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. Z-test tests the mean of a distribution. For each significance level in the confidence interval, the

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}**:

For Null hypothesis ** H_{0}**:

For Null hypothesis ** H_{0}**:

Third, calculate the standard score :

,

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.

- 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 when the sample variance is known. For example, 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 . - The null hypothesis is that the mean value of X is a given number μ
_{0}. We can use*X*as a test-statistic, rejecting the null hypothesis if*X*− μ_{0}is large. - To calculate the standardized statistic , we need to either know or have an approximate value for σ
^{2}, from which we can calculate . 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*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 data are exactly normal. - Difference between Z-test and t-test: Z-test is used when sample size is large (n>50), or the population variance is known. t-test is used when sample size is small (n<50) and population variance is unknown.
- 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: the sample size should be 50 observations or more.
- For large 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.

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.

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 *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 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 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 *Z*-test statistic.

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. Although there is no simple, universal rule stating how large the sample size must be to use a *Z*-test, simulation can give a good idea as to whether a *Z*-test is appropriate in a given situation.

*Z*-tests are employed whenever it can be argued that a test statistic follows a normal distribution under the null hypothesis of interest. Many non-parametric test statistics, such as U statistics, are approximately normal for large enough sample sizes, and hence are often performed as *Z*-tests.

In probability theory, a **normal****distribution** is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is

In statistics, the **standard deviation** is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.

In statistics, the **likelihood-ratio test** assesses the goodness of fit of two competing statistical models based on the ratio of their likelihoods, specifically one found by maximization over the entire parameter space and another found after imposing some constraint. If the constraint is supported by the observed data, the two likelihoods should not differ by more than sampling error. Thus the likelihood-ratio test tests whether this ratio is significantly different from one, or equivalently whether its natural logarithm is significantly different from zero.

In probability and statistics, **Student's t-distribution** is any member of a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown. It was developed by William Sealy Gosset under the pseudonym

In probability theory and statistics, the **chi-square distribution** with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-square distribution is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing and in construction of confidence intervals. This distribution is sometimes called the **central chi-square distribution**, a special case of the more general noncentral chi-square distribution.

The **power** of a binary hypothesis test is the probability that the test rejects the null hypothesis when a specific alternative hypothesis is true. The statistical power ranges from 0 to 1, and as statistical power increases, the probability of making a type II error decreases.

In statistics, a **confidence interval** (**CI**) is a type of estimate computed from the statistics of the observed data. This proposes a range of plausible values for an unknown parameter. The interval has an associated **confidence level** that the true parameter is in the proposed range. Given observations and a confidence level , a valid confidence interval has a probability of containing the true underlying parameter. The level of confidence can be chosen by the investigator. In general terms, a confidence interval for an unknown parameter is based on sampling the distribution of a corresponding estimator.

In statistics, an **effect size** is a quantitative measure of the magnitude of a phenomenon. It can refer to the value of a statistic calculated from a sample of data, the value of a parameter of a hypothetical statistical population, or to the equation that operationalizes how statistics or parameters lead to the effect size value. Examples of effect sizes include the correlation between two variables, the regression coefficient in a regression, the mean difference, or the risk of a particular event happening. Effect sizes complement statistical hypothesis testing, and play an important role in power analyses, sample size planning, and in meta-analyses. The cluster of data-analysis methods concerning effect sizes is referred to as estimation statistics.

In statistical inference, specifically predictive inference, a **prediction interval** is an estimate of an interval in which a future observation will fall, with a certain probability, given what has already been observed. Prediction intervals are often used in regression analysis.

The ** t-test** is any statistical hypothesis test in which the test statistic follows a Student's

In statistics, a **consistent estimator** or **asymptotically consistent estimator** is an estimator—a rule for computing estimates of a parameter *θ*_{0}—having the property that as the number of data points used increases indefinitely, the resulting sequence of estimates converges in probability to *θ*_{0}. This means that the distributions of the estimates become more and more concentrated near the true value of the parameter being estimated, so that the probability of the estimator being arbitrarily close to *θ*_{0} converges to one.

**Sample size determination** is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample. In practice, the sample size used in a study is usually determined based on the cost, time, or convenience of collecting the data, and the need for it to offer sufficient statistical power. In complicated studies there may be several different sample sizes: for example, in a stratified survey there would be different sizes for each stratum. In a census, data is sought for an entire population, hence the intended sample size is equal to the population. In experimental design, where a study may be divided into different treatment groups, there may be different sample sizes for each group.

As with other probability distributions with noncentrality parameters, the **noncentral t-distribution** generalizes a probability distribution – Student's

In statistics, a **pivotal quantity** or **pivot** is a function of observations and unobservable parameters such that the function's probability distribution does not depend on the unknown parameters. A pivot quantity need not be a statistic—the function and its *value* can depend on the parameters of the model, but its *distribution* must not. If it is a statistic, then it is known as an *ancillary statistic.*

**Bootstrapping** is any test or metric that uses random sampling with replacement, and falls under the broader class of resampling methods. Bootstrapping assigns measures of accuracy to sample estimates. This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods.

**Tukey's range test**, also known as the **Tukey's test**, **Tukey method**, **Tukey's honest significance test**, or **Tukey's HSD****test**, is a single-step multiple comparison procedure and statistical test. It can be used to find means that are significantly different from each other.

In statistics, the ** t-statistic** is the ratio of the departure of the estimated value of a parameter from its hypothesized value to its standard error. It is used in hypothesis testing via Student's t-test. The T-statistic is used in a T test to determine if you should support or reject the null hypothesis. It is very similar to the Z-score but with the difference that T-statistic is used when the sample size is small or the population standard deviation is unknown. For example, the T-statistic is used in estimating the population mean from a sampling distribution of sample means if the population standard deviation is unknown. It is also used along with p-value when running hypothesis test where the p-value tells us what the odds are of the results to have happened.

In statistics, a **generalized p-value** is an extended version of the classical

In the comparison of various statistical procedures, **efficiency** is a measure of quality of an estimator, of an experimental design, or of a hypothesis testing procedure. Essentially, a more efficient estimator, experiment, or test needs fewer observations than a less efficient one to achieve a given performance. This article primarily deals with efficiency of estimators.

In statistics and probability theory, the **nonparametric skew** is a statistic occasionally used with random variables that take real values. It is a measure of the skewness of a random variable's distribution—that is, the distribution's tendency to "lean" to one side or the other of the mean. Its calculation does not require any knowledge of the form of the underlying distribution—hence the name nonparametric. It has some desirable properties: it is zero for any symmetric distribution; it is unaffected by a scale shift; and it reveals either left- or right-skewness equally well. In some statistical samples it has been shown to be less powerful than the usual measures of skewness in detecting departures of the population from normality.

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations .(May 2020) (Learn how and when to remove this template message) |

- Sprinthall, R. C. (2011).
*Basic Statistical Analysis*(9th ed.). Pearson Education. ISBN 978-0-205-05217-2. - Casella, G., Berger, R. L. (2002).
*Statistical Inference*. Duxbury Press. ISBN 0-534-24312-6. - Douglas C.Montgomery, George C.Runger.(2014).
*Applied Statistics And Probability For Engineers*.(6th ed.). John Wiley & Sons, inc. ISBN 9781118539712 , 9781118645062.

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