Downloaded from https://www.everyspec.com AMCP 706-106 2.4 CIOODNEMSOF-FIT TESTS tile proposed distribution will fit the Actual one in the regions where there are no data A most important consideration is why It is poor practice to find the "best" dis- the teat It being performed, see paz. 2-1 for tribution by choosing the one which fits a full discussion of this point. The two the sample best. The examples In Figs. loodnesa,-olflft tests dou4ribed in this para- 2-10 and 2-11 ought to dispell that notion graph make a null hypothesis, iLe., the sam- once and for all, pie is from the assumed distribution.
Then a stattistic, evaluated from the sample datia, 2.4.1 C141.SOUARE TESI is calculated and looked-up In at table that shows how lucky/unlucky you wore for that The chd-square test Is performed by di- asample. Thea luck is determined by the size viding the data f'rom the sample Into cells. of the 2-aided tail area. if that tail area is The actual number of data points In each very small (you were very unlucky if the cell Is compared to the predicted number null hypothesis Is true), the null hypothesis for that cell and a combined statistic V~ Is (there is no difference between the actual calculated for all cells; It is then compared and the assumned distribuitions) is rejected.
with xp2 Otherwise, tile null hypothesis is accepted,. iLe., the actital distribution could easily have X2 0 (119 fo!l (2-43) generated that set of data (within thle range Im i of tho data); the test says nothilng about the whomr behavior of the diktrib~tion outside the range of thi dwta, x3 r- random variable having the (Refs. 3, 4, 6, 7. 24, and 25). The two pro- degrees of freedom aented In this paragraph arfe alI-pur~ose (do too observed number In each vell not depend very much on which distribu- tion Is aasumedl) and are reasonably lgood.
)* weiocted number In oach cell 'the chi-square test ought not be used for ' ~ [ If too-small samples (Psy * less than 30) be- k - number or cells cause the assumptions involved aro not There are niany goodnesa-of-fit tests chi-square distrbution with Y likely to be fulfilled. 0 k -I I none of the parameters Ref. 24 (Chap. 30) and Ref. 25 discuss of thle distribution is estifmatted both tests. In practice, many of tile lmpaэ- fromi the data. If sv paruateter tani requirement. aro not fulilled because life estiulited from thle ditto thon k- I ;ov - I - s.r too much of tht analysis Is decided after 4le exact value ofv dopends seeing the data.
But if thle test is used frut o heprmtr siae "44ballpark" confirmation, little harm Is done, oilwer thie.prmtroiiuo Tests-oi-fit are statistical tests, not engi- If Pi is the same for each cell (equal pru- neering tests, No matter what the distri- bability mnethod of choosing cells), then bution or what the test, it Is posolible to n l n take a sample small enough so that virtually n / n no distribution will be rejected, or largeo enough~ so that virtually every distribution X2 P - ~ ol) - a(4.) will be rejected, IN 10 'rests-of-fit do NOT determine how well where N -number In sample, 2-64