# 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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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₯βs

Μ
to each other

β’ Dependent samples t-test

β’ Also known as paired samples t-test

β’ Comparing π₯βs

Μ
of one group before/after Tx

235

All about t

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

All about t

β’ Instead of calculating CIβs 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 tπΌ

tπΌ=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)
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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

240

π‘/01 23 π‘4

Formula for one sample t

β’ one sample t-test:

π‘$%& ππ =

Difference between

2 means

π‘=

β

ππΈβ

π»” : π‘ = 0, πππππ’π π β = 0

(Average difference

expected from sampling

variability if H0 were

true)

241

β

βππΈβ ππΈβ

3 t-tests all have same conceptual form:

&

β

π₯Μ
β π

=π

ππΈβ

. π

Standard Error

π₯ = sample mean

ΞΌ = population mean

s = sample standard deviation

n = sample size

πππ π πππππ π‘ (π£π π) =

β

‘(β

β

β

$*+

”

, #

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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)

Itβs 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

ππΈβ

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

ππΈβ

xbar1: mean of sample 1

xbar2: mean of sample 2

Find way to estimate ππΈβ

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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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249

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 :;