Chapter 11: The t Test for Two Related Samples

Related t Formulas

€

t =DSD

€

= DSDn

€

t =(X1 − X 2 )

S12 + S2

2

n−

2rS1S2

n

€

t =D∑

1n−1

n D2∑ − ( D∑ )2[ ]

Group Assignment of Subjects Matched by IQ

Control Reading Program

Subject IQ Subject IQ

A 120 E 120

B 105 F 105

C 110 G 110

D 95 H 95

Person

Before Treatment

X1

After Treatment

X2 D

A 72 64 -8

B 68 60 -8

C 60 50 -10

D 71 66 -5

E 55 56 +1

€

D∑ = −30

€

D =D∑n

€

=−305

€

=−6

€

t =D−μ DSD

€

s =SSn−1

€

SD =sn

€

df = n−1

(Where n = number of difference scores)

µD = ?D scores

Population of difference scores

Obtain sample

Makeinference about µD

Scores on Depression Inventory Before and After Treatment

Person

Before Treatment

X1

After Treatment

X2 D

A 72 64 -8

B 68 60 -8

C 60 50 -10

D 71 66 -5

E 55 56 +1

€

D = −30∑

€

D =D∑n

€

=−305

€

=−6

Person

Before Treatment

X1

After Treatment

X2 D D2

A 72 64 -8 64

B 68 60 -8 64

C 60 50 -10 100

D 71 66 -5 25

E 55 56 +1 1

-30 254

€

SS = D2 −( D∑ )2

n∑

€

=(254)−(−30)2

5

€

=254 −9005

€

=254 −180 = 74

D D (D - D)2

-8 -6 (-2)2

-8 -6 (-2)2

-10 -6 (-4)2

-5 -6 (+1)2

+1 -6 (+7)2

€

SS = (D−D∑ )2 = 74

€

s =SSn−1

€

= 744

€

= 18.5

€

=4.3

€

SD =sn

€

=4.35

€

= 4.32.236

€

=1.92

€

t =D−μ DSD

€

=(−6)− (0)1.92

€

=−3.125

What is the effect of relaxation training on severity of asthma symptoms:

1. Measure the number of doses of medication needed to counter asthma attacks (over 1 week)

Relaxation Training

2. Measure the number of doses of medication needed to counter asthma attacks (over 1 week)

1. Ho: µD = 0 (No change in symptoms)

H1: µD ≠ 0 (There is a change…)

= 0.05

2. tcrit(4) = + 2.776

3.

€

tobt =D−μ DSD

€

SD =sn

€

s =SSn−1

€

SS = D2∑ −(∑D)2

n

€

tobt = −3.72

4. Reject Ho because tobt of -3.72 < tcrit of -2.776

5. Conclusion: Relaxation training significantly reduced the number of asthma attacks, t(5) = -3.72, p < 0.05.

-2.776 0 +2.776

Reject Ho

Reject Ho

PatientWeek Before

TrainingWeek After

Training D D2

A 9 4 -5 25

B 4 1 -3 9

C 5 5 0 0

D 4 0 -4 16

E 5 1 -4 16

€

D∑ = −16

€

D2 = 66∑

€

D =D∑n

€

=−165

€

=−3.2

€

SS = D2∑ −( D∑ )2

n

€

=66 −(−16)2

5

€

=66 − 51.2

€

=14.8

-2.132

Reject Ho

df = 4

ControlReading Program D D2

Matched pair A 6 15 +9 81

Matched pair B 5 15 +10 100

Matched pair C 11 17 +6 36

Matched pair D 6 13 +7 49

€

D∑ = +32

€

D2 = 266∑

€

D =D∑n

€

=+324

€

=+8

€

SS = D2∑ −( D∑ )2

n

€

=266 −(32)2

4

€

=266 − 256

€

=10

€

=266 −1024

4

Subject Treatment 1 Treatment 2 D

A 10 16 +6

B 20 24 +4

C 30 35 +5

D 40 46 +6

E 50 54 +4

€

D = +5

€

X1 = 30

€

X 2 = 35

€

SS1 =1000

€

SS2 = 964

€

SSD = 964

Assumptions for Related-Samples t test

1. Observations within each treatment must be independent

2. The population distribution of difference scores must be normal

3. Note: #2 is not a concern as long as sample size is 30 or less

Person X1 (Before) X2 (After)

A 15 15

B 11 13

C 10 18

D 11 12

E 14 16

F 10 10

G 11 19

SubjectBefore

TreatmentAfter

Treatment

1 19 13

2 35 37

3 20 14

4 31 25

Control Reading Program

Subject IQ Subject IQ

A 120 E 120

B 105 F 105

C 110 G 110

D 95 H 95