Ztest
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A Ztest 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 Ztest has a single critical value (for example, 1.96 for 5% two tailed) which makes it more convenient than the Student's ttest which has separate critical values for each sample size. Therefore, many statistical tests can be conveniently performed as approximate Ztests if the sample size is large or the population variance is 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 ttest may be more appropriate.
If T is a statistic that is approximately normally distributed under the null hypothesis, the next step in performing a Ztest 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 onetailed and twotailed pvalues can be calculated as Φ(−Z) (for uppertailed tests), Φ(Z) (for lowertailed tests) and 2Φ(−Z) (for twotailed tests) where Φ is the standard normal cumulative distribution function.
Contents
Use in location testing
The term "Ztest" is often used to refer specifically to the onesample location test comparing the mean of a set of measurements to a given constant when the sample variance is known. If the observed data X_{1}, ..., X_{n} are (i) independent, (ii) have a common mean μ, and (iii) have a common variance σ^{2}, then the sample average X has mean μ and variance σ^{2} / n.
The null hypothesis is that the mean value of X is a given number μ_{0}. We can use X as a teststatistic, rejecting the null hypothesis if X − μ_{0} is large.
To calculate the standardized statistic Z = (X − μ_{0}) / s, we need to either know or have an approximate value for σ^{2}, from which we can calculate s^{2} = σ^{2} / n. In some applications, σ^{2} is known, but this is uncommon.
If the sample size is moderate or large, we can substitute the sample variance for σ^{2}, giving a plugin test. The resulting test will not be an exact Ztest 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 ttest can be used to account for the uncertainty in the sample variance when 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 plugin test. Typical rules of thumb: the sample size should be 50 observations or more.
For large sample sizes, the ttest procedure gives almost identical pvalues as the Ztest procedure.
Other location tests that can be performed as Ztests are the twosample location test and the paired difference test.
Conditions
For the Ztest to be applicable, certain conditions must be met.
 Nuisance parameters should be known, or estimated with high accuracy (an example of a nuisance parameter would be the standard deviation in a onesample location test). Ztests focus on a single parameter, and treat all other unknown parameters as being fixed at their true values. In practice, due to Slutsky's theorem, "plugging in" consistent estimates of nuisance parameters can be justified. However if the sample size is not large enough for these estimates to be reasonably accurate, the Ztest may not perform well.
 The test statistic should follow a normal distribution. Generally, one appeals to the central limit theorem to justify assuming that a test statistic varies normally. There is a great deal of statistical research on the question of when a test statistic varies approximately normally. If the variation of the test statistic is strongly nonnormal, a Ztest should not be used.
If estimates of nuisance parameters are plugged in as discussed above, it is important to use estimates appropriate for the way the data were sampled. In the special case of Ztests for the one or two sample location problem, the usual sample standard deviation is only appropriate if the data were collected as an independent sample.
In some situations, it is possible to devise a test that properly accounts for the variation in plugin estimates of nuisance parameters. In the case of one and two sample location problems, a ttest does this.
Example
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. We can ask whether this mean score is significantly lower than the regional mean—that is, are the students in this school comparable to a simple random sample of 55 students from the region as a whole, or are their scores surprisingly low?
First calculate the standard error of the mean:
where is the population standard deviation.
Next calculate the zscore, which is the distance from the sample mean to the population mean in units of the standard error:
In this example, we treat the population mean and variance as known, which would be appropriate if all students in the region were tested. When population parameters are unknown, a t test should be conducted instead.
The classroom mean score is 96, which is −2.47 standard error units from the population mean of 100. Looking up the zscore in a table of the standard normal distribution, we find that the probability of observing a standard normal value below −2.47 is approximately 0.5 − 0.4932 = 0.0068. This is the onesided pvalue for the null hypothesis that the 55 students are comparable to a simple random sample from the population of all testtakers. The twosided pvalue is approximately 0.014 (twice the onesided pvalue).
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. 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 testtakers.
The Ztest tells us that the 55 students of interest have an unusually low mean test score compared to most simple random samples of similar size from the population of testtakers. A deficiency of this analysis is that it does not consider whether the effect size of 4 points is meaningful. If instead of a classroom, we considered a subregion containing 900 students whose mean score was 99, nearly the same zscore and pvalue would be observed. This shows that if the sample size is large enough, very small differences from the null value can be highly statistically significant. See statistical hypothesis testing for further discussion of this issue.
Ztests other than location tests
Location tests are the most familiar Ztests. Another class of Ztests arises in maximum likelihood estimation of the parameters in a parametric statistical model. Maximum likelihood estimates are approximately normal under certain conditions, and their asymptotic variance can be calculated in terms of the Fisher information. The maximum likelihood estimate divided by its standard error can be used as a test statistic for the null hypothesis that the population value of the parameter equals zero. More generally, if is the maximum likelihood estimate of a parameter θ, and θ_{0} is the value of θ under the null hypothesis,
can be used as a Ztest statistic.
When using a Ztest 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. Although there is no simple, universal rule stating how large the sample size must be to use a Ztest, simulation can give a good idea as to whether a Ztest is appropriate in a given situation.
Ztests are employed whenever it can be argued that a test statistic follows a normal distribution under the null hypothesis of interest. Many nonparametric test statistics, such as U statistics, are approximately normal for large enough sample sizes, and hence are often performed as Ztests.
See also
References
 Sprinthall, R. C. (2011). Basic Statistical Analysis (9th ed.). Pearson Education. ISBN 9780205052172.