Independent Group t-Test

(Student's t-test)

**Definition:** Used to
compare the means of two independent groups.

**Assumptions: **Subjects
are randomly assigned to one of two groups. The distribution of the means
being compared are normal with equal variances.

Test: The hypotheses for the
comparison of two independent groups are:

H_{o}: u_{1}
= u_{2} (means of the two groups are equal)

H_{a}: u_{1}
u_{2} (means of the two group are not equal)

The test statistic for is t,
with N_{1} + N_{2} - 2 degrees of freedom, where N_{1}
and N_{2} are the sample sizes for groups 1 and 2. A low p-value for
this test (less than 0.05 for example) means that there is evidence to
reject the null hypothesis in favor of the alternative hypothesis. Or, there
is evidence that the difference in the two means are statistically
significant.

*Note: *One sided t-tests
are not as common. In this case, the alternative hypothesis is directional.
For example:

H_{a}: u_{1} < u_{2} (the mean of group 1 is less than the mean of group 2)

When a one-sided hypothesis is
used, the p-value must be adjusted accordingly.

**Pre-test: Test for variance
assumption:** A test of the equality of variance is used to test the
assumption of equal variances. The test statistic is F with N_{1}-1
and N_{2}-1 degrees of freedom.

1. If the p-value for this test
is not small (>0.05), use the standard t-test.

2. If the p-value for this test
is small, the t-test for unequal variances (Welch's test) should be used
instead of the standard t-test.

**Results of the t-test: ** If the p-value associated with the t-test is small (< 0.05), there is
evidence to reject the null hypothesis in favor of the alternative. In other
words, there is evidence that the means are significantly different at the
significance level reported by the p-value. If the p-value associated with
the t-test is not small (> 0.05), there is not enough evidence to reject the
null hypothesis, and you conclude that there is evidence that the means are
not different.

**Graphical comparison: **
The graphical comparison allows you to visually see the distribution of the
two groups. If the p-value is low, chances are there will be little overlap
between the two distributions. If the p-value is not low, there will be a
fair amount of overlap between the two groups. There are a number of options
available in the comparison graph to allow you to examine the two groups.
These include box plots, means, medians, and error bars.

**Location in WINKS: **The
independent group t-test is located in the Analyze/t-test and ANOVA menu.

**See Also:** When data are
not normally distributed, The Mann-Whitney U test, a non-parametric test
between groups, can be used. This test is available as an option in the
Nonparametric Comparisons menu.

Example: Independent group
t-test

The FERTLIZ.DBF data on disk
contains information on two types of fertilizer, designated as 1 and 2 in
the database. The t-test results for this data are:

---------------------------------------------------------------------------

Independent Group Analysis C:\KSWIN\FERTILIZ.DBF

---------------------------------------------------------------------------

Grouping variable is GROUP

Analysis variable is OBS

Group Means and Standard Deviations

-----------------------------------

1: mean = 51.4571 s.d. = 4.7476 n = 7

2: mean = 54.9667 s.d. = 4.7944 n = 6

Mean Difference = -3.50952 Pooled s.d. = 2.65538

Test for Equality of Variance

-----------------------------

This preliminary test determines which version of the t-test to perform.

Test equality of variance: F = 1.02 with (5, 6) D.F. p = 0.961 (two-tail)

Note: Since the p-value for equality of variance is greater than 0.05,

use the Equal variance t-test results.

Independent Group t-test Hypotheses

------------------------------------

Ho: There is no difference between means.

Ha: The means are different.

Independent Group t-test on OBS

---------------------------------------------------------------------------

Equal variance: Calculated t= -1.32 with 11 D.F. p = 0.213 (two-tail)

Unequal variance: Calculated t= -1.32 with 10.7 D.F. p = 0.214 (two-tail)

(For a one-sided test, you must adjust the p-value according to

the direction of your alternative hypothesis.)

Confidence Interval

-------------------

A 95% Confidence Interval about the mean difference is: ( -9.3541 to 2.335)

Based on a standard error of 2.6554 and a 0.05% t-statistic of 2.201 with 11
d.f.

On this output, you first
examine the pre-test for the equality of variance. In this case the p-value
is large (0.961). Therefore, you will use the equal variance t-test.

The equal variances t-test has
a p-value of 0.213. Since this is large, you do not have evidence to
conclude that the mean growth produced by one fertilizer is different than
the other.

The graph of this data
clarifies why no significance was found. There is a large amount of overlap
between the two groups, as seen in the box and whiskers plot to the right

**Exercise - Independe**nt
Group t-test

Professor Testum wondered if
students tended to make better scores on his test depending if the test were
taken in the morning or afternoon. From a group of 19 similarly talented
students, he randomly selected some to take a test in the morning and some
to take it in the afternoon. The scores by groups were: