﻿ Z-test

# Z-test

The Gaussian test is a term used in mathematical statistics. He refers to a group hypothesis tests with standard normal distributed Testprüfgröße under the null hypothesis. The test is named after Carl Friedrich Gauss.

With the Gaussian test are tested hypotheses about the expected values ​​of those populations by sampling averages, from which the sample.

The Gaussian test has a strong relationship with the t-test. The most important difference lies in the conditions for the application of these tests: While the t-test works with the empirical standard deviations of the samples, the standard deviations of the populations must be known for the Gauss test.

• 3.1 Application
• 3.2 Calculation of Testprüfgröße
• 4.1 application
• 4.2 Calculation of Testprüfgröße

## Mathematical Foundations

Are independent normally distributed random variables with mean and standard deviation, as is their arithmetic mean

Normally distributed with mean and standard deviation.

The sampling function

Is then a standard normal distribution under the null hypothesis and is used as a test statistic.

The test statistic can be written as:

So as a standard normal random variable χ plus a number that shows the distance between the real and the assumed expectation value in a standardized way.

Also chairs independent normally distributed random variables which are also independent of the X - sample, with mean, standard deviation and arithmetic mean

Before, as is normally distributed with mean and standard deviation.

The sampling function

Is then a standard normal distribution under the null hypothesis and is used as a test statistic.

## One sample Gauss test

### Application

The one-sample Gauss test checks be the arithmetical mean of a sample if the expectation value of the corresponding population equal to (or less than or greater ) is a predetermined value.

The sample consisted of the forms of independent random variables and come from a normally distributed population with unknown mean and known standard deviation.

It will be tested at a

• Two-sided test: against
• Right-sided test: against
• Left -sided test: against

The value of is specified by the user.

### Calculation of Testprüfgröße

With the sample mean to calculate the Testprüfgröße.

## Two-sample Gauss test for independent samples

### Application

The two-sample Gauss test for independent samples was verified using the arithmetic mean of the sample if the expectation values ​​of the corresponding populations are different.

The independent samples and are intended to be independent of each other and normally distributed populations with unknown expectation values ​​and known or standard deviations or originate.

It will be tested at a

• Two-sided test: against
• Right-sided test: against
• Left -sided test: against

The value of is specified by the user.

### Calculation of Testprüfgröße

With the sample means and calculating the Testprüfgröße.

## Two-sample Gauss test for dependent ( related ) samples

### Application

For the two-sample Gauss test for dependent samples pairs of measured values ​​must be present, such as back encounters with before-and- after measurements. By means of the pair differences will determine whether these differences in the expected value of the corresponding population equal to (or less than or greater ) is a predetermined value.

The differences are to form an independent sample and are from a normally distributed population with unknown mean and known standard deviation. The used test is the injection sample Gauss test for the independent sample of the differences.

It will be tested at a

• Two-sided test: against
• Right-sided test: against
• Left -sided test: against

Is specified by the user. In most applications, is tested for " imbalance " ( ), then.

### Calculation of Testprüfgröße

The differences a new sample with arithmetic mean. So you can apply the one-sample Gauss test on the sample of differences and receives as Testprüfgröße.

## Decision on the hypotheses

In all three Gaussian tests the general criteria for hypothesis tests are applied to decide on the acceptance or rejection of the hypotheses. Since under the null hypothesis is a standard normal random variable, we obtain the following rules.

Rejection of (ie, acceptance of ) the level of significance, if the following holds:

• The two-sided test: (this is the quantile of the standard normal distribution )
• The right-sided test:
• The left -sided test:

## Gaussian test for non - normally distributed random variables

For large sample sizes (> 30 as a rule of thumb ) can be dispensed with the assumption of normal distribution due to the Central Limit Theorem. So if the law applicable to the Gaussian test requirements are met, the expected values ​​and standard deviations of the random variables involved, it is assumed that the necessary for the calculation of z sums are normally distributed approximate and the Gaussian test to a good approximation provides correct results.

## Example

A certain blood parameter B is in the population in a very good approximation normally distributed with. From a group of chemically related drugs is known that they can shift the distribution of the blood parameter, ie they may alter the expected value (while maintaining the distribution shape).

For a drug P in this group should be examined whether such a change actually adjusted. Random independent samples of size n = 22 results in the following values ​​for B:

Without administration of P xi 12 13 10 12 14 11 14 18 15 13 15 13 11 17 11 12 13 14 15 13 14 13 with administration of P yi 13 14 13 17 13 16 16 19 17 15 17 15 15 20 15 15 14 15 13 15 16 15 Using these values, different hypotheses to be tested. The significance level should be 0.05; the corresponding u values ​​are then (hereinafter all rounded values):

The mean value is calculated.

Now, an attempt should be viewed with dependent samples. With large pre-post studies, a normal distribution was found for the change in the B values ​​by the gift of the affected drugs also, with. In the table of measured values ​​each one above the other readings now had been identified in a before-after trial.

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