Chapter 9, Part B Hypothesis Tests

Transcription

1 SlidesPreared by JOHN S.LOUCKS St.Edward suiversity Slide 1 Chater 9, Part B Hyothesis Tests Poulatio Proortio Hyothesis Testig ad Decisio Makig Calculatig the Probability of Tye II Errors Determiig the Samle Size for Hyothesis Tests About a Poulatio Mea Slide 2 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 : = > H a: Oe-tailed (lower tail) Oe-tailed (uer tail) Two-tailed Slide 3 1

2 Tests About a Poulatio Proortio Test Statistic where: z = ( 1 ) = assumig > 5 ad (1 ) > 5 Slide 4 Tests About a Poulatio Proortio Rejectio Rule: Value Aroach Reject H if value < α Rejectio Rule: Critical Value Aroach H : < H : > H : = Reject H if z > z α Reject H if z < -z α Reject H if z < -z α/2 or z > z α/2 Slide 5 Two-Tailed Test About a Poulatio Proortio Examle: Natioal Safety Coucil For a Christmas ad New Year s week, the Natioal Safety Coucil estimated that 5 eole would be killed ad 25, ijured o the atio s roads. The NSC claimed that 5% of the accidets would be caused by druk drivig. Slide 6 2

3 Two-Tailed Test About a Poulatio Proortio Examle: Natioal Safety Coucil A samle of 12 accidets showed that 67 were caused by druk drivig. Use these data to test the NSC s claim with α =.5. Slide 7 Two-Tailed Test About a Poulatio Proortio Value ad Critical Value Aroaches 1. Determie the hyotheses. 2. Secify the level of sigificace. H : =.5 H a:.5 α =.5 3. Comute the value of the test statistic. a commo error is usig i this formula (1 ).5(1.5) = = = (67/12).5 z = = = Slide 8 Value Aroach Two-Tailed Test About a Poulatio Proortio 4. Comute the -value. For z = 1.28, cumulative robability =.8997 value = 2(1.8997) = Determie whether to reject H. Because value =.26 > α =.5, we caot reject H. Slide 9 3

4 Critical Value Aroach Two-Tailed Test About a Poulatio Proortio 4. Determie the criticals value ad rejectio rule. For α/2 =.5/2 =.25, z.25 = 1.96 Reject H if z < or z > Determie whether to reject H. Because > ad < 1.96, we caot reject H. Slide 1 Hyothesis Testig ad Decisio Makig I may decisio-makig situatios the decisio maker may wat, ad i some cases may be forced, to take actio with both the coclusio do ot reject H ad the coclusio reject H. I such situatios, it is recommeded that the hyothesis-testig rocedure be exteded to iclude cosideratio of makig a Tye II error. Slide 11 Calculatig the Probability of a Tye II Error i Hyothesis Tests About a Poulatio Mea 1. Formulate the ull ad alterative hyotheses. 2. Usig the critical value aroach, use the level of sigificace α to determie the critical value ad the rejectio rule for the test. 3. Usig the rejectio rule, solve for the value of the samle mea corresodig to the critical value of the test statistic. Slide 12 4

5 Calculatig 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 accetace of H ; this defies the accetace regio. 5. Usig the samlig distributio of x for a value of µ satisfyig the alterative hyothesis, ad the accetace regio from ste 4, comute the robability that the samle mea will be i the accetace regio. (This is the robability of makig a Tye II error at the chose level of µ.) Slide 13 Calculatig the Probability of a Tye II Error Examle: Metro EMS (revisited) Recall that the resose times for a radom samle of 4 medical emergecies were tabulated. The samle mea is miutes. The oulatio stadard deviatio is believed to be 3.2 miutes. The EMS director wats to erform a hyothesis test, with a.5 level of sigificace, to determie whether or ot the service goal of 12 miutes or less is beig achieved. Slide 14 Calculatig the Probability of a Tye II Error 1. Hyotheses are: H : µ < 12 ad H a : µ > Rejectio rule is: Reject H if z > Value of the samle mea that idetifies the rejectio regio: x 12 z = / x = We will accet H whe x < Slide 15 5

6 Calculatig the Probability of a Tye II Error 5. Probabilities that the samle mea will be i the accetace regio: µ z = Values of µ 3.2/ 4 β 1-β Slide 16 Calculatig the Probability of a Tye II Error Calculatig the Probability of a Tye II Error Observatios about the recedig table: Whe the true oulatio mea µ is close to the ull hyothesis value of 12, there is a high robability that we will make a Tye II error. Examle: µ = 12.1, β =.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. Examle: µ = 14., β =.14 Slide 17 Power of the Test The robability of correctly rejectig H whe it is false is called the ower of the test. For ay articular value of µ, the ower is 1 β. We ca show grahically the ower associated with each value of µ; such a grah is called a ower curve. (See ext slide.) Slide 18 6

7 Power Curve Probability of Correctly Rejectig Null Hyothesis H False µ Slide 19 Determiig the Samle Size for a Hyothesis Test About a Poulatio Mea The secified level of sigificace determies the robability of makig a Tye I error. By cotrollig the samle size, the robability of makig a Tye II error is cotrolled. Slide 2 Determiig the Samle Size for a Hyothesis Test About a Poulatio Mea Samlig distributio of x whe H is true ad µ = µ c α Reject H H : µ < µ H a : µ > µ x Note: x = µ β Samlig distributio of x whe H is false ad µ a > µ c µ a x Slide 21 7

8 Determiig the Samle Size for a Hyothesis Test About a Poulatio Mea 2 2 α β 2 a ( z + z ) = ( µ µ ) where z α = z value rovidig a area of α i the tail z β = z value rovidig a area of β i the tail = oulatio stadard deviatio µ = value of the oulatio mea i H 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 22 Relatioshi Amog α, β, ad Oce two of the three values are kow, the other ca be comuted. For a give level of sigificace α, icreasig the samle size will reduce β. For a give samle size, decreasig α will icrease β, whereas icreasig α will decrease b. Slide 23 Determiig the Samle Size for a Hyothesis Test About a Poulatio Mea Let s assume that the director of medical services 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 rejectig H. If the mea resose time is.75 miutes over the secificatio (µ = 12.75), I am willig to risk a β =.1 robability of ot rejectig H. Slide 24 8

9 Determiig the Samle Size for a Hyothesis Test About a Poulatio Mea α =.5, β =.1 z α = 1.645, z β = 1.28 µ = 12, µ a = = 3.2 ( ) ( ) (3.2) = = = z α+ z β ( µ µ a ) ( ) Slide 25 Ed of Chater 9, Part B Slide 26 9