Downloaded from https://www.everyspec.com ADS-79D-HDBK normal; that is, it has more probability in the tails than the normal distribution. As the number of degrees of freedom approaches infinity, the limiting form of the t distribution is the standard normal distribution. Similar to the z -distribution, confidence interval of t can be determined given a significance level of α : So, the confidence interval of estimated mean can be calculated by: where is the upper 100α/2 percentage point of the t – distribution with ( n – 1) degrees of freedom. Once the significance level is defined and maximum tolerant error of mean estimate, is specified, the sample size can be determined by:
It should also be noted that the value of t – distribution at given α level stabilizes once the number of degrees of freedom reaches 4, as depicted in Figure I-1, below. 35 30 25 t Value 20 15 10 5 0 0 10 20 30 40 Degree of Freedom [n - 1] FIGURE I-1: Information gain as function of sample size Initially, t-distribution applies when the population standard deviation is unknown and has to be estimated from the data. Quite often, however, some practical problems will treat the population standard deviation as if it were known. These problems are generally of three kinds:
(1) those in which the sample size is so large that one may treat a data-based estimate of the variance as if it were certain; (2) those that illustrate mathematical reasoning, in which the problem of estimating the standard 150 a lph a = 0.0 1 a lph a = 0.0 25 a a lph lph a = a = 0.0 0.1 5 a lph a = 0.2 5