# University of Central Florida Test Performance of Students Questions

9/21/22Confidence intervals
to test H0 involving π
β’ When π falls outside 95% CI around sample mean
β’ if H0 were true, then the p of observing a β this
large between π₯Μ and π just from sampling
variability is β€ π (.05)
β’ When π falls inside 95% CI around sample mean
β’ if H0 were true, then the p of observing a β this
large between π₯Μ and π just from sampling
variability is > π (.05)
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228
Testing H0
95% CIs do
NOT overlap
β’ 1st test of H0: ____ % Confidence Intervals
β’ π₯ vs π
β’ β=0
β’ 95% CI -> π = 0.05 -> p π = 0.10 -> p π = 0.01 -> p CF
Increasing Complexity/Confidence
3 types of t tests
β’ IV/Predictor
β’ Physical Activity
DV/Outcome
Cognitive Function
Γ
Γ
β’ Time points (dv)?
Test scores
LOM = Continuous
Normally Distributed
π₯ π
Group Assignment
LOM = Binary
233
# samples?
β Cross-sectional,
β Cross-sectional,
β Cross-sectional,
β Pre-post,
β Pre-post,
β Pre-post,
One sample t (π₯ π£ π)
1 group/sample
2 groups (bt subjects) Independent samples t
>2 groups (bt subjects) One way ANOVA
1 group (wn subjects) Dependent samples t
2 groups (bt/wn subjects)RM Factorial ANOVA
>2 groups (bt/wn subjects)RM Factorial ANOVA
β Longitudinal,
β Longitudinal,
β Longitudinal,
RM ANOVA
1 group (wn subjects)
2 groups (bt/wn subjects) RM Factorial ANOVA
>2 groups (bt/wn subjects)RM Factorial ANOVA
234
3 types of t tests
β’ One sample t-test (vs. π)
β’ Comparing one group π₯Μ to fixed value
β’ Must have π to carry out this test
β’ Independent samples t-test
β’ Comparing two group π₯βs
Μ to each other
β’ Dependent samples t-test
β’ Also known as paired samples t-test
β’ Comparing π₯βs
Μ of one group before/after Tx
235
β’ Used π‘! to find Confidence Intervals (CI)
β’ π₯Μ Β± (π‘! * SE)
β’ If π fell outside CI around π₯,Μ then
* π»” < 0.05 β’π β β’ Reject π»" β’ If π fell inside CI around π₯,Μ then * π»" > 0.05
β’π β
β’ Fail to Reject π»”
236
β’ Instead of calculating CIβs and looking for
overlap to test H0
β’ Could Calculate π‘#\$% =
β
‘( β
β’ Then directly compare π‘#\$% to π‘!
t=0
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9/21/22
POPULATION (π,π)
Sampling
Variability
Sampling Distribution
Theoretical Distribution of π₯
π₯! (= π); SE
texp vs tπΌ
tπΌ=1.96
(p=0.05)
Samples
π‘012 < π‘+ Sample (π₯,s,n) t=0 Sample (π₯,s,n) Sample (π₯,s,n) Sample (π₯,s,n) Sample (π₯,s,n) Sample (π₯,s,n) Sample (π₯,s,n) Sample (π₯,s,n) Sample (π₯,s,n) 239 π‘012 > π‘+,-.-/
If π‘012 β₯ π‘+,-.-/, then π β π»- β€ 0.05
Reject H0
If π‘012 < π‘+,-.-/, then π β π»- > 0.05
Fail to Reject H0
π‘+,-.-/ β₯ 1.96
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π‘/01 23 π‘4
Formula for one sample t
β’ one sample t-test:
π‘\$%& ππ =
Difference between
2 means
π‘=
β
ππΈβ
π»” : π‘ = 0, πππππ’π π β = 0
(Average difference
expected from sampling
variability if H0 were
true)
241
β
βππΈβ ππΈβ
3 t-tests all have same conceptual form:
&
β
π₯Μ β π
=π
ππΈβ
. π
Standard Error
π₯ = sample mean
ΞΌ = population mean
s = sample standard deviation
n = sample size
πππ π πππππ π‘ (π£π  π) =
β
‘(β
β
β
\$*+

, #
245
Formula for (two) independent
samples t-test
Experimental design
π₯Μ A = 1000 (sd=250, n=50)
π₯Μ B = 1300 (sd=250, n=50)
Itβs computationally
burdensome to derive a
sample of differences.
Easier to use the difference
between the means (π₯\$ β π₯% )
to estimate the
mean difference (β)
246
β
ππΈβ
π‘012 =
1000
β’ Independent samples t-test:
1300
π₯. β π₯/ β β
Need π‘&!’ to calculate π β π»(
π‘&!’ =
mean difference of 300
(sd=290)
β
ππΈβ
xbar1: mean of sample 1
xbar2: mean of sample 2
Find way to estimate ππΈβ
247
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9/21/22
Formula for (two) independent
samples t-test
β’ Independent samples t-test:
Formula for two (independent)
samples t-test
β’ when samples sizes are unequal or equal:
Differences
between groups
Pooled Variance
β’ ONLY WORKS WHEN SAMPLE SIZES FROM
BOTH GROUPS ARE EQUAL (i.e., when n1 = n2)
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Experimental design (PA-> CF)
Experimental design
Compare two groups and observe Ξ in SAT scores
2 reasons observe Ξ
β’ H 0:
Mean from control group is 1000 (sd=100)
β’ β=0
β’ β 91 :; cognitive function
2 reasons for observed difference in sample mean
SAT scores (Ξ=300, nTX = 50, nCX = 50)
β’ H 0:
Experimental design
Mean from school1 is 1000 (sd=250, n=50)
Mean from school2 is 1300 (sd=250, n=50)
300
β’ β=0
β’ β 91 :;

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