Transcript 4-03-08 -- t-distributions and t-test

The t distribution and the
independent sample t-test
I271B
Some Data Considerations for Final Exam:
Two metric variables
Correlation
One binary categorical variable, one metric variable
T-test
Two categorical variables
Chi-Square (crosstabulations)
Polytomous dependent variable, metric IV
ANOVA
Metric dependent variable, multiple IV’s
Linear Regression
How do we test to see if the means
between two sample populations are, in
fact, different?
Null and Alternative Hypotheses
 Null Hypothesis:
 H0: μ1 = μc
 Alternative Hypotheses:
 H1: μ1 < μc
 H2: μ1 > μc
 H3: μ1 ≠ μc
One and Two-Tailed Tests: Defining Critical
Regions
Why use the t distribution?
 Useful for testing mean differences when the N is
not very large
 In very large samples, a t-test is the same as a z-
score test. Why?
The t-test

Where:
M
SDM
N
s
df
= mean
= Standard error of the
difference between means
= number of subjects in group
= Standard Deviation of group
= degrees of freedom
Essentially, the t-value is the difference between group means (mean
difference), divided by the variability of the two groups (standard error
of difference).
Standard Deviations vs. Standard Errors
 Standard deviation is a
measure of spread for a given
measure.
 Standard error of a given
statistic is the standard
deviation of the sampling
distribution of that statistic.
Standard Error
of the mean
Degrees of freedom
 d.f. = the number of independent pieces of information
from the data collected in a study.

Example: Choosing 10 numbers that add up to 100.

This kind of restriction is the same idea: we had 10 choices but
the restriction reduced our independent selections to N-1.

In statistics, further restrictions reduce the degrees of freedom.

In the t-test, since we deal with two means, our degrees of
freedom are reduced by two (df = n1 + n2 – 2)
Z-distribution versus t-distribution
t distribution
 As the degrees of freedom increase (towards infinity), the t
distribution approaches the z distribution (i.e., a normal distribution)
 Because N plays such a prominent role in the calculation of the t-
statistic, note that for very large N’s, the sample standard deviation (s)
begins to closely approximate the population standard deviation (σ)
 http://www.econtools.com/jevons/java/Graphics2D/tDist.html
Assumptions Underlying the Independent Sample t-test
 Assumption of Normality
 Variables are normally distributed
within each group
 Assumption of Homogeneity of
Variance

Variation of scores in the two groups
should not be very different.
What if the Variances are not equal?
 We can test to see if the variances are significantly
different between groups (equality of variance test)
 If they *are* significantly different, we can use a
modified t-test that corrects for this problem
(Welch’s t-test)

Essentially, the Welch’s t-test uses a different estimation of the
degrees of freedom and also avoids using the pooled variance.
Calculating Effect Size
 Note that the SD in the effect size calculation is the pooled
(combined) standard deviation for both groups
 Effect size allows you to compare relative effect of one test to another
(using same dependent variable)
 Conceptually, think of effect size as increasing the probability that
you could guess which group a given person came from.
 Values of .8 or greater are usually large effects, .5 medium, .2 small.
 Effect size is a measure of practical significance, not statistical
significance.
t-tests using STATA
 t-test.do
 auto.dta dataset
 grades.dta dataset