Chapter 9 Hypothesis Tests

Transcription

1 Chater 9 Hyothesis Tests Learig Objectives 1. Lear how to formulate ad test hyotheses about a oulatio mea ad/or a oulatio roortio. 2. Uderstad the tyes of errors ossible whe coductig a hyothesis test. 3. Be able to determie the robability of makig various errors i hyothesis tests. 4. Kow how to comute ad iterret -values. 5. Be able to use critical values to draw hyothesis testig coclusios. 6. Be able to determie the sie of a simle radom samle ecessary to kee the robability of hyothesis testig errors withi accetable limits. 7. Kow the defiitio of the followig terms: ull hyothesis alterative hyothesis Tye I error Tye II error oe-tailed test two-tailed test -value level of sigificace critical value ower curve 9-1

2 Chater 9 Solutios: 1. a. H : µ 6 Maager s claim. H a : µ > 6 We are ot able to coclude that the maager s claim is wrog. c. The maager s claim ca be rejected. We ca coclude that µ > a. H : µ 14 H a : µ > 14 Research hyothesis There is o statistical evidece that the ew bous la icreases sales volume. c. The research hyothesis that µ > 14 is suorted. We ca coclude that the ew bous la icreases the mea sales volume. 3. a. H : µ = 32 Secified fillig weight H a : µ 32 Overfillig or uderfillig exists There is o evidece that the roductio lie is ot oeratig roerly. Allow the roductio rocess to cotiue. c. Coclude µ 32 ad that overfillig or uderfillig exists. Shut dow ad adjust the roductio lie. 4. a. H : µ 22 H a : µ < 22 Research hyothesis to see if mea cost is less tha $22. We are uable to coclude that the ew method reduces costs. c. Coclude µ < 22. Cosider imlemetig the ew method based o the coclusio that it lowers the mea cost er hour. 5. a. The Tye I error is rejectig H whe it is true. I this case, this error occurs if the researcher cocludes that the mea ewsaer-readig time for idividuals i maagemet ositios is greater tha the atioal average of 8.6 miutes whe i fact it is ot. The Tye II error is accetig H whe it is false. I this case, this error occurs if the researcher cocludes that the mea ewsaer-readig time for idividuals i maagemet ositios is less tha or equal to the atioal average of 8.6 miutes whe i fact it is greater tha 8.6 miutes. 6. a. H : µ 1 The label claim or assumtio. H a : µ > 1 Claimig µ > 1 whe it is ot. This is the error of rejectig the roduct s claim whe the claim is true. 9-2

3 Hyothesis Testig c. Cocludig µ 1 whe it is ot. I this case, we miss the fact that the roduct is ot meetig its label secificatio. 7. a. H : µ 8 H a : µ > 8 Research hyothesis to see if the la icreases average sales. Claimig µ > 8 whe the la does ot icrease sales. A mistake could be imlemetig the la whe it does ot hel. c. Cocludig µ 8 whe the la really would icrease sales. This could lead to ot imlemetig a la that would icrease sales. 8. a. H : µ 22 H a : µ < 22 Claimig µ < 22 whe the ew method does ot lower costs. A mistake could be imlemetig the method whe it does ot hel. c. Cocludig µ 22 whe the method really would lower costs. This could lead to ot imlemetig a method that would lower costs. 9. a = = = 2.12 σ / 2/ 5 Area =.483 -value = =.17 c. -value.5, reject H d. Reject H if , reject H 1. a = = = 1.48 σ / 6/ 4 Area =.436 -value = =.694 c. -value >.1, do ot reject H d. Reject H if < 2.33, do ot reject H 11. a = = = 2. σ / 3/ 5 Area =

4 Chater 9 -value = 2( ) =.456 c. -value.5, reject H d. Reject H if or , reject H 12. a = = = 1.25 σ / 12 / 1 -value = =.156 -value >.1, do ot reject H 77 8 = = = 2.5 σ / 12 / 1 -value = =.62 -value.1, reject H c = = = 3.75 σ / 12 / 1 -value -value.1, reject H d = = =.83 σ / 12/ 1 Area to left of =.83 -value = = value >.1, do ot reject H 13. Reject H if a = = = 2.42 σ / 8/ , reject H 51 5 = = =.97 σ / 8/ 6.97 < 1.645, do ot reject H 9-4

5 Hyothesis Testig c = = = 1.74 σ / 8/ , reject H 14. a = = =.87 σ / 1/ 75 -value = 2( ) = value >.1, do ot reject H = = = 2.68 σ / 1/ 75 -value = 2( ) =.74 -value.1, reject H c = = = 1.73 σ / 1/ 75 -value = 2( ) =.836 -value >.1, do ot reject H 15. a. H : µ 156 H a : µ < = = = 1.83 σ / 16 / 4 -value = =.336 c. -value.5, reject H. Coclude the mea refud of last miute filers is less tha $156. d. Reject H if , reject H 16. a. H : µ 895 H a : µ > = = = 1.19 σ / 225 / 18 Area =

6 Chater 9 -value = =.117 c. Do ot reject H. We caot coclude the retal rates have icreased. d. Recommed withholdig judgmet ad collectig more data o aartmet retal rates before drawig a fial coclusio. 17. a. H : µ = 39.2 H a : µ = = = 1.54 σ / 4.8/ 112 -value = 2( ) =.1236 c. -value >.5, do ot reject H. We caot coclude that the mea legth of a work week has chaged. d. Reject H if or 1.96 = -1.54; caot reject H 18. a. H : µ = 26,133 H a : µ 26,133 25, ,133 = = = 2.9 σ / 76 / 55 -value = 2( ) =.366 c. -value.5. Reject H ; Collier Couty has a mea aual wage differet from the state mea aual wage. 19. H : µ H a : µ > = = = 2.15 σ / 1.45/ 75 Area = value = =.158 -value.5, reject H. Coclude that there has bee a icrease i the mea hourly wage of roductio workers. 2. a. H : µ 181,9 9-6

7 Hyothesis Testig H a : µ < 181,9 x µ 166,4 181,9 = = = 2.93 σ / 33,5 / 4 c. -value = =.17 d. -value.1; reject H. Coclude mea sellig rice i South is less tha the atioal mea sellig rice. 21. a. H : µ 15 H a : µ > 15 x µ = = = 2.96 σ / 4/ 35 c. -value = =.15 d. -value.1; reject H ; the remium rate should be charged. 22. a. H : µ = 8 H a : µ = = = 1.71 σ / 3.2 / 12 -value = 2( ) =.872 c. Do ot reject H. Caot coclude that the oulatio mea waitig time differs from 8 miutes. d. x ±.25( σ / ) 8.5 ± 1.96 (3.2 / 12) 8.5 ±.57 (7.93 to 9.7) Yes; µ = 8 is i the iterval. Do ot reject H. 23. a t = = = 2.31 s/ 4.32/ 25 Degrees of freedom = 1 = 24 Usig t table, -value is betwee.1 ad.25. Actual -value =.147 c. -value.5, reject H. 9-7

8 Chater 9 d. With df = 24, t.5 = Reject H if t > 1.711, reject H. 24. a t = = = 1.54 s/ 4.5/ 48 Degrees of freedom = 1 = 47 Area i lower tail is betwee.5 ad.1 Usig t table, -value (two-tail) is betwee.1 ad.2 Actual -value =.134 c. -value >.5, do ot reject H. d. With df = 47, t.25 = 2.12 Reject H if t or t 2.12 t = -1.54; do ot reject H 25. a t = = = 1.15 s/ 5.2/ 36 Degrees of freedom = 1 = 35 Usig t table, -value is betwee.1 ad.2 Actual -value = value >.1, do ot reject H t = = = 2.61 s/ 4.6/ 36 Usig t table, -value is betwee.5 ad.1 Actual -value =.66 -value.1, reject H c t = = = 1.2 s/ 5/ 36 Usig t table, area i uer tail is betwee.1 ad.2 -value (lower tail) is betwee.8 ad.9 Actual -value =

9 Hyothesis Testig -value >.1, do ot reject H 26. a t = = = 2.1 s/ 11.5/ 65 Degrees of freedom = 1 = 64 Usig t table, area i tail is betwee.1 ad.25 -value (two tail) is betwee.2 ad.5 Actual -value =.394 -value.5, reject H t = = = 2.57 s/ 11/ 65 Usig t table, area i tail is betwee.5 ad.1 -value (two tail) is betwee.1 ad.2 Actual -value =.127 -value.5, reject H c t = = = 1.54 s/ 1.5/ 65 Usig t table, area i tail is betwee.5 ad.1 -value (two tail) is betwee.1 ad.2 Actual -value = value >.5, do ot reject H 27. a. H : µ 238 H a : µ < t = = =.88 s/ 8 / 1 Degrees of freedom = 1 = 99 Usig t table, -value is betwee.1 ad.2 Actual -value =.1918 c. -value >.5; do ot reject H. Caot coclude mea weekly beefit i Virgiia is less tha the atioal mea. 9-9

10 Chater 9 d. df = 99 t.5 = Reject H if t > -1.66; do ot reject H 28. a. H : µ 353 H a : µ > t = = = 2.49 s/ 81/ 92 Degrees of freedom = 1 = 91 Usig t table, -value is betwee.5 ad.1 Actual -value =.74 c. -value.1; reject H. The mea attedace er game has icreased. Aticiate a ew all-time high seaso attedace durig the 22 seaso. 29. a. H : µ = 56 H a : µ t = = = 2.26 s/ 52 / 25 Degrees of freedom = 1 = 24 Usig t table, area i tail is betwee.1 ad.25 Actual -value =.332 c. -value.5; reject H. The mea diamod rice i New York City differs. d. df = 24 t.25 = 2.64 Reject H if t < or t > > 2.64; reject H 3. a. H : µ = 6 H a : µ t = = = 1.17 s/ 65/ 4 df = - 1 =

11 Hyothesis Testig Usig t table, area i tail is betwee.1 ad.2 -value is betwee.2 ad.4 Actual -value =.251 c. Withα =.1 or less, we caot reject H. We are uable to coclude there has bee a chage i the mea CNN viewig audiece. d. The samle mea of 612 thousad viewers is ecouragig but ot coclusive for the samle of 4 days. Recommed additioal viewer audiece data. A larger samle should hel clarify the situatio for CNN. 31. H : µ 47.5 H a : µ > t = = = 2.33 s/ 12/ 64 Degrees of freedom = - 1 = 63 Usig t table, -value is betwee.1 ad.25 Actual -value =.114 Reject H ; Atlata customers are ayig a higher mea water bill. 32. a. H : µ = 1,192 H a : µ 1, ,192 t = = = 2.23 s/ 14 / 5 Degrees of freedom = 1 = 49 Usig t table, area i tail is betwee.1 ad.25 -value is betwee.2 ad.5 Actual -value =.32 c. -value.5; reject H. The oulatio mea rice at this dealershi differs from the atioal mea rice $1, a. H : µ 28 H a : µ > = 6.9 yards c t = = = 2.7 s/ 1/

12 Chater 9 Degrees of freedom = 1 = 8 Usig t table, -value is betwee.25 ad.5 Actual -value =.361 c. -value.5; reject H. The oulatio mea distace for the ew driver is greater tha the USGA aroved driver. 34. a. H : µ = 2 H a : µ 2 Σx 22 x = i = = c. d. ( x ) 2 i x.516 Σ s = = t = = = 1.22 s/.516 / 1 Degrees of freedom = - 1 = 9 Usig t table, area i tail is betwee.1 ad.2 -value is betwee.2 ad.4 Actual -value =.2518 e. -value >.5; do ot reject H. No reaso to chage from the 2 hours for cost estimatig uroses. 35. a = = = (1 ).2(1.2) Area i tail = ( ) =.156 -value = 2(.156) =.2112 c. -value >.5; do ot reject H d..25 = 1.96 Reject H if or 1.96 = 1.25; do ot reject H 9-12

13 Hyothesis Testig 36. a = = = (1 ).75(1.75) value = =.26 -value.5; reject H = (1.75) = value = = value >.5; do ot reject H c. = (1.75) = value = =.228 -value.5; reject H d. = (1.75) =.8 3 -value = = value >.5; do ot reject H 37. a. H :.4 H a : > = = = = = (1 ).4(1.4) Area = value = =.31 c. -value.5; reject H. Coclude more tha 4% receive more tha 1 messages er day. 9-13

14 Chater a. H : =.64 H a : = = = = = (1 ).64(1.64) Area = value = 2( ) =.124 c. -value.5; reject H. Proortio differs from the reorted.64. d. Yes. Sice =.52, it idicates that fewer tha 64% of the shoers believe the suermarket brad is as good as the ame brad. 39. a. H : =.7 H a :.7 Wiscosi 252 = = = = = (1 ).7(1.7) Area = value = 2( ) =.4122 Caot reject H. Califoria 189 = =.63 3 = (1.7) = Area =.496 -value = 2( ) =.8 Reject H. Califoria has a differet (lower) ercetage of adults who do ot exercise regularly. 9-14

15 Hyothesis Testig 4. a. 414 = =.272 (27%) 1532 H :.22 H a : > = = = (1 ).22(1.22) value Reject H ; There has bee a sigificat icrease i the itet to watch the TV rograms. c. These studies hel comaies ad advertisig firms evaluate the imact ad beefit of commercials. 41. a. H :.75 H a : < = = = (1 ).75(1.75) Area = value = =.1151 c. -value >.5; do ot reject H. The executive's claim caot be rejected. 42. H :.24 H a : > = = = = = (1 ).24(1.24) Area = value = =.23 -value.5; reject H. I 23, a estimated 31% of eole who moved selected to be coveiet to work as their rimary reaso. This is a icrease comared to

16 Chater a. H : =.48 H a : = = = = = (1 ).48(1.48) Area = value = 2( ) =.892 c. -value >.5; do ot reject H. There is o reaso to coclude the roortio has chaged. 44. a. H :.5 H a : > = = = = = (1 ).5(1.5) value c. -value.1; reject H. Coclude Burger Kig fries are referred by more tha.5 of the oulatio. d. Yes; statistical evidece shows Burger Kig fries are referred. The give-away was a good way to get customers to try the ew fries. 45. a. H : =.44 H a : = = = = = (1 ).44(1.44)

17 Hyothesis Testig Area = value = 2( ) =.177 -value >.5; do ot reject H. No chage. c. 245 = =.49 5 = (1.44) = Area = value = 2( ) =.244 -value.5; reject H. There has bee a chage: a icrease i reeat customers. 46. σ 5 σ x = = = c H a : µ < 1 H : µ x c = (5 / 12 ) = 9.25 Reject H if x 9.25 a. Whe µ = 9, = =.55 5/ 12 Prob (H ) = ( ) =.2912 Tye II error 9-17

18 Chater 9 c. Whe µ = 8, = = / 12 β = ( ) = Reject H if or if 1.96 σ 1 σ x = = =.71 2 H a : µ 2 H : µ = 2 H a : µ c 1 2 c 2 x x c 1 = (1 / 2 ) = c 2 = (1 / 2 ) = a. µ = = =.86 1 / 2 β = =.1949 µ = = = / 2 β = =

19 Hyothesis Testig c. µ = = =.55 1 / 2 β = = a. H : µ 15 H a : µ > 15 Cocludig µ 15 whe this is ot true. Fowle would ot charge the remium rate eve though the rate should be charged. Reject H if 2.33 x 15 = = = 2.33 σ / 4/ 35 Solve for x = Decisio Rule: Accet H if x < Reject H if x For µ = 17, = =.62 4/ 35 β = =.2676 c. For µ = 18, = = 2.1 4/ 35 β = = a. H : µ 25 H a : µ < 25 Reject H if -2.5 x 25 = = = 2.5 σ / 3/ 3 Solve for x = Decisio Rule: 9-19

20 Chater 9 Accet H if x > Reject H if x For µ = 23, = = / 3 β = =.537 c. For µ = 24, = =.22 3/ 3 β = =.5871 d. The Tye II error caot be made i this case. Note that whe µ = 25.5, H is true. The Tye II error ca oly be made whe H is false. 5. a. Accetig H ad cocludig the mea average age was 28 years whe it was ot. Reject H if or if 1.96 x 28 = = σ / 6 / 1 Solvig for x, we fid at = -1.96, x = at = +1.96, x = Decisio Rule: Accet H if < x < Reject H if x or if x At µ = 26, = = / 1 β = =.853 At µ = 27, = =.3 6/ 1 9-2

21 Hyothesis Testig β = =.6179 At µ = 29, = =.3 6 / 1 β = =.6179 At µ = 3, = = / 1 β = =.853 c. Power = 1 - β at µ = 26, Power = =.9147 Whe µ = 26, there is a.9147 robability that the test will correctly reject the ull hyothesis that µ = a. Accetig H ad lettig the rocess cotiue to ru whe actually over - fillig or uder - fillig exists. Decisio Rule: Reject H if or if 1.96 idicates Accet H if < x < Reject H if x or if x For µ = = = / 3 β = =

22 Chater 9 c β x c. Power = =.9251 d. The ower curve shows the robability of rejectig H for various ossible values of µ. I articular, it shows the robability of stoig ad adjustig the machie uder a variety of uderfillig ad overfillig situatios. The geeral shae of the ower curve for this case is Power Possible Values of u 52. c µ.1 σ 4 = + = = At µ = = = 1.2 4/ 5 β = = At µ = 18 = = / 5 β = =.15 Icreasig the samle sie reduces the robability of makig a Tye II error. 9-22

23 Hyothesis Testig 53. a. Accet µ 1 whe it is false. Critical value for test: σ 75 c = µ +.5 = = At µ = = =.4 75/ 4 β = =.484 c. At µ = = =.88 75/ 4 β = =.1894 d. Critical value for test: σ 75 c = µ +.5 = = At µ = 12 = =.74 75/ 8 β = =.2296 At µ = = = / 8 β = =.268 Icreasig the samle sie from 4 to 8 reduces the robability of makig a Tye II error ( α + β) σ ( ) (5) = = = ( µ µ ) (1 9) a ( α + β) σ ( ) (1) = = = ( µ µ ) (2 22) a 56. At µ = 3, α =.1..1 = 2.33 At µ a = , β =.1..1 = 1.28 σ = ( α + β) σ ( ) (.18) = = = ( µ µ ) ( ) a Use

24 Chater At µ = 4, α =.2..2 = 2.5 At µ a = 385, β =.1..1 = 1.28 σ = ( α + β) σ ( ) (3) = = = ( µ µ ) (4 385) a Use At µ = 28, α =.5. Note however for this two - tailed test, α / 2 =.25 = 1.96 At µ a = 29, β = = 1.4 σ = ( α /2 + β) σ ( ) (6) = = = ( µ µ ) (28 29) a 59. At µ = 25, α =.2..2 = 2.5 At µ a = 24, β =.2..2 =.84 σ = ( α + β) σ ( ) (3) = = = ( µ µ ) (25 24) a Use a. H : µ = 16 H a : µ = = = 2.19 σ /.8/ 3 Area = value = 2( ) =.286 -value.5; reject H. Readjust roductio lie. c = = = 1.23 σ /.8/ 3 Area =.397 -value = 2( ) = value >.5; do ot reject H. Cotiue the roductio lie. 9-24

25 Hyothesis Testig d. Reject H if or 1.96 For x = 16.32, = 2.19; reject H For x = 15.82, = -1.23; do ot reject H Yes, same coclusio. 61. a. H : µ = 9 H a : µ 9 x ±.25 σ ± ± 25 (91 to 96) c. Reject H because µ = 9 is ot i the iterval. d = = = 2.75 σ / 18 / 2 -value = 2( ) = a. H : µ 45,25 H a : µ > 45,25 47, 45, 25 = = = 2.71 σ / 63 / 95 -value = =.34 -value.1; reject H. Coclude New York City has higher mea salary. 63. a. H : µ 37, H a : µ > 37, 38, 37, t = = = 1.47 s/ 52 / 48 Degrees of freedom = 1 = 47 Usig t table, -value is betwee.5 ad.1 Actual -value =.747 c. -value >.5, do ot reject H. Caot coclude mea greater tha $37,. A larger samle is desirable. 9-25

26 Chater H : µ = 6 H a : µ t = = =.93 s/ 114 / 32 Degrees of freedom = 1 = 31 Usig t table, area i tail is betwee.1 ad.2 -value is betwee.2 ad.4 Actual -value =.3581 Do ot reject H. There is o evidece to coclude that the mea umber of freshma alicatios has chaged. 65. a. H : µ 3 H a : µ < t = = = 1.57 s/ 1.8/ 5 Degrees of freedom = 5 1 = 49 Usig t table, -value is betwee.5 ad.1 Actual -value =.613 c. -value >.1; do ot reject H. Caot coclude miles er gallo is less tha H : µ 125, H a : µ > 125, 13, 125, t = = = 2.26 s/ 12,5 / 32 Degrees of freedom = 32 1 = 31 Usig t table, -value is betwee.1 ad.25 Actual -value =.154 -value.5; reject H. Coclude that the mea cost is greater tha $125, er lot. 67. a. H : µ = 3 H a : µ

27 Hyothesis Testig Σx x = i = 2.8 c. d. ( x ) 2 i x.7 Σ s = = t = = =.9 s/.7/ 1 Degrees of freedom = 1-1 = 9 Usig t table, area i tail is betwee.1 ad.2 -value is betwee.2 ad.4 Actual -value =.392 e. -value >.5; do ot reject H. There is o evidece to coclude a differece comared to rior year. 68. a. H :.5 H a : >.5 c. 64 = = = = = (1 ).5(1.5) Area i tail = value = =.26 -value.1; reject H. College graduates have a greater sto-smokig success rate. 69. a. H : =.6667 H a :.6667 c. 355 = = = = = (1 ).6667(1.6667) value = 2( ) =

28 Chater 9 -value >.5; do ot reject H ; Caot coclude that the oulatio roortio differs from 2/3. 7. a. H :.5 H a : >.5 c. 67 = =.6381 (64%) = = = (1 ).5(1.5) value = =.23 -value.1; reject H. Coclude that the four 1-hour day schedules is referred by more tha 5% of the office workers. 71. a. 33 = = H : =.78 H a : = = = (1 ).78(1.78) value = 2( ) =.3 c. -value.5; reject H. The o-time arrival record has chaged. It is imrovig. 72. H :.9 H a : <.9 49 = = = = = (1 ).9(1.9) value = =.88 -value >.5; do ot reject H. Claim of at least 9% caot be rejected. 73. a. H :.47 H a : <

29 Hyothesis Testig c. 44 = = = = = (1 ).47(1.47) value = =.41 d. -value.1; reject H. The roortio of foods cotaiig esticides has declied. 74. a. H : µ 72 H a : µ > 72 Reject H if x 72 = = = σ / 2/ 3 Solve for x = 78 Decisio Rule: Accet H if x < 78 Reject H if x 78 For µ = = =.55 2 / 3 β = =.2912 c. For µ = 75, = =.82 2 / 3 β = =.7939 d. For µ = 7, H is true. I this case the Tye II error caot be made. e. Power = 1 - β 9-29

30 Chater 9 1. P o w e r Possible Values of µ H o False 75. H : µ 15, H a : µ < 15, At µ = 15,, α =.2..2 = 2.5 At µ a = 14,, β =.5..1 = ( α + β) σ ( ) (4, ) = = = ( µ µ ) (15, 14, ) a Use H : µ = 12 H a : µ 12 At µ = 12, α =.5. With a two - tailed test, α / 2 =.25 = 1.96 At µ a = 117, β =.2..2 = ( α /2 + β) σ ( ) (5) = = = ( µ µ ) (12 117) a Use 45 Examle calculatio for µ = 118. Reject H if or if 1.96 x 12 = = σ / 5/ 45 Solve for x. At = -1.96, x = At = +1.96, x =

31 Hyothesis Testig Decisio Rule: Accet H if < x < Reject H if x or if x For µ = 118, = =.72 5/ 45 β = =.2358 Other Results: If µ is β