Chapter 11: The t Test for Two Related Samples. Related t Formulas.

Chapter 11: The t Test for Two Related Samples

Related t Formulas

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t = DSD

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= DSDn

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t = (X1 − X 2 )S1

2 + S22

n− 2rS1S2

n

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t = D∑1n−1

n D2∑ − ( D∑ )2[ ]

Group Assignment of Subjects Matched by IQ

Control Reading ProgramSubject IQ Subject IQ

A 120 E 120B 105 F 105C 110 G 110D 95 H 95

Person

Before Treatment

X1

After Treatment

X2 DA 72 64 -8B 68 60 -8C 60 50 -10D 71 66 -5E 55 56 +1

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D∑ = −30

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D = D∑n

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=−305

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=−6

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t = D−μDSD

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s = SSn−1

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SD = sn

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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 DA 72 64 -8B 68 60 -8C 60 50 -10D 71 66 -5E 55 56 +1

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D = −30∑

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D = D∑n

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=−305

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=−6

Person

Before Treatment

X1

After Treatment

X2 D D2

A 72 64 -8 64B 68 60 -8 64C 60 50 -10 100D 71 66 -5 25E 55 56 +1 1

-30 254

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SS = D2 − ( D∑ )2

n∑

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=(254)− (−30)2

5

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=254 − 9005

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=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

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SS = (D−D∑ )2 = 74

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s = SSn−1

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= 744

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= 18.5

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=4.3

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SD = sn

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=4.35

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= 4.32.236

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=1.92

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t =D−μ DSD

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=(−6)− (0)1.92

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=−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.

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tobt = D−μ DSD

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SD = sn

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s = SSn−1

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SS = D2∑ − (∑D)2

n

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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 25B 4 1 -3 9C 5 5 0 0D 4 0 -4 16E 5 1 -4 16

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D∑ = −16

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D2 = 66∑

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D = D∑n

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=−165

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=−3.2

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SS = D2∑ − ( D∑ )2

n

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=66 − (−16)2

5

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=66 − 51.2

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=14.8

-2.132

Reject Ho

df = 4

ControlReading Program D D2

Matched pair A 6 15 +9 81Matched pair B 5 15 +10 100Matched pair C 11 17 +6 36Matched pair D 6 13 +7 49

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D∑ = +32

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D2 = 266∑

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D = D∑n

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=+324

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=+8

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SS = D2∑ − ( D∑ )2

n

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=266 − (32)2

4

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=266 − 256

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=10

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=266 −10244

Subject Treatment 1 Treatment 2 DA 10 16 +6B 20 24 +4C 30 35 +5D 40 46 +6E 50 54 +4

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D = +5

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X1 = 30

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X 2 = 35

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SS1 =1000

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SS2 = 964

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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 15B 11 13C 10 18D 11 12E 14 16F 10 10G 11 19

SubjectBefore

TreatmentAfter

Treatment1 19 132 35 373 20 144 31 25

Control Reading ProgramSubject IQ Subject IQ

A 120 E 120B 105 F 105C 110 G 110D 95 H 95