$\frac{{\sigma}^{}}{}$Because of the central limit theorem, many test statistics are approximately normally distributed for large samples. Therefore, many statistical tests can be conveniently performed as approximate *Z*-tests 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 *t*-test may be more appropriate.

How to perform a Z test when *T* is a statistic that is approximately normally distributed under the null hypothesis is as follows:

First, estimate the expected value μ of *T* under the null hypothesis, and obtain an estimate *s* of the standard deviation of *T*.

Second, determine the properties of *T* : one tailed or two tailed.

For Null hypothesis *H*_{0}: *μ≥μ*_{0} vs alternative hypothesis *H*_{1}: *μ<μ*_{0} , it is upper/right-tailed (one tailed).

For Null hypothesis *H*_{0}: *μ≤μ*_{0} vs alternative hypothesis *H*_{1}: *μ>μ*_{0} , it is lower/left-tailed (one tailed).

For Null hypothesis *H*_{0}: *μ=μ*_{0} vs alternative hypothesis *H*_{1}: *μ≠μ*_{0} , it is two-tailed.

Third, calculate the standard score :

$Z={\frac {({\bar {X}}-\mu _{0})}{s}}$ ,

which one-tailed and two-tailed *p*-values can be calculated as Φ(*Z*) (for upper/right-tailed tests), Φ(−*Z*)(for lower/left-tailed tests) and 2Φ(−|*Z*|) (for two-tailed tests) where Φ is the standard normal cumulative distribution function.

For the *Z*-test 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 one-sample location test).
*Z*-tests 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 *Z*-test 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 non-normal, a
*Z*-test 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 *Z*-tests 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 plug-in estimates of nuisance parameters. In the case of one and two sample location problems, a *t*-test 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:

- $\mathrm {SE} ={\frac {\sigma }{\sqrt {n}}}={\frac {12}{\sqrt {55}}}={\frac {12}{7.42}}=1.62\,\!$

where ${\sigma }$ is the population standard deviation.

Next calculate the *z*-score, which is the distance from the sample mean to the population mean in units of the standard error:

- $z={\frac {M-\mu }{\mathrm {SE} }}={\frac {96-100}{1.62}}=-2.47\,\!$

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 *z*-score in a table of the standard normal distribution cumulative probability, 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 sampled. In the special case of *Z*-tests 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 plug-in estimates of nuisance parameters. In the case of one and two sample location problems, a *t*-test 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:

*t*-test does this.
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 ${\sigma }$ is the population standard deviation.

Next calculate the *z*-score, which is the distance from the sample mean to the population mean in units of the standard error:

- $z={\frac {M-\mu }{\mathrm {SE} }}={\frac {96-100}{1.62}}=-2.47\,\!$
*z*-score, 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 *z*-score in a table of the standard normal distribution cumulative probability, 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 one-sided *p*-value for the null hypothesis that the 55 students are comparable to a simple random sample from the population of all test-takers. 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. 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.

The *Z*-test 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 test-takers. 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 *z*-score and *p*-value would be observed. This shows that if the sample size

The classroom mean score is 96, which is −2.47 standard error units from the population mean of 100. Looking up the *z*-score in a table of the standard normal distribution cumulative probability, 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 one-sided *p*-value for the null hypothesis that the 55 students are comparable to a simple random sample from the population of all test-takers. 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. 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.

The *Z*-test 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 test-takers. 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 *z*-score and *p*-value 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.

Location tests are the most familiar *Z*-tests. Another class of *Z*-tests 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 ${\hat {\theta }}$ is the maximum likelihood estimate of a parameter θ, and θ_{0} is the value of θ under the null hypothesis,

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