# Student's t-Test
> One-sample and paired Student's t-Test.
## 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.
```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
*/
```
## Examples
```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
*/
```
[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