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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

**Statistics** is a branch of mathematics dealing with data collection, organization, analysis, interpretation and presentation. In applying statistics to, for example, a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with all aspects of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics.

A **statistical hypothesis**, sometimes called **confirmatory data analysis**, is a hypothesis that is testable on the basis of observing a process that is modeled via a set of random variables. A **statistical hypothesis test** is a method of statistical inference. Commonly, two statistical data sets are compared, or a data set obtained by sampling is compared against a synthetic data set from an idealized model. A hypothesis is proposed for the statistical relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis that proposes no relationship between two data sets. The comparison is deemed *statistically significant* if the relationship between the data sets would be an unlikely realization of the null hypothesis according to a threshold probability—the significance level. Hypothesis tests are used in determining what outcomes of a study would lead to a rejection of the null hypothesis for a pre-specified level of significance. The process of distinguishing between the null hypothesis and the alternative hypothesis is aided by identifying two conceptual types of errors. The first type occurs when the null hypothesis is falsely rejected. The second type of error occurs when the null hypothesis is falsely assumed to be true. By specifying a threshold probability ('alpha') on, e.g., the admissible risk of making a type 1 error, the statistical decision process can be controlled.

In probability theory and statistics, a **probability distribution** is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. For instance, if the random variable X is used to denote the outcome of a coin toss, then the probability distribution of X would take the value 0.5 for *X* = heads, and 0.5 for *X* = tails. Examples of random phenomena can include the results of an experiment or survey.

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.

In probability theory, the **expected value** of a random variable, intuitively, is the long-run average value of repetitions of the **same experiment** it represents. For example, the expected value in rolling a six-sided die is 3.5, because the average of all the numbers that come up is 3.5 as the number of rolls approaches infinity. In other words, the law of large numbers states that the arithmetic mean of the values almost surely converges to the expected value as the number of repetitions approaches infinity. The expected value is also known as the **expectation**, **mathematical expectation**, **EV**, **average**, **mean value**, **mean**, or **first moment**.

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

In statistics, the **standard score** is the signed fractional number of standard deviations by which the value of an observation or data point is above the mean value of what is being observed or measured. Observed values above the mean have positive standard scores, while values below the mean have negative standard scores.

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. 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*.

A **location test** is a statistical hypothesis test that compares the location parameter of a statistical population to a given constant, or that compares the location parameters of two statistical populations to each other. Most commonly, the location parameter of interest are expected values, but location tests based on medians or other measures of location are also used.

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 *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 *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.

In probability theory, the **normal****distribution** is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be **normally distributed** and is called a **normal deviate**.

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.

In statistics, a **paired difference test** is a type of location test that is used when comparing two sets of measurements to assess whether their population means differ. A paired difference test uses additional information about the sample that is not present in an ordinary unpaired testing situation, either to increase the statistical power, or to reduce the effects of confounders.

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, 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 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 population standard deviation is unknown. It was developed by William Sealy Gosset under the pseudonym

In probability theory and statistics, the **chi-squared 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 or in construction of confidence intervals. When it is being distinguished from the more general noncentral chi-squared distribution, this distribution is sometimes called the **central chi-squared distribution**.

The **power** of a binary hypothesis test is the probability that the test rejects the null hypothesis (H_{0}) when a specific alternative hypothesis (H_{1}) is true. The statistical power ranges from 0 to 1, and as statistical power increases, the probability of making a type II error (wrongly failing to reject the null) decreases. For a type II error probability of β, the corresponding statistical power is 1 − β. For example, if experiment 1 has a statistical power of 0.7, and experiment 2 has a statistical power of 0.95, then there is a stronger probability that experiment 1 had a type II error than experiment 2, and experiment 2 is more reliable than experiment 1 due to the reduction in probability of a type II error. It can be equivalently thought of as the probability of accepting the alternative hypothesis (H_{1}) when it is true—that is, the ability of a test to detect a specific effect, if that specific effect actually exists. That is,

In statistics, a **confidence interval** (**CI**) is a type of interval estimate, computed from the statistics of the observed data, that might contain the true value of an unknown population parameter. The interval has an associated **confidence level** that, loosely speaking, quantifies the level of confidence that the parameter lies in the interval. More strictly speaking, the **confidence level** represents the frequency of possible confidence intervals that contain the true value of the unknown population parameter. In other words, if confidence intervals are constructed using a given confidence level from an infinite number of independent sample statistics, the proportion of those intervals that contain the true value of the parameter will be equal to the confidence level.

In statistics and optimization, **errors** and **residuals** are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its "theoretical value". The **error** of an observed value is the deviation of the observed value from the (unobservable) *true* value of a quantity of interest, and the **residual** of an observed value is the difference between the observed value and the *estimated* value of the quantity of interest. The distinction is most important in regression analysis, where the concepts are sometimes called the **regression errors** and **regression residuals** and where they lead to the concept of studentized residuals.

In statistics, a **sampling distribution** or **finite-sample distribution** is the probability distribution of a given random-sample-based statistic. If an arbitrarily large number of samples, each involving multiple observations, were separately used in order to compute one value of a statistic for each sample, then the sampling distribution is the probability distribution of the values that the statistic takes on. In many contexts, only one sample is observed, but the sampling distribution can be found theoretically.

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

The **standard error** (**SE**) of a statistic is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the parameter or the statistic is the mean, it is called the **standard error of the mean** (SEM).

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 determined based on the expense of data collection, and the need to have sufficient statistical power. In complicated studies there may be several different sample sizes involved in the study: for example, in a stratified survey there would be different sample sizes for each stratum. In a census, data are collected on the entire population, hence the sample size is equal to the population size. In experimental design, where a study may be divided into different treatment groups, this may be different sample sizes for each group.

*Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.*

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.*

In statistics, **bootstrapping** is any test or metric that relies on random sampling with replacement. Bootstrapping allows assigning measures of accuracy to sample estimates. This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods. Generally, it falls in the broader class of resampling methods.

In statistics, the **68–95–99.7 rule**, also known as the **empirical rule**, is a shorthand used to remember the percentage of values that lie within a band around the mean in a normal distribution with a width of two, four and six standard deviations, respectively; more accurately, 68.27%, 95.45% and 99.73% of the values lie within one, two and three standard deviations of the mean, respectively.

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. For example, it is used in estimating the population mean from a sampling distribution of sample means if the population standard deviation is unknown.

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.

- Sprinthall, R. C. (2011).
*Basic Statistical Analysis*(9th ed.). Pearson Education. ISBN 978-0-205-05217-2.

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