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.

When using a Z-test for maximum likelihood estimates, it is important to be aware that the normal approximation may be poor if the sample size is not sufficiently large.If the population variance is unknown (and therefore has to be estimated from the sample itself) and the sample size is not large (n , 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.The two-sided p-value is approximately 0.014 (twice the one-sided p-value).Another way of stating things is that with probability 1 − 0.014 = 0.986, a simple random sample of 55 students would have a mean test score within 4 units of the population mean.

## Single test mann

Suppose that in a particular geographic region, the mean and standard deviation of scores on a reading test are 100 points, and 12 points, respectively.Our interest is in the scores of 55 students in a particular school who received a mean score of 96.In some situations, it is possible to devise a test that properly accounts for the variation in plug-in estimates of nuisance parameters.In the case of one and two sample location problems, a t-test does this.We could also say that with 98.6% confidence we reject the null hypothesis that the 55 test takers are comparable to a simple random sample from the population of test-takers.

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