Philosophy of Science by Gerhard Schurz

Author:Gerhard Schurz
Language: eng
Format: epub, pdf
ISBN: 9781134101221
Publisher: Taylor and Francis

The mean (or expectation) value of the sample means is identical with the population mean; it is a so-called unbiased estimator (Hays/Winkler 1970, 308). The standard deviation of the sample mean shrinks with increasing sample size n, proportionally to the square root of n, and converges to zero for n→∞. This fact implies the laws of large numbers for continuous variables (which were explained for binary variables in sec. 3.13.2). Equally important is the central limit theorem, according to which for every arbitrarily distributed variable, the form of the distribution of its sample means converges with increasing sample size n to a normal distribution with mean value μ(X) and standard deviation (Bauer 1996, sec. 51). The central limit theorem justifies the approximation of the distribution of the sample means of an arbitrarily distributed variable by a normal distribution, provided the sample size is sufficiently large (n ≥ 30). This explains the importance of the normal distribution for the methods of test and inference statistics: it turns out to be the general form of a distribution of random errors around a central parameter value.

Statisticians also compute the distribution of sample variances. Following from the squaring operation, this distribution is not symmetrical, but left-skewed (a so-called χ2-distribution; see Hays/Winkler 1970, 310). Therefore, the sample variance is not an unbiased estimator of the population variance. But the so-called corrected sample variance is an unbiased estimator, which is used to estimate the population variance.

Based on these facts we now know how to compute the acceptance intervals and significant differences that were introduced in sec. 4.3. Here is an example. Assume the variable X measures the age of the first love affair of girls; our hypothesis H asserts that the mean of X, μ(X), is 15 years. We draw a 25-element random sample of girls with a corrected sample variance of . We thus estimate the population variance as 2.5. We look up the boundaries of the symmetric 95% interval in the integral table of the z-distribution; they are z = ±1.96. Based on the ztransformation we now calculate the 95% acceptance interval for X as follows:

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