# ttest

One-sample and paired-sample t-test

## Description

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h = ttest(x) returns a test decision for the null hypothesis that the data in x comes from a normal distribution with mean equal to zero and unknown variance, using the one-sample t-test. The alternative hypothesis is that the population distribution does not have a mean equal to zero. The result h is 1 if the test rejects the null hypothesis at the 5% significance level, and 0 otherwise.

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h = ttest(x,y) returns a test decision for the null hypothesis that the data in x – y comes from a normal distribution with mean equal to zero and unknown variance, using the paired-sample t-test.

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h = ttest(x,y,Name,Value) returns a test decision for the paired-sample t-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.

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h = ttest(x,m) returns a test decision for the null hypothesis that the data in x comes from a normal distribution with mean m and unknown variance. The alternative hypothesis is that the mean is not m.

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h = ttest(x,m,Name,Value) returns a test decision for the one-sample t-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.

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

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[h,p,ci,stats] = ttest(___) also returns the confidence interval ci for the mean of x, or of x – y for the paired t-test, and the structure stats containing information about the test statistic.

## Examples

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Load the sample data. Create a vector containing the third column of the stock returns data.

x = stocks(:,3);

Test the null hypothesis that the sample data comes from a population with mean equal to zero.

[h,p,ci,stats] = ttest(x)
h = 1
p = 0.0106
ci = 2×1

-0.7357
-0.0997

stats = struct with fields:
tstat: -2.6065
df: 99
sd: 1.6027

The returned value h = 1 indicates that ttest rejects the null hypothesis at the 5% significance level.

Load the sample data. Create a vector containing the third column of the stock returns data.

x = stocks(:,3);

Test the null hypothesis that the sample data are from a population with mean equal to zero at the 1% significance level.

h = ttest(x,0,'Alpha',0.01)
h = 0

The returned value h = 0 indicates that ttest does not reject the null hypothesis at the 1% significance level.

Load the sample data. Create vectors containing the first and second columns of the data matrix to represent students’ grades on two exams.

Test the null hypothesis that the pairwise difference between data vectors x and y has a mean equal to zero.

[h,p] = ttest(x,y)
h = 0
p = 0.9805

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

Load the sample data. Create vectors containing the first and second columns of the data matrix to represent students’ grades on two exams.

Test the null hypothesis that the pairwise difference between data vectors x and y has a mean equal to zero at the 1% significance level.

[h,p] = ttest(x,y,'Alpha',0.01)
h = 0
p = 0.9805

The returned value of h = 0 indicates that ttest does not reject the null hypothesis at the 1% significance level.

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

Test the null hypothesis that sample data comes from a distribution with mean m = 75.

h = ttest(x,75)
h = 0

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

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

Test the null hypothesis that the data comes from a population with mean equal to 65, against the alternative that the mean is greater than 65.

h = ttest(x,65,'Tail','right')
h = 1

The returned value of h = 1 indicates that ttest rejects the null hypothesis at the 5% significance level, in favor of the alternate hypothesis that the data comes from a population with a mean greater than 65.

## Input Arguments

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Sample data, specified as a vector, matrix, or multidimensional array. ttest performs a separate t-test along each column and returns a vector of results. If y sample data is specified, x and y must be the same size.

Data Types: single | double

Sample data, specified as a vector, matrix, or multidimensional array. If y sample data is specified, x and y must be the same size.

Data Types: single | double

Hypothesized population mean, 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 conducts a right-tailed hypothesis test at the 1% significance level.

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

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

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.

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

Example: 'Tail','right'

## Output Arguments

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Hypothesis test result, returned as 1 or 0.

• 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-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.

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.

Test statistics, returned as a structure containing the following:

• tstat — Value of the test statistic.

• df — Degrees of freedom of the test.

• sd — Estimated population standard deviation. For a paired t-test, sd is the standard deviation of x – y.

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### One-Sample t-Test

The one-sample t-test is a parametric test of the location parameter when the population standard deviation is unknown.

The test statistic is

$t=\frac{\overline{x}-\mu }{s/\sqrt{n}},$

where $\overline{x}$ is the sample mean, μ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size. Under the null hypothesis, the test statistic has Student’s t distribution with n – 1 degrees of freedom.

### 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.

## Version History

Introduced before R2006a