EDUC 200C
Two sample t-tests
November 9, 2010
Review: What are the following?
•
•
•
•
•
Sampling Distribution
Standard Error of the Mean
Central Limit Theorem
Confidence Interval
Standard Error of the difference between the
means
– Pooled variance
20 samples randomly drawn from a population with mean = 0 and sd = 2
-4
-2
0
Sample means
2
4
-4
-2
N=5
-4
-2
0
Sample means
N = 20
0
Sample means
2
4
2
4
N = 10
2
4
-4
-2
0
Sample means
N = 50
.5
1
Density
.4
0
0
.2
Density
.6
1.5
.8
2
As the number of samples increases, the distribution of sample means approaches
a normal distribution with a mean equal to the population mean
-.5
0
mn
.5
-.6
-.4
0
.2
.4
20 Samples
Density
.6
.5
.4
0
.2
0
Density
1
.8
1
1.5
10 Samples
-.2
mn
-1.5
-1
-.5
0
mn
50 Samples
.5
1
-1
-.5
0
mn
100 Samples
.5
1
Hypothesis testing cheat sheet
Number
of
samples
Null
hypothesis
example
Population
standard
deviation
known?
Variance of test sample
One
H0: μ=0
Yes – σ
σ2
𝜎𝑋 =
One
H0: μ=0
No – s
s2
𝑠𝑋 =
Two
H0: μ1- μ2=0
No – s1, s2
2
𝑠𝑝𝑜𝑜𝑙𝑒𝑑
=
2
𝑁1 − 1 𝑠1 + 𝑁2 − 1 𝑠22
𝑁1 + 𝑁2 − 2
Standard error of the
mean
𝜎
𝑁
𝑠
𝑁
𝑠𝑋1 −𝑋2 =
2
𝑠𝑝𝑜𝑜𝑙𝑒𝑑
1
1
+
𝑁1 𝑁2
Test statistic
Confidence interval
𝑧=
𝑋−𝜇
𝜎𝑋
𝑋 ± 𝑧𝛼 𝑠𝑋
𝑡=
𝑋−𝜇
𝑠𝑋
𝑋 ± 𝑡𝛼 𝑠𝑋
𝑡=
𝑋1 − 𝑋2 − (𝜇1 − 𝜇2 ) (𝑋1 − 𝑋2 ) ± 𝑡𝛼 𝑠𝑋1 −𝑋2
𝑠𝑋1 −𝑋2
Practice Problem 1
• A high school social studies teacher decides to conduct
action research in her classroom by investigating the
effects of immediate testing on memory. She randomly
divides her class into two groups. Group 1 studies a
short essay for 20 minutes, while group 2 studies the
essay for 20 minutes and then immediately takes a 10minutes test on the essay. The results below are from a
final exam on the essay, taken one month later:
Group 1 (studied only)
Group 2 (Studied and tested)
– Set up the appropriate statistical hypotheses
– Perform the test (α=0.05)
– Draw final conclusions.
Independent sample t-tests
• So far, we’ve talked about and calculated the
difference in the mean of a certain variable between
two independent populations.
– Here, “independent” tells us that there was no
connection between the two groups we were comparing
• Our test statistic took the form
• This allowed us to make claims about how the
mean of one population compared to the mean of
the other.
Matched pairs t-tests
• Used when the observations are not independent
– Same person measured twice
– Relevant connection between observation (e.g. parents
and children)
– Observations deliberately matched on particular
attributes (e.g. intelligence)
• Tells whether the mean difference within
observations is significantly different from the null
hypothesis value (usually 0).
Matched pairs t-statistic
• Based on how we’ve constructed our tests of
significance so far, how would we go about
testing a matched pairs hypothesis?
Matched pairs t-statistic
Stata
• Performing a matched pairs t-test is the same as
performing and independent samples t-test, just
leave off the “unpaired” option
ttest var1==var2
Questions??