ZTest
ZTest[data]
tests whether the mean of the data is zero.
ZTest[{data_{1},data_{2}}]
tests whether the means of data_{1} and data_{2} are equal.
ZTest[dspec,σ^{2}]
tests for zero or equal means assuming a population variance σ^{2}.
ZTest[dspec,σ^{2},μ_{0}]
tests the mean against μ_{0}.
ZTest[dspec,σ^{2},μ_{0},"property"]
returns the value of "property".
Details and Options
 ZTest tests the null hypothesis against the alternative hypothesis :

data {data_{1},data_{2}}  where μ_{i} is the population mean for data_{i}.
 By default, a probability value or value is returned.
 A small value suggests that it is unlikely that is true.
 The data in dspec can be univariate {x_{1},x_{2},…} or multivariate {{x_{1},y_{1},…},{x_{2},y_{2},…},…}.
 Given one dataset, the argument σ^{2} can be any positive real number or a positive definite matrix with dimension equal to the dimension of data.
 Given two datasets, the argument σ^{2} can be any positive real number, a positive definite matrix with dimension equal to the dimension of dspec, or two such numbers or matrices.
 The argument μ_{0} can be a real number or a real vector with length equal to the dimension of the data.
 ZTest assumes that the data is normally distributed and that the variance is known and not estimated from the data.
 If variances or covariance matrices are not provided, ZTest treats the sample estimate as the known variance or covariance.
 ZTest[dspec,σ^{2},μ_{0},"HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
 ZTest[dspec,σ^{2},μ_{0},"property"] can be used to directly give the value of "property".
 Properties related to the reporting of test results include:

"DegreesOfFreedom" the degrees of freedom of a test "PValue" list of values "PValueTable" formatted table of values "TestData" list of pairs of test statistics and values "TestDataTable" formatted table of values and test statistics "TestStatistic" list of test statistics "TestStatisticTable" formatted table of test statistics  If a known variance σ^{2} is not provided, ZTest performs a test assuming the sample variance is the known variance for univariate data, and Hotelling's test assuming the sample covariance is the known covariance for multivariate data.
 Options include:

AlternativeHypothesis "Unequal" the inequality for the alternative hypothesis SignificanceLevel 0.05 cutoff for diagnostics and reporting VerifyTestAssumptions All what assumptions to verify  For tests of location, a cutoff is chosen such that is rejected if and only if . The value of used for the "TestConclusion" and "ShortTestConclusion" properties is controlled by the SignificanceLevel option. This value is also used in diagnostic tests of assumptions, including tests for normality, equal variance, and symmetry. By default, is set to 0.05.
 Named settings for VerifyTestAssumptions in ZTest include:

"Normality" verify that all data is normally distributed
Examples
open allclose allBasic Examples (4)
Scope (17)
Testing (14)
Setting the known variance to Automatic is equivalent to using the sample variance:
Specify a single common variance for two groups, or give a list of variances:
For multivariate data, covariance matrices can be given:
Separate covariance matrices for each dataset:
The values are typically large when the mean is close to :
The values are typically small when the location is far from :
Using Automatic is equivalent to testing for a mean of zero:
The values are typically large when the mean is close to μ_{0}:
The values are typically small when the location is far from μ_{0}:
Test whether the mean vector of a multivariate population is the zero vector:
Alternatively, test against {0.1,0,0.05,0}:
The values are generally small when the locations are not equal:
The values are generally large when the locations are equal:
The order of the datasets affects the test results:
Test whether the mean difference vector of two multivariate populations is the zero vector:
Alternatively, test against {1,0,1,0}:
Create a HypothesisTestData object for repeated property extraction:
The properties available for extraction:
Extract some properties from a HypothesisTestData object:
The value and test statistic:
Options (9)
AlternativeHypothesis (3)
SignificanceLevel (2)
VerifyTestAssumptions (4)
Control whether to verify the assumption of normality:
Unlisted assumptions are not tested:
The result is the same but a warning is issued:
Bypassing diagnostic tests can save compute time:
It is often useful to bypass diagnostic tests for simulation purposes:
The assumptions of the test hold by design, so a great deal of time can be saved:
Applications (1)
A factory creates 10inch fuse casings on two separate lines. The length of these casings has a known variance of 0.005 inches squared. In a quality control test, 25 casings were randomly selected from each of the two lines. A line is shut down and recalibrated if there is a significant deviation from 10 inches:
The first line performs as expected; the second needs to be recalibrated:
Suppose it is known that the factory machinery produces longer casings after prolonged use. The onesided alternative "Greater" is appropriate:
Properties & Relations (6)
The univariate test statistic for one sample:
The test statistic follows a NormalDistribution[0,1] under :
The multivariate test statistic for one sample:
Under , zChiSquareDistribution[p] where p is the dimension of the data:
If variance is not specified, the test statistic is equivalent to that of TTest:
The values are not equivalent:
In general, ZTest is more powerful than TTest:
ZTest works with the values only when the input is a TimeSeries:
ZTest works with all the values together when the input is a TemporalData:
Possible Issues (2)
The test statistic does not follow a NormalDistribution[0,1] if the sample variance is used:
TTest gives the appropriate value when the sample variance is used:
The data should follow a NormalDistribution:
Use medianbased tests if the data is not normally distributed:
LocationTest will pick an appropriate test automatically:
Text
Wolfram Research (2010), ZTest, Wolfram Language function, https://reference.wolfram.com/language/ref/ZTest.html.
CMS
Wolfram Language. 2010. "ZTest." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ZTest.html.
APA
Wolfram Language. (2010). ZTest. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ZTest.html