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: u₁ = u₂ (means of the two groups are equal)

H_a: u₁ u₂ (means of the two group are not equal)

The test statistic for is t, with N₁ + N₂ - 2 degrees of freedom, where N₁ and N₂ 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: u₁ < u₂ (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 N₁-1 and N₂-1 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.

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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

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Exercise - Independent 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:

Morning	Afternoon
89.8	87.3
90.2	87.6
98.1	87.3
91.2	91.8
88.9	86.4
90.3	86.4
99.2	93.1
94.0	89.2
88.7	90.1
83.9

1. Perform an independent group t-test on this data. Remember that the program expect two fields, a Group field and an Observation field. What does the test of equality of variance tell you?

2. What version of the t-test did you use? What was the result?

3. From this evidence, does it appear that time of day makes a difference in performance on a test?

4. Display a graphical comparison of this data.

 

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