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Errors in Hypothesis Testing

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

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  1. Errors in Hypothesis Testing

  2. 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 

  3. 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 

  4. 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

  5. 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

  6. 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

  7. DO NOT ACCEPT HA WRONG Prob = =.7910 ACCEPT HA RIGHT! “>” TestDetermining  When  = 25.5 .7910 25.5 25.987 0 .81 Z

  8. 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!)

  9. DO NOT ACCEPT HA WRONG Prob = =.0455 ACCEPT HA RIGHT! “>” TestDetermining  When  = 27 .0455 25.987 27 -1.69 0 Z

  10. 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

  11. 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

  12. DO NOT ACCEPT HA WRONG Prob = =.1949 ACCEPT HA RIGHT! .1949 “<” TESTDetermining  When  = 25.5 .8051 25.5 26.013 0 .86 Z

  13. 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

  14. 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

  15. 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

  16. 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 - 

  17. Power Curve Characteristics • The power increases with: • Sample Size, n • The distance the true value of μ is from the hypothesized value of μ

  18. n = 49 n = 25 α = .05 Power Curves For HA: μ 26With n = 25 and n = 49

  19. 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)

  20. 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)

  21. 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)

  22. =B3+NORMSINV(1-B2)*(B5/SQRT(B6)) =(B8-B7)/(B5/SQRT(B6)) =NORMSDIST(B9) =1-B10 β for “>” Tests

  23. =B3-NORMSINV(1-B2)*(B5/SQRT(B6)) =(B8-B7)/(B5/SQRT(B6)) =1-NORMSDIST(B9) =1-B10 β for “<” Tests

  24. =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

  25. 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

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