The t-test assesses whether the means of two
groups are *statistically*
different from each other.

**Requirements:**

Theoretically, the t-test can be used even if
the sample sizes are very small, as long as the variables are ** normally
distributed** within each group and the variation in the two groups is
not reliably different (

The normality assumption can be evaluated by
looking at the distribution of the data (via histograms, QQ plot) or by
performing a normality test.

The equality of variances assumption can be
verified with the *F* test.

If these conditions are not met, then you can
evaluate the differences in means between two groups using one of the
nonparametric alternatives to the *t*- test (see *nonparametrics** two sample
t-test*).**
**

If equal variance
does not meet,** Welch's t test,** an
adaptation of Student's

**Two sides or
one side:**

The *p*-level reported with a *t*-test
represents the probability of error associated with rejecting the hypothesis of
no difference between the two group when, in fact, the hypothesis is true.

Some researchers suggest that if the
difference is in the predicted direction, you can consider only one half (one
"tail") of the probability distribution and thus divide the standard *p*-level
reported with a *t*-test (a "two-tailed" probability) by two.
Others, however, suggest that you should always report the standard, two-tailed
t-test probability.

**Paired data**

For paired data, we use paired *t*-test, which focuses on the
difference between the paired data and reports the probability that the actual
mean difference is consistent with zero. This comparison is aided by the
reduction in variance achieved by taking the differences.

According to the data structure, if data for each pair
was represented by two variables in one observation (eg,
x1, x2), we can use Empower “*create
new variables*” function to create a new variable which is the
difference of these two variables (eg. x=x1-x2); if
data for each pair was represented by one variable in two observation, and a pair
ID variables was used for identifying the pairs, we can use Empower “*Transpose
multiple observations to multiple variables*” function and then use “create
new variables” function to calculate the difference for each pair. Once the difference for each pair was calculated,
we can use “One
sample t-test” function to compare the mean difference to zero.

Below is the **sample
input window**

Below is the **sample
output and explanation** of the above model:

t test comparing means
from two samples

Two Sample t-test

**(1) ****T
test for SBP comparison smoker (SMOKE.NEW=1) versus non-smoker (SMOKE.NEW=0)**

data: SBP by SMOKE.NEW

t
= -4.7558, df = 651, p-value
= 2.435e-06

alternative
hypothesis: true difference in means is not equal to 0

**(2) ****95%
confidence interval of the mean difference**

95
percent confidence interval:

-4.919987 -2.044465

**(3) ****Means
in each group**

sample
estimates:

mean in group 0 mean in
group 1

126.4062 129.8885

**Empower also
output following bar plot and box plot to show the means and distribution for
each group.**