Null hypothesis were true, we would get such a large z-score more than 1% of the time. 01, we fail to reject the null hypothesis at the 1% level if the Note that for these two tailed tests we are using the absolute value of the Probability of such a large or larger z-score is. Since one could have been as far below 85, the
161 in "The Emergence of Mathematical Statistics: A Historical Sketch with Particular Reference to the United States","On the History of Statistics and Probability", ed. This number is called the level of significance” Neyman 1976, p. The first demand of the mathematical theory is to deduce such test criteria as would ensure that the probability of committing an error of the first kind would equal (or approximately equal, or not exceed) a preassigned number α, such as α = 0.05 or 0.01, etc. As Neyman wrote: “The error that a practising statistician would consider the more important to avoid (which is a subjective judgment) is called the error of the first kind. If we set the significance level alpha to 0.05, and only reject the null hypothesis if the p-value is less than or equal to 0.05, then our hypothesis test will indeed have significance level (maximal type 1 error rate) 0.05. This definition ensures the complementarity of p-values and alpha-levels. In these circumstances (the case of a so-called composite null hypothesis) the p-value is defined by taking the least favourable null-hypothesis case, which is typically on the border between null and alternative. For each possible value of the theoretical mean, the Z-test statistic has a different probability distribution. In the just mentioned example that would be the Z-statistic belonging to the one-sided one-sample Z-test. For example, when testing the null hypothesis that a distribution is normal with a mean less than or equal to zero against the alternative that the mean is greater than zero (variance known), the null hypothesis does not specify the probability distribution of the appropriate test statistic. In contrast, in a composite hypothesis the parameter's value is given by a set of numbers. In parametric hypothesis testing problems, a simple or point hypothesis refers to a hypothesis where the parameter's value is assumed to be a single number. A p-curve can be used to assess the reliability of scientific literature, such as by detecting publication bias or p-hacking. The distribution of p-values for a group of studies is sometimes called a p-curve. By contrast, if the alternative hypothesis is true, the distribution is dependent on sample size and the true value of the parameter being studied. In statistics, every conjecture concerning the unknown probability distribution of a collection of random variables representing the observed data X, and the underlying random variable is continuous, then the probability distribution of the p-value is uniform on the interval.