Slides Preared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 Chater 9, Part B Hyothesis Tests Poulatio Proortio Hyothesis Testig ad Deisio Makig Calulatig the Probability of Tye II Errors Determiig the Samle Size for Hyothesis Tests About a Poulatio Mea Slide 2 Chater 9, Part B Hyothesis Tests Not Your Average Cosultat! Slide 3 1
A Summary of Forms for Null ad Alterative Hyotheses About a Poulatio Proortio The equality art of the hyotheses always aears i the ull hyothesis. I geeral, a hyothesis test about the value of a oulatio roortio must take oe of the followig three forms (where is the hyothesized value of the oulatio roortio). H : H a: < H H a: : : > H a: Oe-tailed (lower tail) Oe-tailed (uer tail) Two-tailed Slide 4 Tests About a Poulatio Proortio Test Statisti where: z ( 1 ) assumig > 5 ad (1 ) > 5 Slide 5 Tests About a Poulatio Proortio Rejetio Rule: Value Aroah Rejet if value < α Rejetio Rule: Critial Value Aroah : < : > : Rejet if z > z α Rejet if z < -z α Rejet if z < -z α/2 or z > z α/2 Slide 6 2
Two-Tailed Test About a Poulatio Proortio Eamle: Natioal Safety Couil For a Christmas ad New Year s week, the Natioal Safety Couil estimated that 5 eole would be killed ad 25, ijured o the atio s roads. The NSC laimed that 5% of the aidets would be aused by druk drivig. Slide 7 Two-Tailed Test About a Poulatio Proortio Eamle: Natioal Safety Couil A samle of 12 aidets showed that 67 were aused by druk drivig. Use these data to test the NSC s laim with α.5. Slide 8 Two-Tailed Test About a Poulatio Proortio Value ad Critial Value Aroahes 1. Determie the hyotheses. 2. Seify the level of sigifiae. :.5 H : a.5 α.5 3. Comute the value of the test statisti. a ommo error is usig i this formula (1 ).5(1.5).45644 12 (67/12).5 z 1.28.45644 Slide 9 3
Value Aroah Two-Tailed Test About a Poulatio Proortio 4. Comute the -value. For z 1.28, umulative robability.8997 value 2(1.8997).26 5. Determie whether to rejet. Beause value.26 > α.5, we aot rejet. Slide 1 Critial Value Aroah Two-Tailed Test About a Poulatio Proortio 4. Determie the ritial values ad rejetio rule. For α/2.5/2.25, z.25 1.96 Rejet if z < -1.96 or z > 1.96 Rejet if <.41 or >.59 5. Determie whether to rejet. Beause 1.278 > -1.96 ad < 1.96, we aot rejet. Or Beause.56 >.41 ad.56 <.59, we aot rejet. Slide 11 Tye I Error Beause hyothesis tests are based o samle data, we must allow for the ossibility of errors. A Tye I error is rejetig whe it. The robability of makig a Tye I error whe the ull hyothesis as a equality is alled the level of sigifiae. Aliatios of hyothesis testig that oly otrol the Tye I error are ofte alled sigifiae tests. Slide 12 4
Tye II Error A Tye II error is aetig whe it is false. It is diffiult to otrol for the robability of makig a Tye II error. Statistiias avoid the risk of makig a Tye II error by usig do ot rejet ad ot aet. Slide 13 Tye I ad Tye II Errors Poulatio Coditio Colusio Aet Rejet If Is True Corret Deisio Tye I Error α If Is False Tye II Error Corret Deisio Slide 14 Hyothesis Testig ad Deisio Makig I may deisio-makig situatios the deisio maker may wat, ad i some ases may be fored, to take atio with both the olusio do ot rejet ad the olusio rejet. I suh situatios, it is reommeded that the hyothesis-testig roedure be eteded to ilude osideratio of makig a Tye II error. Slide 15 5
Calulatig the Probability of a Tye II Error i Hyothesis Tests About a Poulatio Mea 1. Formulate the ull ad alterative hyotheses. 2. Usig the ritial value aroah, use the level of sigifiae α to determie the ritial value ad the rejetio rule for the test. 3. Usig the rejetio rule, solve for the value of the samle mea orresodig to the ritial value of the test statisti. Slide 16 Calulatig the Probability of a Tye II Error i Hyothesis Tests About a Poulatio Mea 4. Use the results from ste 3 to state the values of the samle mea that lead to the aetae of ; this defies the aetae regio. 5. Usig the samlig distributio of for a value of µ satisfyig the alterative hyothesis, ad the aetae regio from ste 4, omute the robability that the samle mea will be i the aetae regio. (This is the robability of makig a Tye II error at the hose level of µ.) Slide 17 Calulatig the Probability of a Tye II Error Eamle: Metro EMS (revisited) Reall that the resose times for a radom samle of 4 medial emergeies were tabulated. The samle mea is 13.25 miutes. The oulatio stadard deviatio is believed to be 3.2 miutes. The EMS diretor wats to erform a hyothesis test, with a.5 level of sigifiae, to determie whether or ot the servie goal of 12 miutes or less is beig ahieved. Slide 18 6
Calulatig the Probability of a Tye II Error 1. Hyotheses are: : µ < 12 ad H a : µ > 12 2. Rejetio rule is: Rejet if z > 1.645 3. Value of the samle mea that idetifies the rejetio regio: 12 z 1.645 3.2 / 4 3.2 12 + 1.645 12.8323 4 4. We Rejet whe > 12.8323 We will aet whe < 12.8323 We solve for the Critial Value ( ) Slide 19 Calulatig the Probability of a Tye II Error 5. Probabilities that the samle mea will be i the aetae regio: 12.8323 µ z Values of µ 3.2/ 4 1-14. -2.31.14.9896 13.6-1.52.643.9357 13.2 -.73.2327.7673 12.8323..5.5 12.8.6.5239.4761 12.4.85.823.1977 12.1 1.645.95.5 Slide 2 Calulatig the Probability of a Tye II Error Calulatig the Probability of a Tye II Error Observatios about the reedig table: Whe the true oulatio mea µ is lose to the ull hyothesis value of 12, there is a high robability that we will make a Tye II error. Eamle: µ 12.1,.95 Whe the true oulatio mea µ is far above the ull hyothesis value of 12, there is a low robability that we will make a Tye II error. Eamle: µ 14.,.14 Slide 21 7
Power of the Test The robability of orretly rejetig whe it is false is alled the ower of the test. For ay artiular value of µ, the ower is 1. We a show grahially the ower assoiated with eah value of µ ; suh a grah is alled a ower urve. (See et slide.) Slide 22 Power Curve Probability of Corretly Rejetig Null Hyothesis 1..9.8.7.6.5.4.3.2.1. False 11.5 12. 12.5 13. 13.5 14. 14.5 µ Slide 23 Determiig the Samle Size for a Hyothesis Test About a Poulatio Mea The seified level of sigifiae determies the robability of makig a Tye I error. By otrollig the samle size, the robability of makig a Tye II error is otrolled. Slide 24 8
Determiig the Samle Size for a Hyothesis Test About a Poulatio Mea Samlig distributio ad µ µ If α Rejet : µ < µ H a : µ > µ Note: µ If is false Samlig distributio is false ad µ a > µ µ a Slide 25 Determiig the Samle Size for a Hyothesis Test About a Poulatio Mea Samlig distributio ad µ µ If Samlig distributio If is false is false α ad µ a > µ µ µ a Aet Note: Rejet : µ < µ H a : µ > µ Slide 26 Derease i Tye II Error whe Atual Mea is Far Away from Hyothesized Mea Samlig distributio ad µ µ If Samlig distributio If is false is false α ad µ a > µ µ µ a Aet Note: Rejet : µ < µ H a : µ > µ Slide 27 9
Derease i Tye II Error whe Samle Size is Ireased Samlig distributio ad µ µ If α Samlig distributio If is false is false ad µ a > µ µ µ a Note: Aet Rejet : µ < µ H a : µ > µ Slide 28 Determiig the Samle Size for a Hyothesis Test About a Poulatio Mea 2 2 α 2 a ( z + z ) ( µ µ ) where z α z value rovidig a area of α i the tail z z value rovidig a area of i the tail oulatio stadard deviatio µ value of the oulatio mea i µ a value of the oulatio mea used for the Tye II error Note: I a two-tailed hyothesis test, use z α /2 ot z α Slide 29 Relatioshi Amog α,, ad Oe two of the three values are kow, the other a be omuted. For a give level of sigifiae α, ireasig the samle size will redue. For a give samle size, dereasig α will irease, whereas ireasig α will derease b. Slide 3 1
Determiig the Samle Size for a Hyothesis Test About a Poulatio Mea Let s assume that the diretor of medial servies makes the followig statemets about the allowable robabilities for the Tye I ad Tye II errors: If the mea resose time is µ 12 miutes, I am willig to risk a α.5 robability of rejetig. If the mea resose time is.75 miutes over the seifiatio (µ 12.75), I am willig to risk a.1 robability of ot rejetig. Slide 31 Determiig the Samle Size for a Hyothesis Test About a Poulatio Mea α.5,.1 z α 1.645, z 1.28 µ 12, µ a 12.75 3.2 ( ) (1.645 1.28) (3.2) 155.75 156 2 2 2 2 z α+ z + 2 2 ( µ µ a ) (12 12.75) Slide 32 Ed of Chater 9, Part B Slide 33 11