# Z-Test

One-sample z-Test.

## Usage

``````var ztest = require( '@stdlib/math/stats/ztest' );
``````

### ztest( x, sigma[, opts] )

The function performs a one-sample z-test for the null hypothesis that the data in array or typed array `x` is drawn from a normal distribution with mean zero and known standard deviation `sigma`.

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

var rnorm = normal( 0.0, 2.0, {
'seed': 5776
});

var arr = new Array( 300 );
var i;
for ( i = 0; i < arr.length; i++ ) {
arr[ i ] = rnorm();
}

var out = ztest( arr, 2.0 );
/* e.g., returns
{
rejected: false,
pValue: ~0.155,
statistic: -1.422,
ci: [~-0.391,~0.062],
// ...
}
*/
``````

The returned object comes with a `.print()` method which when invoked will print a formatted output of the hypothesis test results. `print` accepts a `numdigits` option that controls the number of decimal digits displayed for the outputs.

``````var table = out.print({
'numdigits': 3
});
console.log( table );
/* e.g., =>
One-sample z-test

Alternative hypothesis: True mean is not equal to 0

pValue: 0.155
statistic: -1.422
95% confidence interval: [-0.391,0.062]

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

The `ztest` 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 the mean of `x` is larger than `mu` (`greater`), smaller than `mu` (`less`) or equal to `mu` (`two-sided`). Default: `two-sided`.
• mu: `number` denoting the hypothesized true mean 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 table;
var out;
var arr;

arr = [ 2, 4, 3, 1, 0 ];

out = ztest( arr, 2.0, {
'alpha': 0.01
});
table = out.print();
/* e.g., returns
One-sample z-test

Alternative hypothesis: True mean is not equal to 0

pValue: 0.0253
statistic: 2.2361
99% confidence interval: [-0.3039,4.3039]

Test Decision: Fail to reject null in favor of alternative at 1% significance level
*/

out = ztest( arr, 2.0, {
'alpha': 0.1
});
table = out.print();
/* e.g., returns
One-sample z-test

Alternative hypothesis: True mean is not equal to 0

pValue: 0.0253
statistic: 2.2361
90% confidence interval: [0.5288,3.4712]

Test Decision: Reject null in favor of alternative at 10% significance level
*/
``````

To test whether the data comes from a distribution with a mean different than zero, set the `mu` option.

``````var out;
var arr;

arr = [ 4, 4, 6, 6, 5 ];

out = ztest( arr, 1.0, {
'mu': 5.0
});
/* e.g., returns
{
rejected: false,
pValue: 1,
statistic: 0,
ci: [ ~4.123, ~5.877 ],
// ...
}
*/
``````

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 table;
var out;
var arr;

arr = [ 4, 4, 6, 6, 5 ];

out = ztest( arr, 1.0, {
'alternative': 'less'
});
table = out.print();
/* e.g., returns
One-sample z-test

Alternative hypothesis: True mean is less than 0

pValue: 1
statistic: 11.1803
95% confidence interval: [-Infinity,5.7356]

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

out = ztest( arr, 1.0, {
'alternative': 'greater'
});
table = out.print();
/* e.g., returns
One-sample z-test

Alternative hypothesis: True mean is greater than 0

pValue: 0
statistic: 11.1803
95% confidence interval: [4.2644,Infinity]

Test Decision: Reject null in favor of alternative at 5% significance level
*/
``````

## Examples

``````var normal = require( '@stdlib/random/base/normal' ).factory;
var ztest = require( '@stdlib/math/stats/ztest' );

var rnorm;
var arr;
var out;
var i;

rnorm = normal( 5.0, 4.0, {
'seed': 37827
});
arr = new Array( 500 );
for ( i = 0; i < arr.length; i++ ) {
arr[ i ] = rnorm();
}

// Test whether true mean is equal to zero:
out = ztest( arr, 4.0 );
console.log( out.print() );
/* e.g., =>
One-sample z-test

Alternative hypothesis: True mean is not equal to 0

pValue: 0
statistic: 28.6754
95% confidence interval: [4.779,5.4802]

Test Decision: Reject null in favor of alternative at 5% significance level
*/

// Test whether true mean is equal to five:
out = ztest( arr, 4.0, {
'mu': 5.0
});
console.log( out.print() );
/* e.g., =>
One-sample z-test

Alternative hypothesis: True mean is not equal to 5

pValue: 0.4688
statistic: 0.7245
95% confidence interval: [4.779,5.4802]

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