testing two related means

7/21/2019 Testing Two Related Means

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Hypothesis Testing :Two Related Means


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Comparing Two Related Means

Before-after study :

blood pressure of a patient before/after intake of a drug performance of staff before/after receiving training

Name Before AfterA

B

C

D

E

F

210

185

215

198

187

225

196

192

204

193

181

233


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Repeated measure study :

blood pressure of a patient measured several times in a day

Name BloodPressure (am)

BloodPressure (pm)

A

B

C

D

:

Y

Z

135

136

121

121

:

116

117

126

129

124

119

:

123

120


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Experiment with natural pairing :

comparing the heights of husband and wife comparing the IQ of twins

Pair Husband Wife

1

2

3

45

6

2.1

1.8

1.5

1.61.8

1.7

1.9

1.9

1.4

1.31.8

2.3


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Notations & Assumptions

Pair 1st sample 2nd sample Difference

1

2

:

n

X11X12:

X1n

X21X22:

X2n

D1 = X11 X21D2 = X12 X22

:

Dn = X1n X2n

The two random samples come from normal populations

The differences form a random sample from a

normal population with mean and varianceD

2

D


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n n

i 1i 2i

i 1 i 11 2

D (X X )

D X Xn n

Hypotheses :

H0: D H1: D (upper-tail test)

H0: D =

H1: D

(two-tail test)

D is estimated by D where


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Paired Z Test

Used when :

populations normal

D

known

Test statistic :

D

DZn


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Paired t Test

Used when :

populations normal

D

unknown

Test statistic :

where

which has a t distribution with (n-1) df

D

D

tS n

2

1 ( )

1

n

ii

D

D D

Sn


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User Old System (1) New System (2) DifferenceC.B. 9.98 s 9.88 s 0.10

T.F. 9.88 9.86 0.02M.H. 9.84 9.75 0.09

R.K. 9.99 9.80 0.19

M.O. 9.94 9.87 0.07

D.S. 9.84 9.84 0.00

S.S. 9.86 9.87 - 0.01C.T. 10.12 9.98 0.14

K.T. 9.90 9.83 0.07

S.Z. 9.91 9.86 0.05

Paired t Test : Example

Is the new computer system faster at the 0.05 level ? You collect

the processing times of 10 jobs, assumed coming from normalpopulations :


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Step 1: Define hypotheses

(new system is not faster)

(new system is faster)

(continued)

0

1

: 0

: > 0

D

D

H

H


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Step 2: Determine the rejection region

(continued)

- Significance level is 0.05

- Critical Value:

- Reject H0 if

8331.11-01,05.01, tt n

8331.1t

8331.1110,05.0 t

RejectionRegion


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Step 3: Compute test statistic

Test Statistic


2

0.072

0.062151

i

i

D

DD

n

D DS

n

(continued)

0 0.072 03.66

0.06215 10D

Dt

S n


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Paired t Test : Example (continued)

Step 4: Make decision Since t = 3.66 is in

the rejection region,

we reject the nullhypothesis.

There is sufficientevidence that the

new system is fasterthan the existing

system

66.3t

Rejectionregion

8331.1110,05.0 t-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

Test Statistic 3.66 is in the Rejection Region


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Paired tTest Using EXCEL


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Paired tTest Using EXCEL


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Paired tTest Using SPSS


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Relation Between and

22

1

2

1 1 2 2

12 2

1 1 2 2 1 1 2 2

1 1 1

2 2

1 2 12

2 2

1 2 1 2

( )

1

(( ) ( )) =

1( ) ( ) ( )( )

21 1 1

2

2

n

i

D

i

n

i i

i

n n n

i i i i

i i i

D D

S n

X X X X

n

X X X X X X X X

n n n

S S S

S S rS S

1 2, ,DS S S r

testing two related means

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