time-to-botec/squiggle/node_modules/@stdlib/stats/ttest/README.md

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# Student's t-Test

> One-sample and paired Student's t-Test.

<section class="usage">

## Usage

```javascript
var ttest = require( '@stdlib/stats/ttest' );
```

#### ttest( x\[, y]\[, opts] )

The function performs a one-sample t-test for the null hypothesis that the data in [array][mdn-array] or [typed array][mdn-typed-array] `x` is drawn from a normal distribution with mean zero and unknown variance.

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

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

rnorm = normal( 0.0, 2.0, {
    'seed': 5776
});
arr = new Array( 100 );
for ( i = 0; i < arr.length; i++ ) {
    arr[ i ] = rnorm();
}
out = ttest( arr );
/* e.g., returns
    {
        'rejected': false,
        'pValue': ~0.722,
        'statistic': ~0.357,
        'ci': [~-0.333,~0.479],
        // ...
    }
*/
```

When [array][mdn-array] or [typed array][mdn-typed-array] `y` is supplied, the function tests whether the differences `x - y` come from a normal distribution with mean zero and unknown variance via the paired t-test.

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

var rnorm;
var out;
var i;
var x;
var y;

rnorm = normal( 1.0, 2.0, {
    'seed': 786
});
x = new Array( 100 );
y = new Array( 100 );
for ( i = 0; i < x.length; i++ ) {
    x[ i ] = rnorm();
    y[ i ] = rnorm();
}
out = ttest( x, y );
/* e.g., returns
    {
        'rejected': false,
        'pValue': ~0.191,
        'statistic': ~1.315,
        'ci': [ ~-0.196, ~0.964 ],
        // ...
    }
*/
```

The returned object comes with a `.print()` method which when invoked will print a formatted output of the hypothesis test results. `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.

<!-- run-disable -->

```javascript
console.log( out.print() );
/* e.g., =>
    Paired t-test

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

        pValue: 0.1916
        statistic: 1.3148
        df: 99
        95% confidence interval: [-0.1955,0.9635]

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

The `ttest` 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.

```javascript
var table;
var out;
var arr;

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

out = ttest( arr, {
    'alpha': 0.01
});
table = out.print();
/* e.g., returns
    One-sample t-test

    Alternative hypothesis: True mean is not equal to 0

        pValue: 0.0474
        statistic: 2.8284
        df: 4
        99% confidence interval: [-1.2556,5.2556]

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

out = ttest( arr, {
    'alpha': 0.1
});
table = out.print();
/* e.g., returns
    One-sample t-test

    Alternative hypothesis: True mean is not equal to 0

        pValue: 0.0474
        statistic: 2.8284
        df: 4
        90% confidence interval: [0.4926,3.5074]

    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.

```javascript
var out;
var arr;

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

out = ttest( arr, {
    'mu': 5
});
/* e.g., returns
{
    'rejected': false,
    'pValue': 1,
    'statistic': 0,
    'ci': [ ~3.758, ~6.242 ],
    // ...
}
*/
```

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

```javascript
var table;
var out;
var arr;

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

out = ttest( arr, {
    'alternative': 'less'
});
table = out.print();
/* e.g., returns
    One-sample t-test

    Alternative hypothesis: True mean is less than 0

        pValue: 0.9998
        statistic: 11.1803
        df: 4
        95% confidence interval: [-Infinity,5.9534]

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

out = ttest( arr, {
    'alternative': 'greater'
});
table = out.print();
/* e.g., returns
    One-sample t-test

    Alternative hypothesis: True mean is greater than 0

        pValue: 0.0002
        statistic: 11.1803
        df: 4
        95% confidence interval: [4.0466,Infinity]

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

</section>

<!-- /.usage -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

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

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

rnorm = normal( 5.0, 4.0, {
    'seed': 37827
});
arr = new Array( 100 );
for ( i = 0; i < arr.length; i++ ) {
    arr[ i ] = rnorm();
}

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

    Alternative hypothesis: True mean is not equal to 0

        pValue: 0
        statistic: 15.0513
        df: 99
        95% confidence interval: [4.6997,6.127]

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

// Test whether true mean is equal to five:
out = ttest( arr, {
    'mu': 5.0
});
console.log( out.print() );
/* e.g., =>
    One-sample t-test

    Alternative hypothesis: True mean is not equal to 5

        pValue: 0.2532
        statistic: 1.1494
        df: 99
        95% confidence interval: [4.6997,6.127]

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

</section>

<!-- /.examples -->

<section class="links">

[mdn-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array

[mdn-typed-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Typed_arrays

</section>

<!-- /.links -->