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A ''Z''-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. Because of the central limit theorem, many test statistics are approximately normally distributed for large samples. For each significance level, the ''Z''-test has a single critical value (for example, 1.96 for 5% two tailed) which makes it more convenient than the Student's ''t''-test which has separate critical values for each sample size. Therefore, many statistical tests can be conveniently performed as approximate ''Z''-tests if the sample size is large or the population variance known. If the population variance is unknown (and therefore has to be estimated from the sample itself) and the sample size is not large (n < 30), the Student's ''t''-test may be more appropriate. If ''T'' is a statistic that is approximately normally distributed under the null hypothesis, the next step in performing a ''Z''-test is to estimate the expected value θ of ''T'' under the null hypothesis, and then obtain an estimate ''s'' of the standard deviation of ''T''. After that the standard score ''Z'' = (''T'' − θ) / ''s'' is calculated, from which one-tailed and two-tailed ''p''-values can be calculated as Φ(−''Z'') (for upper-tailed tests), Φ(''Z'') (for lower-tailed tests) and 2Φ(−|''Z''|) (for two-tailed tests) where Φ is the standard normal cumulative distribution function. ==Use in location testing== The term "''Z''-test" is often used to refer specifically to the one-sample location test comparing the mean of a set of measurements to a given constant. If the observed data ''X''1, ..., ''X''n are (i) uncorrelated, (ii) have a common mean μ, and (iii) have a common variance σ2, then the sample average ''X'' has mean μ and variance σ2 / ''n''. If our null hypothesis is that the mean value of the population is a given number μ0, we can use ''X'' −μ0 as a test-statistic, rejecting the null hypothesis if ''X'' − μ0 is large. To calculate the standardized statistic ''Z'' = (''X'' − μ0) / ''s'', we need to either know or have an approximate value for σ2, from which we can calculate ''s''2 = σ2 / ''n''. In some applications, σ2 is known, but this is uncommon. If the sample size is moderate or large, we can substitute the sample variance for σ2, giving a ''plug-in'' test. The resulting test will not be an exact ''Z''-test since the uncertainty in the sample variance is not accounted for—however, it will be a good approximation unless the sample size is small. A ''t''-test can be used to account for the uncertainty in the sample variance when the sample size is small and the data are exactly normal. There is no universal constant at which the sample size is generally considered large enough to justify use of the plug-in test. Typical rules of thumb range from 20 to 50 samples. For larger sample sizes, the ''t''-test procedure gives almost identical ''p''-values as the ''Z''-test procedure. Other location tests that can be performed as ''Z''-tests are the two-sample location test and the paired difference test. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Z-test」の詳細全文を読む スポンサード リンク
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