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Chapter 8 MATH 4753
By the end of this lesson you should know: •What hypothesis testing is. •What type 1 and 2 errors are. •What power is.
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Development of Hypothesis testing (R script)
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Test errors
•Type 1 error happens when you reject 𝑯𝒐 when it is true. –Probability of type 1 error =𝜶 •Type 2 error happens when you accept 𝑯𝒐 when it is false. –Probability of type 2 error=𝜷
The Courtroom
Power •Probability of rejecting 𝑯𝟎 when it is false is 1 − 𝛽 •This is called the power of a test. •Power is not the probability of an error! •Power=P(Rejecting 𝑯𝒐 |𝑯𝒐 is False)=1-P(Accepting 𝑯𝒐 |𝑯𝒐 is False)
What is the importance of power? •The higher the value of the power the more we can detect departures from 𝐻𝑜 .
Cut off and Acceptance, Rejection regions
Investigating POWER
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H1
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xcut= 12
xcut= 14
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H1
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Density
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Density
H0
xcut= 13
Density
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Density
xcut= 11
𝐻0 : 𝜇 = 10 𝐻1 : 𝜇 = 15
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Can you dig it?
•A) Yes •B) No
Environmental Protection Agency
Test
• 𝐻0 : 𝜇 = 3 • 𝐻1 : 𝜇 > 3
Power calculation
Hypothesis Testing: Errors
Use R
Population of Wisconsin lakes has 𝜇 = 15
qt(1-0.1/2, 25-1) [1] 1.710882
Probability the test will detect a mean that differs from 15 gm/m^3 if 𝜇𝑎 = 14
Taken from Montgomery and Runger App. Stat pg 311 Fourth edition
See BBD for Bayesian Example
𝑃 𝑥 =
𝑥
𝛼−1
1−𝑥 𝐵 𝛼, 𝛽
𝛽−1