Download Lecture 15: Levin & Fox ch 7

Independent Sample T-test
• Often used with experimental designs
• N subjects are randomly assigned to two groups
(Control * Treatment).
• After treatment, the individuals are measured on
the dependent variable.
• A test of differences in means between groups
provides evidence for the treatment's effect.
Measures of Variation
• A lot of statistical techniques (using interval
data) use measures of variation in some manner
• What is the difference between a standard
deviation, the standard error of the mean, and
the standard error of the difference between
means? Or How are they related?
Using Measures of Variation
• Leaned how to measure variation in data,
i.e., variance, standard deviation (Ch.4)
• Used the normal curve & standard
deviation to calculate z-scores and
probabilities (Ch.5)
• Used the normal curve & the z-score & the
Standard error of the mean to calculate
confidence intervals (Ch.6)
• Used the concept of the confidence
interval and the standard error of the
differences between means to calculate
the t-test (Ch.7)
Standard Error of the Differences
between Means
• Similar to the idea behind the SE of the
mean
• Lets say that in the population men and
women IQ scores are (on average) equal.
• If we took a 1000 pairs of sample means
for men and women, calculated the
difference between those means and
plotted those 1000 differences, the plot
would look like a normal curve.
•
•
•
•
Some differences will be at or near zero
Some will be a little below or above zero
A few will be noticeably different from zero
Even though the true population difference
between men and women IQs is zero,
because of sampling error, we will get
differences that are above or below zero.
• What if we don’t know the true population
difference?
• Create confidence intervals to estimate
what the true population difference is
Null Hypothesis
• The two groups come from the same
population or that the two means are equal
• μ 1 = μ2
Levels of Significance
• What does an α = .05 level of significance
mean?
• We decide to reject the null if the
probability is very small (5% or less) that
the sample difference is a product of
sampling error.
• The observed difference is outside the
95% confidence interval of the difference
Choosing a Level of Significance
• Convention
• Minimize type I error – Reject null
hypothesis when the null is true
• Minimize type II error – fail to reject null
when the null is false
• Making alpha smaller reduces the
likelihood of making a type I error
• Making alpha larger reduces the
probability of a type II error
Assumptions of the t-test
• 1. All observations must be independent of each other
(random sample should do this)
• 2. The dependent variable must be measured on an interval
or ratio scale
• 3. The dependent variable must be normally distributed in
the population (for each group being compared).
(NORMALITY ASSUMPTION) [this usually occurs when N is
large and randomly selected]
• 4. The distribution of the dependent variable for one of the
groups being compared must have the same variance as the
distribution for the other group being compared.
(HOMOGENEITY OF VARIANCE ASSUMPTION)
Independent Sample T-test
Formula
t=
X1  X 2
s X1  X 2
 N1s1  N 2 s2
 
 N1  N 2  2
2
s x1  x2
2
 N1  N 2 


 N N 
 1 2 
Let’s play with some fake data
• Go to our webpage
• http://faculty.unlv.edu/kfernandez/methods1.htm
• Open the IQ SPSS file
SPSS & the Independent Sample T-Test
Group Statistics
VAR00001
VAR00002
1.00
2.00
N
10
10
Mean
102.0000
98.0000
Std. Error
Mean
10.03328
9.63789
Std. Deviation
31.72801
30.47768
Independent Samples Test
Levene's Test for
Equality of Variances
F
VAR00001
Equal variances
ass umed
Equal variances
not as sumed
.073
Sig.
.789
t-tes t for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
.288
18
.777
4.00000
13.91242 -25.22892
33.22892
.288
17.971
.777
4.00000
13.91242 -25.23230
33.23230
Now a hand calculation with
more fake data
t=
X1  X 2
s X1  X 2
 N1s1  N 2 s2
 
 N1  N 2  2
2
s x1  x2
2
 N1  N 2 


 N N 
 1 2 
Real world example
• Matland (1994) – available on WebCampus
• 528 high school students in Norway
• Students given documents (speeches) from
the Conservative and Labor party. 50% of
the speeches were associated with a male
name the other a female name.
• Hypothesis: Norway has so many female
politicians that students should evaluate
speeches/candidates equally