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18 TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS . The table presented by Dixon and Massey (1957, table A-5, p. 384) gives values for one- half of the distribution. For 95-percent limits there would be 0.025 in each tail, and the value is taken in the colunm headed to.,," Notice that The confidence limits are: for infinite degrees of freedom the t and normal distributions are the same. In the normal distribution, 1.96u on each side of the meau includes 95 percent of the items. In table A-5, 1.96 is listed under to..,.. b,- (t".O.05) (8,,) <13,<b,+ (t",o.oo) (8,,) 1.08851- (2.03) (0.0344) <1l2<1.08851 + (2.03) (0.0344) 1.0187<13,<1.1583 -0.03938- (2.03) (0.0394)<.8,< -0.03938+ (2.03) (0.0394) -0.1194<.8,< +0.0406 1.42885- (2.03) (0.1234) <13,< 1.42885 + (2.03) (0.1234) 1.1783<.8,< 1.6793 .8 is considered the true slope. Therefore the confidence limits give the range within which .8 lies with 95-percent probability. In this ex- ample the limits of Il2 include zero. This indi- cates that .8, is not significantly different from zero at 95-percent level and that the parameter A should be eliminated from the regression. Regressions having various numbers of independent variables Examples have been given of computations for regressions having one and three inde- pendent variables, and the normal equations for a regression of two independent variables have also been given. The method of solution involving two variables is similar to that for three independent variables, but is much shorter. Normal equations for regressions of four or more independent variables have been given by Ezekiel and Fox (1959, 1'.181-183). Because computation of such regressions on a desk calculator is very time consuming, digital com- puters are being used. Use of digital computers Programs for regression computations are available for most computers, and regressions of more than two independent variables should ordinarily be made by digital computer rather than on a desk calculator. Simple regressions and regressions of two independent variables may be made quite rapidly on a desk calcu- - lator; use of a desk calculator for computations of these sizes may be advantageous. Regression programs for digital computers vary but usually require listing of the data in floating decimal notation. These values are then punched on cards which are entered in the computer. Results are printed by the com- puter. A wide variety of options as to output is available. Detailed instructions for prepara- tion of data and instructions to the computer should be obtained for the particular computer and program to be used. Although no knowledge of regression analysis is necessary for preparing data for a computer program, some experience is needed to appraise the results. Opportunities for errors to be intro- duced into the process exist in the listin'g of the data and in its transferral to cards. Questionable results may also be obtained if too few significant figures are carried through the computations. Only a person who has made regression computations the hard way can ade- quately judge whether the results of a regres- sion analysis by digital computer (or any other method) are correct. The availability of digital computers has permitted ready computation of regressions using many variables, which has sometimes led to substitution of the computer for the analyst's brain. The problem should be solved by the analyst; the computer does the arithmetic. . Application of the regression method An analytical problem to be solved by regression involves (1) selection of factors which .