Significance level

For example, if an effect has to be of magnitude 15% to be of practical importance, and our data gives us as our estimate of it 5% 2%, it is clear that the effect is statistically significant but not of practical importance. Alternatively if our result was 5% 10%, then it is not statistically significant on our present data but may still be of practical importance. 

The application of statistical methods generally in-volves a certain amount of computation. A large part of this is the summing of numbers and of their squares this can easily be performed with a table of squares ( Barlow s Tables of Squares, etc. (E. and F. M. Spon) is a very complete set), and -with a-simple adding machine. For much correlation work on calculating machine (one that will multiply and di-vide) is almost essential. There are a -variety of modds available it can be safely said that the better the machine the more accurately and rapidly can the work be carried out. An electric machine with automatic division and multiplication is really desirable. 

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If the r-value falls short of the formal significance level, this is not to be interpreted as proving the absence of a systematic error. Perhaps the data were insufficient in precision or in number to establish the presence of a constant error. Especially when the calculated value for t is only slightly short of the tabulated value, some additional data may suffice to build up the evidence for a constant error (or the lack thereof). 

Since significance tests are based on probabilities, their interpretation is naturally subject to error. As we have already seen, significance tests are carried out at a significance level, a, that defines the probability of rejecting a null hypothesis that is true. For example, when a significance test is conducted at a = 0.05, there is a 5% probability that the null hypothesis will be incorrectly rejected. This is known as a type 1 error, and its risk is always equivalent to a. Type 1 errors in two-tailed and one-tailed significance tests are represented by the shaded areas under the probability distribution curves in Figure 4.10. 

The critical value for F(0.05, 6, 4) is 9.197. Since Fexp is less than F(0.05, 6, 4), the null hypothesis is retained. There is no evidence at the chosen significance level to suggest that the difference in precisions is significant. 

The value of fexp is compared with a critical value, f(a, v), as 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. 

The critical value for f(0.05, 10), from Appendix IB, is 2.23. Since fexp is less than f(0.05, 10) the null hypothesis is retained, and there is no evidence that the two sets of pennies are significantly different at the chosen significance level. 

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. 

Significance tests, however, also are subject to type 2 errors in which the null hypothesis is falsely retained. Consider, for example, the situation shown in Figure 4.12b, where S is exactly equal to (Sa)dl. In this case the probability of a type 2 error is 50% since half of the signals arising from the sample s population fall below the detection limit. Thus, there is only a 50 50 probability that an analyte at the lUPAC detection limit will be detected. As defined, the lUPAC definition for the detection limit only indicates the smallest signal for which we can say, at a significance level of a, that an analyte is present in the sample. Failing to detect the analyte, however, does not imply that it is not present. 

Determine if there are any potential outliers in Sample 1, Sample 2, or Sample 3 at a significance level of a = 0.05. 

Reevaluate the data in problem 24 in Chapter 4 using the same significance level as in the original problem. ... 

This value of fexp is compared with the critical value for f(a, v), where the significance level is the same as that used in the ANOVA calculation, and the degrees of freedom is the same as that for the within-sample variance. Because we are interested in whether the larger of the two means is significantly greater than the other mean, the value of f(a, v) is that for a one-tail significance test. 

A major concern when remediating wood-treatment sites is that pentachlorophenol was often used in combination with metal salts, and these compounds, such as chromated copper—arsenate, are potent inhibitors of at least some pentachlorophenol degrading organisms (49). Sites with significant levels of such inorganics may not be suitable candidates for bioremediation. 

Similar to oil-fired plants, either low NO burners, SCR, or SNCR can be appHed for NO control at PC-fired plants. Likewise, fabric filter baghouses or electrostatic precipitators can be used to capture flyash (see Airpollution controlmethods). The collection and removal of significant levels of bottom ash, unbumed matter that drops to the bottom of the furnace, is a unique challenge associated with coal-fired faciUties. Once removed, significant levels of both bottom ash and flyash may require transport for landfilling. Some beneficial reuses of this ash have been identified, such as in the manufacture of Pordand cement. 

Wavelength dispersive x-ray fluorescence spectrometric (xrf) methods using the titanium line at 0.2570 nm may be employed for the determination of significant levels of titanium only by carefiil matrix-matching. However, xrf methods can also be used for semiquantitative determination of titanium in a variety of products, eg, plastics. Xrf is also widely used for the determination of minor components, such as those present in the surface coating, in titanium dioxide pigments. 

Vinyl acetate is a colorless, flammable Hquid having an initially pleasant odor which quickly becomes sharp and irritating. Table 1 Hsts the physical properties of the monomer. Information on properties, safety, and handling of vinyl acetate has been pubUshed (5—9). The vapor pressure, heat of vaporization, vapor heat capacity, Hquid heat capacity, Hquid density, vapor viscosity, Hquid viscosity, surface tension, vapor thermal conductivity, and Hquid thermal conductivity profile over temperature ranges have also been pubHshed (10). Table 2 (11) Hsts the solubiHty information for vinyl acetate. Unlike monomers such as styrene, vinyl acetate has a significant level of solubiHty in water which contributes to unique polymerization behavior. Vinyl acetate forms azeotropic mixtures (Table 3) (12). 

Unesterified tocopherols are found in a variety of foods however, concentration and isomer distribution of tocopherols vary gready with source. Typically, meat, fish, and dairy contain <40 mg/100 g of total tocopherols. Almost all (>75%) of this is a-tocopherol for most sources in this group. The variation in the content of meat and dairy products can be related to the content of the food ingested by the animal. A strong seasonal variation can also be observed. Vegetable oils contain significant levels of y-, P-, and 5-tocopherol, along with a-tocopherol (Table 3). 

Fig. 12. The relationship between the mean oceanic residence time, T, yr, and the seawater—cmstal rock partition ratio,, of the elements adapted from Ref. 29. , Pretransition metals I, transition metals , B-metals , nonmetals. Open symbols indicate T-values estimated from sedimentation rates. The sohd line indicates the linear regression fit, and the dashed curves show the Working-Hotelling confidence band at the 0.1% significance level. The horizontal broken line indicates the time required for one stirring revolution of the ocean, T. ...

Ot = significance level, usually set at. 10,. 05, or. 01 t = tabled t value corresponding to the significance level Ot. For a two-tailed test, each corresponding tail would have an area of Ot/2, and for a one-tailed test, one tail area would be equal to Ot. If O" is known, then z would be used rather than the t. t = (x- il )/ s/Vn) = sample value of the test statistic. 

The decision rules for each of the three forms are defined as follows If the sample t falls within the acceptance region, accept Hq for lack of contrary evidence. If the sample t falls in the critical region, reject Hq at a significance level of lOOot percent. 

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