determining the size of motor Sugar centrifuge motors Motors for deep-well pumps Motors for agricultural application Surface-cooled motors Torque motors or actuator motors Vibration and noise level Service factors Motors for hazardous locations Specification of motors for Zone 0 locations Specification of motors for Zone I locations Motors for Zone 2 locations Motors for mines, collieries and quarries Intrinsically safe circuits, type Ex. f Testing and certifying authorities Additional requirements for ciritical installations Motors for thermal power station auxiliaries Selection of a special-purpose motor... 

Fracture mechanics can be applied to this problem, much as it is to fatigue. We use only the final result, as follows. If the standard test which was used to measure (Tts takes a time f(test), then the stress which the sample will support safely for a time t is... 

Table 18. Statistical comparison (F-test [125]) of the methods. Standard deviation Sxo of the calibration curves for diethylstilbestrol and ethinylestradiol [114].

Figure 10.24a and the allosteric model in Figure 10.24b. The circled data points were changed very slightly to cause an F-test to prefer either model for each respective model, illustrating the fallacy of relying on computer fitting of data and statistical tests to determine molecular mechanism. As discussed in Chapter 7, what is required to delineate orthosteric versus allosteric...

The same conclusion can be drawn from another statistical test for model comparison namely, through the use of Aikake s information criteria (AIC) calculations. This is often preferred, especially for automated data fitting, since it is more simple than F tests and can be used with a wider variety of models. In this test, the data is fit to the various models and the SSq determined. The AIC value is then calculated with the following formula... 

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