Search

Consider sampling in two stages for testing a simple hypothesis about a population whose elements may be classified into a finite number of classes. The joint asymptotic distribution of the two Pearson chi[sup 2] statistics based on the first and combined samples is derived and used to construct a two-stage test. The advantage of this test relative to the conventional one-stage test is discussed in terms of its asymptotic power. Tables of critical values for 1 percent and 5 percent level tests for 1-10 degrees of freedom ore given as well as modifications for cases involving unknown parameters. [ABSTRACT FROM AUTHOR]

Results are given of an empirical sampling study of the sensitivities of nine statistical procedures for evaluating the normality of a complete sample. The nine statistics are W (Shapiro and Wilk, 1965), square root of b[sub I] (standard third moment), b[sub 2] (standard fourth moment), KS (Kolmogorov-Smirnov), CM (Cramer-Ven Mises), WCM (weighted CM), D (modified KS), CS (chi-squared) and u (Studentized range). Forty-five alternative distributions in twelve families and five sample sizes were studied. Results are included on the comparison of the statistical procedures in relation to groupings of the alternative distributions, on means and variances of the statistics under the various alternatives, on dependence of sensitivities on sample size, on approach to normality as measured by the W statistic within some classes of distribution, and on the effect of misspecification of parameters on the performance of the simple hypothesis test statistics. The general findings include: (I) The W statistic provides a generally superior omnibus measure of non-normality; (ii) the distance tests (KS, CM, WCM, D) are typically very insensitive; (iii) the u statistic is excellent against symmetric, especially short-tailed, distributions but has virtually no sensitivity to asymmetry; (iv) a combination of both square root of b[sub 1] and b[sub 2] usually provides a sensitive judgment but even their combined performance is usually dominated by W; (v) with sensitive procedures, good indication of extreme non-normality (e.g., the exponential distribution) can be... [ABSTRACT FROM AUTHOR]

Confidence intervals for the population mean of a normal distribution can be determined from the distribution of the variate H = Square root of n(x - mu)/d, analogue to Student's t-distribution but based on the mean absolute deviation d instead of the standard deviation. The H-distribution is derived: the frequency function is symmetric about zero, with central ordinate (1/pi) Square root of (1-n[sup -1]); asymptotically it is normal, N(0, Square root of (pi/2)). An approximate formula for the calculation of the percent values is developed and numerical factors tabulated by which the mean absolute deviation of a normal sample of size n is to be multiplied in order to obtain 95 percent and 50 percent confidence limits of the mean, for n = 2(1)15(5)30, 40, 60,120. It is shown that the increase in length against confidence intervals from the standard deviation and Student's t is negligible. The usefulness of confidence intervals from the mean absolute deviation for estimating the precision of measurements in Physics is discussed. [ABSTRACT FROM AUTHOR]

This article presents information on the robustness of the studentized range statistic to violations of the normality and homogeneity of variance assumptions. Multiple comparison procedures in recent years have earned a prominent role in the analysis and interpretation of experimental research in the behavioral sciences. Most of these procedures are designed either to test individual contrasts between means after the null hypothesis of no treatment differences in analysis of variance has been rejected or to test a selected set of mean contrasts which are of a priori interest to an investigator in an experiment. Three popular techniques which have primarily been employed for the first purpose are the J.W Tukey's WSD method, the D. Newman-M. Keuls test and the D. B. Duncan's multiple range test. All of these tests have as their parent statistic the studentized range statistic q. It is generally conceded that the q statistic is less powerful overall than the corresponding F statistic, but this finding assumes normal distributions with equal variances. The present study is directed at determining the extent to which Type I error rate is affected by violations in the basic assumptions of the test based on the q statistic. Monte Carlo methods were employed and a variety of departures from the assumptions were examined.