One-Sample T-Test

The One-sample t-test compares the mean of a normally
distributed variable with a hypothesized value (*true mean*). The
test uses the standard deviation of the sample *s* to estimate the
population standard deviation .
To run t-test for multiple variables at the same time, simply select the *variable*
with hypothesized values *instead* of entering the hypothesized *value*.

# Assumptions

The test assumes the population is normally distributed.

*In case the **population is not normally distributed, the
test statistic has different and unknown distribution and, strictly speaking,
the t-test cannot be used. However, according to the central limit theorem, if
the sample size is large enough, we can use the t-test even if the normal
distribution requirement is not met. Unfortunately there is no easy way to
determine what N is large enough. In each particular case there is a separate
limit that depends on how strongly the distribution differs from the normal
distribution. Some sources quote N=30 as large enough, but even this sample
size may turn out to be not large enough. Non-parametric tests may work as an
alternative in this case.*

# How To

Run **Statistics→Basic Statistics→One
Sample T-Test**...

Select one or more variables.

Select a variable with hypothesized values (if you have a distinct hypothesized value for each variable) or enter a hypothesized value as a number (if you have the same hypothesized value for each variable or you are running the t-test for one variable).

Null hypothesis is
defined as H_{0}: . Optionally,
you can define the alternative hypothesis H_{1 }in the Advanced Options:

o H_{1}: less than (lower-tailed) ,

o H_{1}: not equal (two-tailed),

o H_{1}: greater than (upper-tailed).

Default value is H_{1}: Not equal (two-tailed).

# Results

Report includes descriptive statistics summary and results of the one sample t-test for each variable.

Mean, Mean LCL, Mean UCL, Standard Error (of Mean), Sample Size – see the Descriptive Statistics procedure for more information.

Hypothesized value (Test Constant,
H_{0} Value) - the hypothesized value (fixed estimate).

Difference – difference between the sample mean and the hypothesized value.

Test statistic– the number defined as

where is the standard deviation of the sample, – sample mean, – sample size.

Test statistic either exactly follows or closely approximates
a t-distribution with *N-1* degrees of freedom under the null hypothesis.

d.f. - appropriate degrees of freedom for each variable.

p-value (*-tailed) –p-value for
the test, corresponding to the selected H_{1}.

H1: Not equal (two-tailed): If the p-value is less than default (default value 0.05), statistically, the sample mean is significantly different from the hypothesized value.