h = ttest2(x,y) returns
a test decision for the null hypothesis that the data in vectors x and y comes
from independent random samples from normal distributions with equal
means and equal but unknown variances, using the two-sample t-test.
The alternative hypothesis is that the data in x and y comes
from populations with unequal means. The result h is 1 if
the test rejects the null hypothesis at the 5% significance level,
and 0 otherwise.

h = ttest2(x,y,Name,Value) returns
a test decision for the two-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 the
test without assuming equal variances.

[h,p,ci,stats]
= ttest2(___) also returns the confidence
interval on the difference of the population means, ci,
and the structure stats containing information
about the test statistic.

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

load examgrades;
x = grades(:,1);
y = grades(:,2);

Test the null hypothesis that the two data vectors are
from populations with equal means, without assuming that the populations
also have equal variances.

[h,p] = ttest2(x,y,'Vartype','unequal')

h =
0
p =
0.9867

The returned value of h = 0 indicates that ttest2 does
not reject the null hypothesis at the default 5% significance level
even if equal variances are not assumed.

Sample data, specified as a vector, matrix, or multidimensional
array. ttest2 treats NaN values
as missing data and ignores them.

If x and y are
specified as vectors, they do not need to be the same length.

If x and y are
specified as matrices, they must have the same number of columns. ttest2 performs
a separate t-test along each column and returns
a vector of results.

Sample data, specified as a vector, matrix, or multidimensional
array. ttest2 treats NaN values
as missing data and ignores them.

If x and y are
specified as vectors, they do not need to be the same length.

If x and y are
specified as matrices, they must have the same number of columns. ttest2 performs
a separate t-test along each column and returns
a vector of results.

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,'Vartype','unequal' specifies
a right-tailed test at the 1% significance level, and does not assume
that x and y have equal population
variances.

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.

Variance type, specified as the comma-separated pair consisting
of 'Vartype' and one of the following.

'equal'

Conduct test using the assumption that x and y are
from normal distributions with unknown but equal variances.

'unequal'

Conduct test using the assumption that x and y are
from normal distributions with unknown and unequal variances. This
is called the Behrens-Fisher problem. ttest2 uses
Satterthwaite's approximation for the effective degrees of
freedom.

Vartype must be a single string, even when x is
a matrix or a multidimensional array.

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 difference in population means of x and y,
returned as a two-element vector containing the lower and upper boundaries
of the 100 × (1 – Alpha)% confidence
interval.

Test statistics for the two-sample t-test,
returned as a structure containing the following:

tstat — Value of the test
statistic.

df — Degrees of freedom
of the test.

sd — Pooled estimate of
the population standard deviation (for the equal variance case) or
a vector containing the unpooled estimates of the population standard
deviations (for the unequal variance case).

where $$\overline{x}$$ and $$\overline{y}$$ are the sample means, s_{x} and s_{y} are
the sample standard deviations, and n and m are
the sample sizes.

In the case where it is assumed that the two data samples are
from populations with equal variances, the test statistic under the
null hypothesis has Student's t distribution
with n + m – 2 degrees
of freedom, and the sample standard deviations are replaced by the
pooled standard deviation

In the case where it is not assumed that the two data samples
are from populations with equal variances, the test statistic under
the null hypothesis has an approximate Student's t distribution
with a number of degrees of freedom given by Satterthwaite's approximation.
This test is sometimes called Welch's t-test.

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.