Department

Economics for Management StudiesCourse Code

MGEB12H3Professor

Vinh QuanThis

**preview**shows pages 1-2. to view the full**6 pages of the document.**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

n

x

/12

120

−

≤ -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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