Given two populations $\pi_1$ and $\pi_2$ each characterized by a distribution density $f(x, \theta)$ which is assumed to be known, except for the value of the parameter $\theta$. It is desired to test the composite hypothesis $\theta_1 < \theta_2$ against the alternative hypothesis $\theta_1 > \theta_2$ where $\theta_i$ is the value of the parameter in the distribution density of $\pi_i, (i = 1, 2)$. The criterion proposed for testing this hypothesis is based on the sequential probability ratio and consists of the following: Choose two positive constants $a$ and $b$ and two values of $\theta$, say $\theta^0_1$ and $\theta^0_2$. Take pairs of observations $x_{1\alpha}$ from $\pi_1$ and $x_{2\alpha}$ from $\pi_2, (\alpha = 1,2, \ldots)$, in sequence and compute $Z_j = \sum^j_{\alpha = 1} z_\alpha$ where $z_\alpha = \log \big\lbrack \frac{f(x_{2\alpha}, \theta^0_1)f(x_{1\alpha}, \theta^0_2)} {f(x_{2\alpha}, \theta^0_2)f(x_{1\alpha}, \theta^0_1)}\big\rbrack.$ The hypothesis tested is accepted or rejected depending on whether $Z_n \geq a$ or $Z_n \leq - b$ where $n$ is the smallest integer $j$ for which either one of these relationships is satisfied. The boundaries $a$ and $b$ are partly given in terms of the desired risks of making an erroneous decision. The values $\theta^0_1$ and $\theta^0_2$ define the magnitude of the difference between the values of $\theta$ in $\pi_1$ and in $\pi_2$ which is considered worth detecting. It is shown that the power of this test is constant on a curve $h(\theta_1, \theta_2) =$ constant. If $E\big(\log \frac{f(x, \theta^0_2)}{f(x, \theta^0_1)}\big)$ is a monotonic function of $\theta$, then the test is unbiased in the sense that all points $(\theta_1, \theta_2)$ which lie on the curve $h(\theta_1, \theta_2) =$ constant are such that either every $\theta_1 < \theta_2$ or every $\theta_1 > \theta_2$. For a large class of known distributions the quantity $h$ is shown to be an appropriate measure of the difference between $\theta_1$ and $\theta_2$ and the test procedure for this class of distributions is simple and intuitively sensible. For the case of the binomial, the exact power of this test as well as the distribution of $n$ is given.

Research for the Army demobilization point system evolved a new approach to paired comparisons and rank order. Each of $N$ individuals compares or ranks $n$ things; the problem is to determine a numerical value for each of the $n$ things that will best represent the comparisons in some sense. The new criterion adopted is that the numerical values be determined so as best to distinguish between those things judged higher and those judged lower for each individual. Least-squares is employed in the analysis, and the solution appears in the form of the latent vector associated with the largest root of a matrix obtained from the comparisons or rankings. This approach applies to the conventional problem of ordinary comparisons, the numerical solution being easily obtainable by simple iterations; the conventional use of hypothetical variables and unverified hypotheses is avoided. The Army point system is an example of a new and more complicated class of problems; the same principle for the solution applies here, only more details occur in the derivations and computations.

A type of population frequently encountered in extensive samplings is one in which the variance within a group of elements increases steadily as the size of the group increases. This class of populations may be represented by a model in which the elements are serially correlated, the correlation between two elements being a positive and monotone decreasing function of the distance apart of the elements. For populations of this type, the relative efficiencies are compared for a systematic sample of every $k$th element, a stratified random sample with one element per stratum and a random sample. The stratified random sample is always at least as accurate on the average as the random sample and its relative efficiency is a monotone increasing function of the size of the sample. No general result is valid for the relative efficiency of the systematic sample. In fact, there are populations in the class in which the systematic sample is more accurate than the stratified sample for one sampling rate, but is less accurate than the random sample for another sampling rate. If, however, the correlogram is in addition concave upwards, the systematic sample is on the average more accurate than the stratified sample for any size of sample. Some numerical results are given for the cases in which the correlogram is (i) linear (ii) exponential.

Methods making possible quick calculation of operating characteristics of power curves of common tests of significance involving the $\chi^2, F, t,$ and normal distributions are presented. In addition, a comprehensive set of curves illustrating graphically the power of each test for the 5% significance level are included. We are interested in the power of: (1) the $\chi^2$-test to determine whether an unknown population standard deviation is greater or less than a standard value, (2) the $F$ test to determine whether one unknown population standard deviation is greater than another (one-sided alternative), and (3) the $t$-test and normal test to determine whether an unknown population mean differs from a standard or two unknown population means differ from each other. Such operating characteristics have application for the quality control engineer and statistician in the design of sampling inspection plans using variables where they may be used to determine the sample size that will guarantee a specified consumer's and producer's risk. On the other hand they are of use in displaying the power of a test if the sample size has already been set. Finally, they are a necessary adjunct to the proper interpretation of the common tests of significance.

When a curve is fitted to a set of data by moments, the usual procedure used in testing the hypothesis that the population is of the given form with the parameters as computed from the moments is to compare the higher moments with their expected values as determined by the hypothesis. Generally speaking, moments about the mean are computed although the reason for this is not clear. To shed some light on this question, the sample given in the introduction is fitted to two curves. Moments about various points are compared with their expected values and the discrepancy in standard units examined. This discrepancy is found to vary widely and to have a maximum. The notion of equivalent moment tests is introduced, and on this basis the most efficient moment test is defined in such a way that of all equivalent moment tests, this one is most likely to reject a false hypothesis. For any moment it is shown that there is a point about which its variance is a minimum. The conditions are found which determine the position of this point for second and third moments. It is proved that for symmetrical populations the variance is minimal when the moments are computed about the mean of the population. If the population is an asymmetrical Pearson frequency function, it is proved that the point about which the third moment variance is minimal differs more from the mean that does the corresponding point for second moments. The condition is pointed out for which this is true in the general case. The third and fourth standard semi-invariants of second moments of minimal variance are computed and compared to those of the second moment about the mean. The ratios of these are displayed for some populations to illustrate how this may be used to investigate when the approach to normality is more rapid in one case than in the other. Some examples are presented to contrast these and other tests.

The problem of constructing tolerance limits for a normal universe is considered. The tolerance limits are required to be such that the probability is equal to a preassigned value $\beta$ that the tolerance limits include at least a given proportion $\gamma$ of the population. A good approximation to such tolerance limits can be obtained as follows: Let $\bar x$ denote the sample mean and $s^2$ the sample estimate of the variance. Then the approximate tolerance limits are given by $\bar x - \sqrt\frac{n}{\chi^2_{n,\beta}} rs \text{and} \bar x + \sqrt\frac{n}{\chi^2_{n,\beta}} rs$ where $n$ is one less than the number $N$ of observations, $\chi^2_{n,\beta}$ denotes the number for which the probability that $\chi^2$ with $n$ degrees of freedom will exceed this number is $\beta$, and $r$ is the root of the equation $\frac{1}{\sqrt{2\pi}} \int^{1/\sqrt{N} + r}_{1/\sqrt{N}-r} e^{-t^2/2} dt = \gamma.$ The number $\chi^2_{n,\beta}$ can be obtained from a table of the $\chi^2$ distribution and $r$ can be determined with the help of a table of the normal distribution.

It is generally assumed in the application of distribution theory that, if the actual population function is not very different from the one used in the theory, then the true sampling distribution of a statistic will not be very different from the one obtained in the theory. But elsewhere in mathematics we do not assert that a conclusion will be only slightly modified by a small deviation in the hypothesis. This paper presents some theorems which are useful in determining the maximum effect on a sampling distribution of certain kinds of small changes in the population function. In particular, if the population is denoted by the function $\phi(t)$, if a sample of $n$ independent measurements $(t_1, \cdots, t_n)$ is taken from this population, if a statistic $x = g(t_1, \cdots, t_n)$ is formed from the sample, and if $D(x)$ denotes the distribution of this statistic; then, when $\phi (t)$ is changed by a small proportionate amount to $\phi_1(t), D(x)$ will be changed to $D_1(x)$, and the relation between $D$ and $D_1$ will be subject to the inequality: $\| \int^b_a (D - D_1)dx\| \leqq \epsilon \int^b_a D(x)dx,$ where $\epsilon = (1 + \delta)^n - 1,\quad \text{and}\quad\mid\phi_1/\phi - 1 \mid < \delta.$

When the response to a drug is a linear function of arithmetic dosage units, the relative potency of two preparations can be computed as a slope-ratio assay. Their dosage-response curves are computed by solving three simultaneous equations to obtain the common intercept $a'$, the slope of the standard, $b_1$, and the slope of the unknown, $b_2$. The method is applicable to certain microbiological assays for the vitamins. Usually several unknowns are assayed at one time with a single standard. Their calculation is simplified when such assays meet the following requirements: (1) restriction of treatments to the zone within which the response is related linearly to the dose, (2) equal spacing of doses on an arithmetic scale beginning with the negative control, (3) an equal number $(k)$ of doses of standard and of each unknown and (4) $r$ replicates for each dose of unknown, $h'$ replicates for the negative control and $h$ replicates for each dose of the standard.

Let $x_1, x_2, x_3, x_4$ represent the values of a sample of size four drawn from a normal population. There is no loss of generality in assuming that the distribution function of this population has zero mean and unit vriance. Denote it by $N(0, 1)$. Let $x_(i)$ be the ith largest of $x_1, x_2, x_3, x_4$. The purpose of this note is to determine the joint distribution of $x{(4)} + x_{(3)} - x_{(2)} - x_{(1)}, x_{(4)} - x_{(3)} + x_{(2)} -x_{(1)}$, and $x_{(4)} - x_{(3)} - x_{(2)} + x_{(1)}$, and derive from this joint distribution the joint distributions of these statistics taken in pairs, also the distribution of each statistic itself.