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.

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.

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.

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.

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.

[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.

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 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.

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.

Hypothesized population mean, 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 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

'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.

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.

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.

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.