z-Test

Two-sample z-Test.

Usage

var ztest2 = require( '@stdlib/math/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.

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/math/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 );