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# ztest

## Description

example

h = ztest(x,m,sigma) returns a test decision for the null hypothesis that the data in the vector x comes from a normal distribution with mean m and a standard deviation sigma, using the z-test. The alternative hypothesis is that the mean is not m. The result h is 1 if the test rejects the null hypothesis at the 5% significance level, and 0 otherwise.

example

h= ztest(x,m,sigma,Name,Value) returns a test decision for the z-test with additional options specified by one or more name-value pair arguments. For example, you can change the significance level or conduct a one-sided test.

example

[h,p] = ztest(___) also returns the p-value of the test, using any of the input arguments from previous syntaxes.

example

[h,p,ci,zval] = ztest(___) also returns the confidence interval of the population mean, ci, and the value of the test statistic, zval.

## Examples

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### Test for a Hypothesized Mean

Load the sample data. Create a vector containing the first column of the students' exam grades data.

```load examgrades;

Test the null hypothesis that the data comes from a normal distribution with mean m = 75 and standard deviation sigma = 10.

`[h,p,ci,zval] = ztest(x,75,10)`
```h =
0

p =
0.9927

ci =
73.2191
76.7975

zval =
0.0091```

The returned value of h = 0 indicates that ztest does not reject the null hypothesis at the default 5% significance level.

### One-Sided Hypothesis Test

Load the sample data. Create a vector containing the first column of the students' exam grades data.

```load examgrades;

Test the null hypothesis that the data comes from a normal distribution with mean m = 65 and standard deviation sigma = 10, against the alternative that the mean is greater than 65.

`[h,p] = ztest(x,65,10,'Tail','right')`
```h =
1

p =
2.8596e-28```

The returned value of h = 1 indicates that ztest rejects the null hypothesis at the default 5% significance level, in favor of the alternative hypothesis that the population mean is greater than 65.

## Input Arguments

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### x — Sample datavector | matrix | multidimensional array

Sample data, specified as a vector, matrix, or multidimensional array.

• If x is specified as a vector, ztest returns a single value for each output argument.

• If x is specified as a matrix, ztest performs a separate z-test along each column of x and returns a vector of results.

• If x is specified as a multidimensional array, ztest works along the first nonsingleton dimension of x.

In all cases, ztest treats NaN values as missing data and ignores them.

Data Types: single | double

### m — Hypothesized meanscalar value

Hypothesized mean, specified as a scalar value.

Data Types: single | double

### sigma — Population standard deviationscalar value

Population standard deviation, specified as a scalar value.

Data Types: single | double

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'Tail','right','Alpha',0.01 specifies a right-tailed hypothesis test at the 1% significance level.

### 'Alpha' — Significance level0.05 (default) | scalar value in the range (0,1)

Significance level of the hypothesis test, specified as the comma-separated pair consisting of 'Alpha' and a scalar value in the range (0,1).

Example: 'Alpha',0.01

Data Types: single | double

### 'Dim' — Dimensionfirst nonsingleton dimension (default) | positive integer value

Dimension of the input matrix along which to test the means, specified as the comma-separated pair consisting of 'Dim' and a positive integer value. For example, specifying 'Dim',1 tests the column means, while 'Dim',2 tests the row means.

Example: 'Dim',2

Data Types: single | double

### 'Tail' — Type of alternative hypothesis'both' (default) | 'right' | 'left'

Type of alternative hypothesis to evaluate, specified as the comma-separated pair consisting of 'Tail' and one of the following.

 'both' Test the alternative hypothesis that the population mean is not equal to m. 'right' Test the alternative hypothesis that the population mean is greater than m. 'left' Test the alternative hypothesis that the population mean is less than m.

Example: 'Tail','right'

## Output Arguments

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### h — Hypothesis test result1 | 0

Hypothesis test result, returned as a logical value.

• If h = 1, this indicates the rejection of the null hypothesis at the Alpha significance level.

• If h = 0, this indicates a failure to reject the null hypothesis at the Alpha significance level.

### p — p-valuescalar value in the range [0,1]

p-value of the test, returned as a scalar value in the range [0,1]. p is the probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis. Small values of p cast doubt on the validity of the null hypothesis.

### ci — Confidence intervalvector

Confidence interval for the true population mean, returned as a two-element vector containing the lower and upper boundaries of the 100 × (1 – Alpha)% confidence interval.

### zval — Test statisticnonnegative scalar value

Test statistic, returned as a nonnegative scalar value.

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### z-Test

The z-test is a parametric hypothesis test used to determine whether a sample data set comes from a population with a particular mean. The test assumes the sample data comes from a population with a normal distribution and a known standard deviation.

The test statistic is

$z=\frac{\overline{x}-\mu }{\sigma /\sqrt{n}},$

where $\overline{x}$ is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. Under the null hypothesis, the test statistic has a standard normal distribution.

### Multidimensional Array

A multidimensional array has more than two dimensions. For example, if x is a 1-by-3-by-4 array, then x is a three-dimensional array.

### First Nonsingleton Dimension

The first nonsingleton dimension is the first dimension of an array whose size is not equal to 1. For example, if x is a 1-by-2-by-3-by-4 array, then the second dimension is the first nonsingleton dimension of x.