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z-Test

Two-sample z-Test.

Installation

npm install @stdlib/stats-ztest2

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

var ztest2 = require( '@stdlib/stats-ztest2' );

ztest2( x, y, sigmax, sigmay[, opts] )

By default, the function performs a two-sample z-test for the null hypothesis that the data in arrays or typed arrays x and y is independently drawn from normal distributions with equal means and known standard deviations sigmax and sigmay.

var x = [ 2.66, 1.5, 3.25, 0.993, 2.31, 2.41, 1.76, 2.57, 2.62, 1.23 ]; // Drawn from N(2,1)
var y = [ 4.88, 2.93, 2.96, 4.5, -0.0603, 4.62, 3.35, 2.98 ]; // Drawn from N(3,2)

var out = ztest2( x, y, 1.0, 2.0 );
/* e.g., returns
    {
        'rejected': false,
        'pValue': ~0.141,
        'statistic': ~-1.471,
        'ci': [ ~-2.658, ~0.379 ],
        // ...
    }
*/

The returned object comes with a .print() method which when invoked will print a formatted output of the results of the hypothesis test. print accepts a digits option that controls the number of decimal digits displayed for the outputs and a decision option, which when set to false will hide the test decision.

console.log( out.print() );
/* e.g., =>
    Two-sample z-test

    Alternative hypothesis: True difference in means is not equal to 0

        pValue: 0.1412
        statistic: -1.4713
        95% confidence interval: [-2.6578,0.3785]

    Test Decision: Fail to reject null in favor of alternative at 5% significance level
*/

The function accepts the following options:

• alpha: number in the interval [0,1] giving the significance level of the hypothesis test. Default: 0.05.
• alternative: Either two-sided, less or greater. Indicates whether the alternative hypothesis is that x has a larger mean than y (greater), x has a smaller mean than y (less) or the means are the same (two-sided). Default: two-sided.
• difference: number denoting the difference in means under the null hypothesis. Default: 0.

By default, the hypothesis test is carried out at a significance level of 0.05. To choose a different significance level, set the alpha option.

var out = ztest2( x, y, 1.0, 2.0, {
    'alpha': 0.2
});
var table = out.print();
/* e.g., returns
    Two-sample z-test

    Alternative hypothesis: True difference in means is not equal to 0

        pValue: 0.1412
        statistic: -1.4713
        80% confidence interval: [-2.1323,-0.147]

    Test Decision: Reject null in favor of alternative at 20% significance level
*/

By default, a two-sided test is performed. To perform either of the one-sided tests, set the alternative option to less or greater.

var out = ztest2( x, y, {
    'alternative': 'less'
});
var table = out.print();
/* e.g., returns
    Two-sample z-test

    Alternative hypothesis: True difference in means is less than 0

        pValue: 0.0706
        statistic: -1.4713
        95% confidence interval: [-Infinity,0.1344]

    Test Decision: Fail to reject null in favor of alternative at 5% significance level
*/

out = ztest2( x, y, {
    'alternative': 'greater'
});
table = out.print();
/* e.g., returns
    Two-sample z-test

    Alternative hypothesis: True difference in means is greater than 0

        pValue: 0.9294
        statistic: -1.4713
        95% confidence interval: [-2.4138,Infinity]

    Test Decision: Fail to reject null in favor of alternative at 5% significance level
*/

To test whether the difference in the population means is equal to some other value than 0, set the difference option.

var normal = require( '@stdlib/random-base-normal' ).factory;

var rnorm = normal({
    'seed': 372
});

var x = new Array( 100 );
var i;
for ( i = 0; i < x.length; i++ ) {
    x[ i ] = rnorm( 2.0, 1.0 );
}
var y = new Array( 100 );
for ( i = 0; i < x.length; i++ ) {
    y[ i ] = rnorm( 1.0, 1.0 );
}

var out = ztest2( x, y, 1.0, 1.0, {
    'difference': 1.0
});
/* e.g., returns
    {
        'rejected': false,
        'pValue': ~0.74,
        'statistic': ~0.332,
        'ci': [ ~0.77, ~1.324 ],
        // ...
    }
*/

var table = out.print();
/* e.g., returns
    Two-sample z-test

    Alternative hypothesis: True difference in means is not equal to 1

        pValue: 0.7395
        statistic: 0.3325
        95% confidence interval: [0.7698,1.3242]

    Test Decision: Fail to reject null in favor of alternative at 5% significance level
*/

Examples

var rnorm = require( '@stdlib/random-base-normal' );
var ztest2 = require( '@stdlib/stats-ztest2' );

var table;
var out;
var x;
var y;
var i;

// Values drawn from a Normal(4,2) distribution
x = new Array( 100 );
for ( i = 0; i < 100; i++ ) {
    x[ i ] = rnorm( 4.0, 2.0 );
}
// Values drawn from a Normal(3,2) distribution
y = new Array( 80 );
for ( i = 0; i < 80; i++ ) {
    y[ i ] = rnorm( 3.0, 2.0 );
}

out = ztest2( x, y, 2.0, 2.0 );
table = out.print();
console.log( table );

out = ztest2( x, y, 2.0, 2.0, {
    'difference': 1.0
});
table = out.print();
console.log( table );

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See Also

• @stdlib/stats-ztest: one-sample and paired z-Test.

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Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

Community

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License

See LICENSE.

Copyright

Copyright © 2016-2025. The Stdlib Authors.