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Errors in Hypothesis Testing. 2 TYPES OF ERRORS. TRUE CASE H A is true H A is false WE Accept H A SAY Do not Accept H A. TYPE I ERROR. CORRECT. PROB = α. TYPE II ERROR. CORRECT. PROB = β. α is set by the decision maker. β varies and depends on:

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errors in hypothesis testing
Errors in

Hypothesis Testing

2 types of errors
2 TYPES OF ERRORS

TRUE CASE

HAis true HAis false

WE Accept HA

SAY Do not Accept HA

TYPE I

ERROR

CORRECT

PROB = α

TYPE II

ERROR

CORRECT

PROB = β

α is set by the decision maker

β varies and depends on:

(1) α; (2) n; (3) the true value of 

relationship between and
Relationship Between  and 
  •  is the Probability of making a Type II error
    • i.e. the probability of not concluding HA is true when it is
  •  depends on the true value of 
  • The closer the true value of  is to its hypothesized value, the more likely we are of not concluding that HA is true -- i.e.  is large (closer to 1)
  •  is calculated BEFOREa sample is taken
    • We do not use the results of a sample to calculate 
calculating

The Hypothesis Test

CALCULATING 
  • Example: If we take a sample of n = 49, with  = 4.2, “What is the probability we will get a sample from which we would not conclude  > 25 when  really = 25.5?” (Use  = .05)

REWRITE REJECTION REGION IN TERMS OF

calculating cont d

That’s a

TYPE II ERROR!!

P(Making this error) = 

CALCULATING  (cont’d)
  • So when  = 25.5,
    • If we get an > 25.987, we will correctly conclude that  > 25
    • If we get an < 25.987 we will not conclude that  > 25 even though  really = 25.5
calculating cont d1

β

CALCULATING  (cont’d)
  • So what is P(not getting an > 25.987 when  really = 25.5? That is P(getting an < 25.987)?

Calculate z = (25.987 - 25.5)/(4.2/ )  .81

  •  is the area to theleftof .81 for a “>” test
  • P(Z < .81) = .7910
test determining when 25 5

DO NOT

ACCEPT HA

WRONG

Prob = =.7910

ACCEPT HA

RIGHT!

“>” TestDetermining  When  = 25.5

.7910

25.5 25.987

0 .81 Z

what is when 27

β

What is  When  = 27?
  • So what is P(not getting an > 25.987 when  really = 27? That is P(getting an < 25.987)?

Calculate z = (25.987 - 27)/(4.2/ )  -1.69

  •  is the area to theleftof -1.69 for a “>” test
  • P(Z < -1.69) = .0455

This shows that the further the true value of  is from the

hypothesized value of , the smallerthe value of β; that is we

are less likely to NOT conclude that HA is true (and it is!)

test determining when 27

DO NOT

ACCEPT HA

WRONG

Prob = =.0455

ACCEPT HA

RIGHT!

“>” TestDetermining  When  = 27

.0455

25.987 27

-1.69 0 Z

for tests

The Hypothesis Test

 for “<” Tests
  • For n = 49,  = 4.2, “What is the probability of not concluding that  < 27, when  really is 25.5? (With  = .05)
  • This time  is the area to the right of
what is when 25 5

β

What is  When  = 25.5?
  • So what is P(not getting an < 26.013 when  really = 25.5? That is P(getting an > 26.013)?

Calculate z = (26.013 – 25.5)/(4.2/ )  .86

  •  is the area to therightof .86 for a “<” test
  • P(Z > .86) = 1 - .8051 = .1949
test determining when 25 51

DO NOT

ACCEPT HA

WRONG

Prob = =.1949

ACCEPT HA

RIGHT!

.1949

“<” TESTDetermining  When  = 25.5

.8051

25.5 26.013

0 .86 Z

for tests1

The Hypothesis Test

 for “” Tests
  • For n = 49,  = 4.2, “What is the probability of not concluding that   26, when  really is 25.5? (With  = .05)
  • This time  is the area in the middle between the two critical values of
what is when 25 51

β

What is  When  = 25.5?
  • So what is P(not getting an < 24.824 or > 27.176 when  really = 25.5? That is P(24.824 < < 27.176)?

Calculate z’s = (24.824 – 25.5)/(4.2/ )  -1.13

and = (27.176 – 25.5)/(4.2/ )  2.79

  •  is the areain between -1.13 and 2.79 for a “” test
  • P(Z < 2.79) = .9974
  • P(Z < -1.13) = .1292

P(-1.13 < Z < 2.79 = .9974 - .1292 = .8682

test determining when 25 52

DO NOT

ACCEPT HA

WRONG

Prob = =.9974 –

.1292 =.8682

ACCEPT HA

RIGHT!

.8682

.9974

.1292

“” TESTDetermining  When  = 25.5

24.824 25.5 27.176

-1.13 0 2.79 Z

the power of a test 1
The Power of a Test = 1 - 
  •  is the Probability of making a Type II error
    • i.e. the probability of not concluding HA is true when it is
  •  depends on the true value of  and sample size, n
  • The Power of the test for a particular value of  is defined to be the probability of concluding HA is true when it is -- i.e. 1 - 
power curve characteristics
Power Curve Characteristics
  • The power increases with:
    • Sample Size, n
    • The distance the true value of μ is from the hypothesized value of μ
calculating using excel tests
Calculating  Using Excel“> Tests”

Suppose H0 is  = 25;  = 4.2, n = 49,  = .05

“>” TESTS: HA:  > 25 and we want  when the true value of  = 25.5

1) Calculate the criticalx-bar value

= 25 + NORMSINV(.95)*(4.2/SQRT(49))

2) Calculate z =(criticalx-bar -25.5)/ (4.2/SQRT(49))

3) Calculate the the probability of getting a z- value < than this critical z value: -- this is =NORMSDIST(z)

calculating using excel tests1
Calculating  Using Excel“< Tests”

Suppose H0 is  = 27;  = 4.2, n = 49,  = .05

“< TESTS”: HA:  < 27 and we want  when the true value of  = 25.5

1) Calculate the critical x-bar value

= 27 - NORMSINV(.95)*(4.2/SQRT(49))

2) Calculate z =(criticalx-bar -25.5)/ (4.2/SQRT(49))

3) Calculate the the probability of getting a z- value > than the critical value: -- this is 

=1-NORMSDIST(z)

calculating using excel tests2
Calculating  Using Excel“ Tests”

Suppose H0 is  = 26;  = 4.2, n = 49,  = .05

 TESTS: HA:   26 and we want  when the true value of  = 25.5

1) Calculate the critical upperx-barU value and the lower criticalx-barL value

= 26 - NORMSINV(.975)*(4.2/SQRT(49)) (x-barL)

= 26 + NORMSINV(.975)*(4.2/SQRT(49)) (x-barU)

2) Calculate zU=(x-barU-25.5)/ (4.2/SQRT(49)) and zL=(x-barL-25.5)/ (4.2/SQRT(49))

3) Calculate the the probability of getting an z- value in between zL and zU - this is =NORMSDIST(zU) - NORMSDIST(zL)

slide22

=B3+NORMSINV(1-B2)*(B5/SQRT(B6))

=(B8-B7)/(B5/SQRT(B6))

=NORMSDIST(B9)

=1-B10

β for “>” Tests

slide23

=B3-NORMSINV(1-B2)*(B5/SQRT(B6))

=(B8-B7)/(B5/SQRT(B6))

=1-NORMSDIST(B9)

=1-B10

β for “<” Tests

slide24

=B3-NORMSINV(1-B2/2)*(B5/SQRT(B6))

=B3+NORMSINV(1-B2/2)*(B5/SQRT(B6))

=(B8-B7)/(B5/SQRT(B6))

=(B9-B7)/(B5/SQRT(B6))

=NORMSDIST(B11)-NORMSDIST(B10)

=1-B12

β for “” Tests

review
REVIEW
  • Type I and Type II Errors
  •  = Prob (Type I error)
  •  = Prob (Type II error) -- depends on , n and α
  • How to calculate  for:
    • “>” Tests
    • “<” Tests
    • “” Tests
  • Power of a Test at  = 1- 
  • How to calculate  using EXCEL