239 lines
6.0 KiB
Markdown
239 lines
6.0 KiB
Markdown
<!--
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@license Apache-2.0
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Copyright (c) 2018 The Stdlib Authors.
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Licensed under the Apache License, Version 2.0 (the "License");
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you may not use this file except in compliance with the License.
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You may obtain a copy of the License at
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http://www.apache.org/licenses/LICENSE-2.0
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Unless required by applicable law or agreed to in writing, software
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distributed under the License is distributed on an "AS IS" BASIS,
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WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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See the License for the specific language governing permissions and
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limitations under the License.
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-->
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# z-Test
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> Two-sample z-Test.
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<section class="usage">
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## Usage
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```javascript
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var ztest2 = require( '@stdlib/stats/ztest2' );
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```
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#### ztest2( x, y, sigmax, sigmay\[, opts] )
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By default, the function performs a two-sample z-test for the null hypothesis that the data in [arrays][mdn-array] or [typed arrays][mdn-typed-array] `x` and `y` is independently drawn from normal distributions with _equal_ means and known standard deviations `sigmax` and `sigmay`.
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```javascript
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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)
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var y = [ 4.88, 2.93, 2.96, 4.5, -0.0603, 4.62, 3.35, 2.98 ]; // Drawn from N(3,2)
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var out = ztest2( x, y, 1.0, 2.0 );
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/* e.g., returns
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{
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'rejected': false,
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'pValue': ~0.141,
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'statistic': ~-1.471,
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'ci': [ ~-2.658, ~0.379 ],
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// ...
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}
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*/
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```
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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.
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<!-- run-disable -->
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```javascript
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console.log( out.print() );
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/* e.g., =>
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Two-sample z-test
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Alternative hypothesis: True difference in means is not equal to 0
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pValue: 0.1412
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statistic: -1.4713
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95% confidence interval: [-2.6578,0.3785]
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Test Decision: Fail to reject null in favor of alternative at 5% significance level
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*/
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```
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The function accepts the following `options`:
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- **alpha**: `number` in the interval `[0,1]` giving the significance level of the hypothesis test. Default: `0.05`.
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- **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`.
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- **difference**: `number` denoting the difference in means under the null hypothesis. Default: `0`.
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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.
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<!-- run-disable -->
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```javascript
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var out = ztest2( x, y, 1.0, 2.0, {
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'alpha': 0.2
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});
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var table = out.print();
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/* e.g., returns
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Two-sample z-test
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Alternative hypothesis: True difference in means is not equal to 0
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pValue: 0.1412
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statistic: -1.4713
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80% confidence interval: [-2.1323,-0.147]
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Test Decision: Reject null in favor of alternative at 20% significance level
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*/
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```
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By default, a two-sided test is performed. To perform either of the one-sided tests, set the `alternative` option to `less` or `greater`.
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<!-- run-disable -->
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```javascript
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var out = ztest2( x, y, {
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'alternative': 'less'
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});
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var table = out.print();
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/* e.g., returns
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Two-sample z-test
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Alternative hypothesis: True difference in means is less than 0
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pValue: 0.0706
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statistic: -1.4713
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95% confidence interval: [-Infinity,0.1344]
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Test Decision: Fail to reject null in favor of alternative at 5% significance level
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*/
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out = ztest2( x, y, {
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'alternative': 'greater'
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});
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table = out.print();
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/* e.g., returns
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Two-sample z-test
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Alternative hypothesis: True difference in means is greater than 0
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pValue: 0.9294
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statistic: -1.4713
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95% confidence interval: [-2.4138,Infinity]
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Test Decision: Fail to reject null in favor of alternative at 5% significance level
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*/
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```
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To test whether the difference in the population means is equal to some other value than `0`, set the `difference` option.
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```javascript
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var normal = require( '@stdlib/random/base/normal' ).factory;
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var rnorm = normal({
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'seed': 372
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});
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var x = new Array( 100 );
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var i;
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for ( i = 0; i < x.length; i++ ) {
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x[ i ] = rnorm( 2.0, 1.0 );
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}
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var y = new Array( 100 );
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for ( i = 0; i < x.length; i++ ) {
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y[ i ] = rnorm( 1.0, 1.0 );
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}
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var out = ztest2( x, y, 1.0, 1.0, {
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'difference': 1.0
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});
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/* e.g., returns
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{
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'rejected': false,
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'pValue': ~0.74,
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'statistic': ~0.332,
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'ci': [ ~0.77, ~1.324 ],
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// ...
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}
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*/
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var table = out.print();
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/* e.g., returns
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Two-sample z-test
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Alternative hypothesis: True difference in means is not equal to 1
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pValue: 0.7395
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statistic: 0.3325
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95% confidence interval: [0.7698,1.3242]
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Test Decision: Fail to reject null in favor of alternative at 5% significance level
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*/
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```
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</section>
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<!-- /.usage -->
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<section class="examples">
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## Examples
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<!-- eslint no-undef: "error" -->
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```javascript
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var rnorm = require( '@stdlib/random/base/normal' );
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var ztest2 = require( '@stdlib/stats/ztest2' );
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var table;
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var out;
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var x;
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var y;
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var i;
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// Values drawn from a Normal(4,2) distribution
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x = new Array( 100 );
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for ( i = 0; i < 100; i++ ) {
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x[ i ] = rnorm( 4.0, 2.0 );
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}
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// Values drawn from a Normal(3,2) distribution
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y = new Array( 80 );
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for ( i = 0; i < 80; i++ ) {
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y[ i ] = rnorm( 3.0, 2.0 );
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}
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out = ztest2( x, y, 2.0, 2.0 );
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table = out.print();
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console.log( table );
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out = ztest2( x, y, 2.0, 2.0, {
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'difference': 1.0
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});
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table = out.print();
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console.log( table );
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```
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</section>
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<!-- /.examples -->
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<section class="links">
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[mdn-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array
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[mdn-typed-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Typed_arrays
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</section>
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<!-- /.links -->
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