TESTING THE RESIDUALS

Once S(6) has been minimized for a given model and data collection, the assumptions of the calculation should be tested. In this step, one examines the weighted residuals 

Systematic trends of the residuals may suggest alterations of the model or the weighting. Systematic differences of parameters fitted to different parts of the data can give similar information, as Hunter and Mezaki (1964) showed in their study of model building for methane oxidation. 

When possible, one should also test the residual sum of squares against its predictive probability distribution. If the weighted residuals are distributed in the manner assumed in Section 6.1, then the standardized sum of squares = 5/cr has the predictive probability density 

A more powerful criterion of goodness of fit is the F-test, pioneered by Fisher (1925), of the variance ratio 

Here 5b is an experimental error sum of squares with i/ degrees of freedom, normally determined from i e independent differences of replicate observations or residuals. Si is a lack-of-fit sum of squares, given by 

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If a phenol is not indicated, the solution may contain an aliphatic acid. Transfer to a distilling-flask, make definitely acid with dih H2SO4, and distil the volatile formic and acetic acids if present will distil over. If the distillation gives negative reactions, test the residual solution in the flask for oxalic, succinic, lactic, tartaric and citric acids and glycine, remembering that the solution is strongly acid. 

Hydrolysis of benzanilide. Place 5 g. of benzanilide and 50 ml. of 70 per cent, sulphuric acid in a small flask fitted with a reflux condenser, and boU gently for 30 minutes. Some of the benzoio acid will vapourise in the steam and solidify in the condenser. Pour 60 ml. of hot water down the condenser this will dislodge and partially dissolve the benzoic acid. Cool the flask in ice water filter off the benzoic acid (anifine sulphate does not separate at this dilution), wash well with water, drain, dry upon filter paper, and identify by m.p. (121°) and other tests. Render the filtrate alkaline by cautiously adding 10 per cent, sodium hydroxide solution, cool and isolate the aniline by ether extraction. Recover the ether and test the residue for anifine (Section IV,100). 

Dissolve the substance in water and warm gently. Effervescence occurs and a dark coloured oil separates, which has the aincll of phenol. When effervescence ceases, cool and shake ii j with a little ether. Decant the ether into a dry test-tube. Evapoiate the ether and test the residue for phenol, see p. 179. 

The solution is diluted with 200 c.c. of water and boiled until a sample, when mixed with dichromate solution, no longer smells of nitrosobenzene but of quinone (ten to fifteen minutes). To the cooled solution 2 g. of dichromate dissolved in water are added, a downward condenser is attached to the flask containing the mixture, and steam is passed through. The quinone is carried over with the steam. On the mechanism of its formation in this reaction compare p. 310. Test the residue in the flask for ammonia. 

In cases which are not very complex, some indication as to the nature of the resin may be obtained by evaporating the solvent and testing the residue with the specific reactions of the various resins, observing its behaviour towards solvents and determining its constants, especially the... 

Test the residue with red litmus paper and blue litmus paper and record the results. 

Here, we want to emphasize that one is able to calculate the fraction of the experimental error only if replicate measurements (at least at one point x ) have been taken. It is then possible to compare model and experimental errors and to test the sources of residual errors. Then, in addition to the GOF test one can perform the test of lack of fit, LOF, and the test of adequacy, ADE, (commonly used in experimental design). In the lack of fit test the model error is tested against the experimental error and in the adequacy test the residual error is compared with the experimental error. 

If HC1 is omitted, the cyanohydrin reverts to HCN and propionaldehyde upon attempted distillation. The checkers found it necessary to ensure that the residue was acidic by adjusting the pH to 5 by testing the residue with wet pH paper. 

It is convenient to test the residue R for oxalate here. Dissolve it by pouring hot dilute sulphuric acid into the filter. Treat the hot filtrate with a few drops of 0 004m potassium permanganate solution. If the permanganate solution is reduced, oxalate is present. If no reduction occurs, the presence of fluoride is indicated. 

Add solid potassium or ammonium peroxodisulphate to the solution of the mixed halides contained in a conical flask strongly acidify with dilute sulphuric acid heat the flask to about 80°, and aspirate a current of air through the solution with the aid of a filter pump (see Fig. IV.2 in Section IV.45, 5) until the solution is colourless. Add more solid peroxodisulphate or water as may be found necessary. Test the residual colourless liquid for chloride with silver nitrate solution and dilute nitric acid. 

Janesville [Wisconsin] authorities and the FBI are investigating a break-in at a water facility reservoir. Earher this week, it was discovered that barbed wire on a chain Unk fence was cut and a padlock on the reservoir had been forcefully removed. The Department of Natural Resomces and the State Lab of Hygiene tested the water and so far all tests have been negative for any contaminants. Nonetheless, the decision was made by Janesville officials to isolate and drain the 5 irulhon gallon reservoir and test the residue. (State of Wisconsin, 2002)... 

Experiment 69. — Put about 50 cc. of ammonium hydroxide in an evaporating dish or casserole, and add dilute hydrochloric acid, drop by drop, until the solution is just neutral or faintly alkaline. A convenient dropping apparatus is shown in Fig. 61. Evaporate to dryness on a water bath or very slowly over a piece of wire gauze. Test the residue as follows —... 

Manipulator. To test the residual moisture in a product in addition to the measurement of desorption rate, the freeze dryer could be equipped with a manipulator (Figure 31) and a vacuum lock, which permits one to take out the sample or to close marked vials with a stopper inside the chamber. 

The assumptions of the structural model regarding distribution of the random effects should be examined. Most population parameters are modeled assuming the random effects are log-normal. The random effects are assumed to be independent and have a normal distribution with mean 0 and variance co2. These assumptions should be tested. Other assumptions that should be tested include testing the residuals for homoscedasticity, normality, and lack of systematic deviations in the residuals over time. More about assumption testing will be presenting later in the chapter. If the assumptions are violated, remedial measures should be taken. 

Pour a few drops of the distillate on a glass plate and determine whether or not it will burn. Test the residue in the flask in the same way. (8) What were the results (9) What... 

The gaseous reaction products were sampled on a specified time schedule and analyzed by chromatography. After the system had cooled following the test, the residual gases were collected, and the residues, consisting mainly of carbon and ash, were removed, weighed, and analyzed. 

In classical balUstic tests, a projectile is shot at a stationary target which consists of the material to be tested. The residual kinetic energies are studied. A recently developed technique for lightweight materials uses the reverse ballistic impact. In contrast to the classical tests, the target is moving and the projectile is a rest. Nitrogen from a commercially compressed gas bottle can be used as acceleration gas. Velocities of up to... 

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