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Lecture

lec4_


Department
Economics for Management Studies
Course Code
MGEB12H3
Professor
Vinh Quan

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Department of Management, UTSC
ECMB12 Quantitative Methods in Economics II - Lecture 04
Chapter 9 - Probability of making a type II error
Consider testing of mean µ with population variance known (z-test) eg.
Ho: µ = µ0
Ha: µ µ0
= test significance level = P[type I error] = p[reject Ho | Ho is true]
= P[type II error] = p[fail to reject Ho | Ho is false]
Power of test = 1- = p[reject Ho | Ho is false]
Can calculate for any z-tests. How to calculate ? 
Example 1
Given = 12, sample size n = 36, = 0.05 and hypothesis test
Ho: µ 120
Ha: µ < 120
Reject Ho if Z0 - Z
If true mean µa = 112 and not 120 what is prob. of making a type II error?
Test Statistic
n
x
Z
/
0
0
σ
µ
=
=
36/12
120
x
Z = Z0.05 = 1.645 so -Z = -Z0.05 = -1.645
Reject Ho if Z0 - Z
36/12
120
x
-1.645 or
x
116.71
Therefore fail to reject Ho or commit type II error if
x
> 116.71
= P[
x
> 116.71] = P[
x
116.71] due to
x
is continuous RV
P[
xx
xExEx
σσ
][71.116][
] = P[
36/12
11271.116
/
n
x
a
σ
µ
] = P[z 2.36] = P[z -2.36] = 0.0091
1
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Type II error – choice of sample size n for testing of mean, variance known
If we want a specific value of , what should sample size be?
Hypothesis mean = µ0, true mean = µa, unknown
Ho: µ µ0, Ha: µ < µ0
Or
Ho: µ µ0, Ha: µ > µ0
given , find Z where P[Z Z ] =
given find Z where P[Z Z ] =
sample size
2
0
22
)(
)(
a
ZZ
n
µµ
σ
βα
+
=
Ho: µ = µ0, Ha: µ µ0
given , find Z/2 where P[Z Z /2 ] = /2
given find Z where P[Z Z ] =
sample size
2
0
22
2/
)(
)(
a
ZZ
n
µµ
σ
βα
+
=
Example
Given = 12, n = 36, = 0.05 and hypothesis test
Ho: µ 120
Ha: µ < 120
If we prob. of making a type II error when the true mean is 115 to be 0.1, what is the sample size required?
= 0.05, find P[Z Z0.05 ] = 0.05 P[Z Z0.05 ] = 0.95 get Z0.05 = 1.645
= 0.1, find P[Z Z0.1 ] = 0.1 P[Z Z0.1 ] = 0.9 get Z0.1 = 1.28
sample size
2
0
22
)(
)(
a
ZZ
n
µµ
σ
βα
+
=
=
2
22
)115120(
12)28.1645.1(
+
= 49.3 = 50 (always round up)
Alternate Method for Sample Size
From previous example, for sample size n: Reject Ho if Z0 - Z
-1.645 or
nx/74.19120
Therefore fail to reject Ho or commit type II error if
nx/74.19120
>
2
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