t-tests
• Quantitative Data
• One group  1-sample t-test
• Two independent groups  2-sample t-test
• Two dependent groups  Matched Pairs t-test
t-Tests
Slide #1
A Full Reality
• No longer know what s is!!!!!
• What should be used instead?
– Our best guess at s  s
• Changes details, not the big picture
t 
x  0
s
n
t-Tests
Slide #2
Student’s t-distribution
Compared to a standard normal (Z):
• Similarities
10521 df
– symmetric about 0
– approximately bell-shaped
• Differences
-4
-2
0
Z or t
– more probability in the tails
– less probability in the center
– Exact shape depends on degrees-of-freedom (df)
2
4
• See HO for R work
t-Tests
Slide #3
1-sample t-test
• H o:  = o
(where o = specific value)
• Statistic: x
• Test Statistic: t 
• Assume:
x  0
s
n
df = n-1
– s is UNknown
– n is large (so that the test stat follows a t-distribution)
• n > 40, OR
• n > 15 and histogram is not strongly skewed, OR
• Histogram is approximately normal
• When: Quantitative variable, one population
sampled, s is UNknown.
t-Tests
Slide #4
A Full Example
• In Health magazine reported (March/April 1990)
that the average saturated fat in one pound
packages of butter was 66%. A food company
wants to determine if its brand significantly differs
from this overall mean. They analyzed a random
sample of 96 one pound packages of its butter.
Test the company’s hypothesis at the 1% level.
Variable n
%SatFat 96
Mean St. Dev.
65.6
1.41
Min
60.2
...
...
t-Tests
Slide #5
Practical Significance
• Is there a real difference between 66% and 65.6%
saturated fat?
• If the sample size is large enough, any hypothesis
can be rejected.
x
t-Tests
Slide #6
R Handout
t-Tests
Slide #7