ztest

z-test

Syntax

h = ztest(x,m,sigma)

h = ztest(x,m,sigma,Name,Value)

[h,p] = ztest(___)

[h,p,ci,zval] = ztest(___)

Description

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.

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.

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

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

Examples

z-Test for a Hypothesized Mean

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

load examgrades
x = grades(:,1);

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 = 2×1

   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 z-Test

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

load examgrades
x = grades(:,1);

Plot a histogram of the exam grades data and fit a normal density function.

histfit(x)
xlabel("Grade")
ylabel("Frequency")

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

[h,~,~,zval] = ztest(x,65,10,"Tail","right")

h = 
1

zval = 
10.9636

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

Plot the standard normal distribution, the returned z-statistic, and the critical z-value. Calculate the critical z-value for the default confidence level of 95% by using norminv.

k = linspace(-15,15,300);
y = normpdf(k);
zvalpdf = normpdf(zval);
zcrit = norminv(0.95);

plot(k,y);
hold on
scatter(zval,zvalpdf,"filled")
xline(zcrit,"--")
legend(["Standard Normal pdf","z-Statistic", ...
    "Critical Cutoff"])

The orange dot represents the z-statistic and is located to the right of the dashed black line that represents the critical z-value.

Input Arguments

x — Sample data

vector | 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 mean

scalar value

Hypothesized mean, specified as a scalar value.

Data Types: single | double

sigma — Population standard deviation

scalar value

Population standard deviation, specified as a scalar value.

Data Types: single | double

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

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

Alpha — Significance level

0.05 (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 — Dimension

first 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:

• 'both' — Test against the alternative hypothesis that the population mean is not m.

• 'right' — Test against the alternative hypothesis that the population mean is greater than m.

• 'left' — Test against the alternative hypothesis that the population mean is less than m.

ztest tests the null hypothesis that the population mean is m against the specified alternative hypothesis.

Example: 'Tail','right'

Output Arguments

h — Hypothesis test result

1 | 0

Hypothesis test result, returned as 1 or 0.

• A value of 1 indicates the rejection of the null hypothesis at the Alpha significance level.

• A value of 0 indicates a failure to reject the null hypothesis at the Alpha significance level.

p — p-value

scalar 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 that is as extreme as, or more extreme than, the observed value under the null hypothesis. A small value of p indicates that the null hypothesis might not be valid.

ci — Confidence interval

vector

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 statistic

scalar value

Test statistic, returned as a scalar value.

More About

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 that 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.

Tips

• Use sampsizepwr to calculate:

  • The sample size that corresponds to specified power and parameter values;

  • The power achieved for a particular sample size, given the true parameter value;

  • The parameter value detectable with the specified sample size and power.

Extended Capabilities

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced before R2006a