Z-Test

One sample z-test of the null hypothesis that data in the sample are a random sample from a normal distribution with specified supposed mean and standard deviation, against the alternative that the mean is different.

This topic contains the following sections:

Z-Test Specification
Implementation
Code Sample
See Also

Z-Test Specification

For one sample z-test we consider the time series HT Time Series

Test statistic is calculated as follows:

where standard error SE:

Default supposed values of mean and variance are 0 and 1 correspondingly.

Implementation

The following constructors create an instance of ZTest class.

Constructor	Description	Performance
two-tailed test, default significance level	Constructor without parameters. Creates ZTest instance with default significance level for two-tailed test. ZTest
default significance level	Creates ZTest instance with default significance level and user defined tail. ZTest(Tail)
two-tailed test, user defined significance level	Creates ZTest instance for two-tailed test and user-defined significance level. ZTest(Double)
user defined tail and significance level	Creates ZTest instance with user defined significance level and tail. ZTest(Double, Tail)

The class provides the following methods:

Method	Description	Performance
update	Update sample by double array part, user specifies offset and number of the elements to use: Update(Double, Int32, Int32)
update mean and variance	Updates test statistic using provided sample and supposition about mean and variance. Double array sample: Update(Double, Double, Double) Vector sample: Update(Vector, Double, Double)

Method

Description

Performance

update

Update sample by double array part, user specifies offset and number of the elements to use:

method Update(Double, Int32, Int32)

update mean and variance

Updates test statistic using provided sample and supposition about mean and variance.

Double array sample:

method Update(Double, Double, Double)

Vector sample:

method Update(Vector, Double, Double)

The class provides the following properties:

Property	Description	Performance
region of acceptance	We fail to reject null hypothesis if test statistics is between left and right borders of region of acceptance. Region of acceptance left border: AcceptanceRegionLeft Region of acceptance right border: AcceptanceRegionRight
p-value	The probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. PValue

Property

Description

Performance

region of acceptance

We fail to reject null hypothesis if test statistics is between left and right borders of region of acceptance.

Region of acceptance left border:

Property AcceptanceRegionLeft

Region of acceptance right border:

Property AcceptanceRegionRight

p-value

The probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true.

Property PValue

Code Sample

The example of ZTest class usage:

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 1using System;
 2using FinMath.LinearAlgebra;
 3using FinMath.Statistics.HypothesisTesting;
 4using FinMath.Statistics.Distributions;
 5
 6namespace FinMath.Samples
 7{
 8    class ZTestSample
 9    {
10        static void Main()
11        {
12            // Create an instance of normal distribution.
13            Normal distr = new Normal(3, 1);
14
15            // Generate random sample from normal distribution.
16            Vector series = Vector.Random(100, distr);
17
18            // Create an instance of ZTest.
19            ZTest test = new ZTest(0.05, Tail.Both);
20            test.Update(series, 3, 1);
21
22            Console.WriteLine("Test Result:");
23            // Test decision
24            Console.WriteLine($"  The null hypothesis failed to be rejected: {test.Decision}");
25            // The statistic of ZTest test.
26            Console.WriteLine($"  Statistics = {test.Statistics:0.000}");
27            // The p-value of the test statistic.
28            Console.WriteLine($"  P-Value = {test.PValue:0.000}");
29        }
30    }
31}

Reference

FinMath.Statistics.HypothesisTesting