Downloaded from https://www.everyspec.com ADS-79D-HDBK I.3.4 Confidence Interval on Sample Mean . Suppose that a set of n random samples are drawn from a given distribution of CI/HI readings with unknown mean μ and known variance σ 2 . According to the Central Limit Theorem, the estimated sample mean is normally distributed with a mean μ and variance σ 2 , if the sample size of CI/HI readings is sufficiently large. We may standardize by subtracting the mean and dividing by the standard deviation, which results in the variable: Now Z has a standard normal distribution. Given a significance level of α , confidence interval of Z can be expressed as:
So, the confidence interval of an estimated mean can be calculated by: where is the upper 100α/2 percentage point of the standard normal distribution. Once the significance level is defined and maximum tolerant error of mean estimate, is specified, the sample size can be determined by: In regression analysis, Student's t-distribution (or simply the t-distribution) is of primary interest. The t-distribution arises in the problem of estimating the mean of a normally distributed population when the sample size is small (e.g. less than 30). It is also the basis for establishing confidence intervals of the regression model.
Again, let us assume that a set of n random samples drawing from a normal distribution. In this case, both mean μ and variance σ 2 are unknown. The random variable: has a t - distribution with n - 1 degrees of freedom. and are the estimated mean and variance obtained from the sample data. In general, the general appearance of the t - distribution is similar to the z (standard normal) distribution. Both distributions are symmetric and unimodal, and the maximum ordinate value is reached at their mean value. However, the t - distribution has heavier tails than the 149 TABLE I-I.
Decision in hypothesis test Decision H is True 0 H is False 0 Reject H 0 Fail to Reject H 0 Type I Error No error No Error Type II Error