anthony greene1 two sample t-test: hypothesis of differences between two groups 1.is group “a”...
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Anthony Greene 1
Two Sample t-test: Hypothesis of Differences
Between Two Groups1. Is Group “A” Different Than Group “B”?
2. Does an Experimental Manipulation Have an Effect?
• Is an experimental group different than a control group?
• If so, then the experimental manipulation had an effect
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Anthony Greene 2
Use of the Two Independent Sample t-test
This is the most universally used inferential statistic
Why?
• Population parameters μ and σ are almost never known
• Most experiments require a comparison, and that requires at least two groups
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Anthony Greene 3
Significant Differences?
M1= 40 M2=60
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Anthony Greene 4
Significant Differences?
M1= 40 M2=60
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Anthony Greene 5
Where You’ve Been Thus Far
1. Computation of descriptives
2. Probability theory, especially standard normal distributions (z-scores)
3. Hypothesis Testing
• Using z-scores
• Single sample t-test
• Two sample t-test
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Anthony Greene 6
Overview of Procedure
Two independent sample t-testa) 1 and 2 are hypothesized or predicted
(not computed and generally not known): M1 and M2 are computed
b) M1 and M2 are unknown ( is unknown) :
sM1 and sM2 are computed
c) Degrees freedom (df) is computed
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Anthony Greene 7
The Basic Idea• A new distribution is used that is
normally distributed
• This time the parent distribution is
2121 MM
)/()/( 2221
2121
nnMM
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Anthony Greene 8
The Basic Idea• So the sampling distribution is
has the following mean and standard error:
21 MM
2
2
1
2
21 n
s
n
ss pp
MM
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Anthony Greene 9
The Basic Idea
• So the basic t-test has the form:
21
2121
soerror standard
mean edhypothesiz -mean observed
MMs
MMt
t
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Anthony Greene 10
Alternate Forms
)/1()/1(
)()(
)/()/(
)()(
)()(
21p
2121
22
12
2121
)(
2121
21
nns
MM
nsns
MM
MMt
pp
MMs
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Anthony Greene 11
The Basic Idea
21
21
MMs
MMt
Since the usual H0 is that μ1=μ2 OR μ1 - μ2= 0
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Anthony Greene 12
The Basic Idea
What is sp?
It’s the pooled variance and its meant to allow you to make a comparison of means even if the σs aren’t equal
21
212
dfdf
SSSSsp
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Anthony Greene 13
Annual salaries ($1000s) for30 faculty members in public institutions and 35 faculty members in private institutions
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Anthony Greene 14
Process for comparing two population means using independent samples
Compare M1 and M2
Based on the Pooled VarianceMake a Decision
Compute M1
Compute M2
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Anthony Greene 15
Notation for parameters and statistics when considering two populations
M M
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Anthony Greene 16
The t-test for two population means(Slide 1 of 3)
Step 1 The null hypothesis is H0: 1 = 2 or 1 - 2 = 0; the alternative hypothesis is one of the following:
Ha: 1 2 Ha: 1 < 2 Ha: 1 > 2 (Two Tailed) (Left Tailed) (Right Tailed)
Step 2 Decide on the significance level, Step 3 The critical values are
±t/2 -t +t(Two Tailed) (Left Tailed) (Right Tailed)
with df = n1+n2 - 2.
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Anthony Greene 17
The t-test for two population means (Slide 2 of 3)
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Anthony Greene 18
The t-test for two population means(Slide 3 of 3)
Step 4 Compute the value of the test statistic
Where
Step 5 If the value of the test statistic falls in the rejection region, reject H0, otherwise do not reject H0.
)/1()/1(
)()(
21p
2121
nns
MMt
21
21
dfdf
SSSSsp
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Anthony Greene 19
Summary statistics
Public Institution
Private Institution
Sample Mean
57.48 66.39
Sum of Squares
16634 1982
Sample Size 30 35
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Anthony Greene 20
Computing the t-value
18.224.017
91.8
351
30117
39.6648.57
1729563
18616
21
21
t
dfdf
SSSSsp
21
21
MMs
MMt
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Anthony Greene 21
Criterion for deciding whether or not to reject the null hypothesis
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Anthony Greene 22
Criterion for deciding whether or not to reject the null hypothesis
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Anthony Greene 23
Sample Problem:Minutes required to comprehend the self-study manual
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X1 (X-μ)2 X2 (X-μ)2
139 61.73 142 663.06
118 832.73 109 52.56
164 293.88 130 189.06 M1-M2 30.61
151 17.16 107 85.56 Sp22.44
182 1235.02 155 1501.56 sM1-M211.61
140 47.02 88 798.06
134 165.31 95 451.56 t 2.64
104 150.06
M 146.86 116.25
SS 2652.86 3891.50
Sample Problem
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Remember the goal is always to see if the effect is large compared to random variation.
1.In the t-test this is done by the ratio: (diff. in means)/(randomness). For a reasonable sample this ratio must exceed ~1.96.
2.In the graph this is done by comparing the mean difference to the error bars which are each 1 s.e. If the difference is greater than 2 s.e. (~1.96) then the difference is significant.
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WithoutExperience
WithExperience
Sample Problem