Transcript
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Hypothesis Test

1. A consumer group, concerned about the mean fat content of a certain grade of steakburger submits to an independent laboratory a random sample of 12 steak burgers foranalysis. The percentage of fat in each of the steak burgers is as follows: 21 18 19 1 182! 22 19 2! 1! 18 1" The manufacturer claims that the mean fat content of this gradeof steak burger is less than 2#\$. Assuming percentage fat content to be normallydistributed with a standard de%iation of &, carry out an appropriate hypothesis test in orderto ad%ise the consumer group as to the %alidity of the manufacturer's claim

2. (uring a particular week, 1& babies were born in a maternity unit. )art of the standardprocedure is to measure the length of the baby. *i%en below is a list of the lengths, incentimeters, of the babies born in this particular week. !9 "# !" "1 !+ !9 !8 "! "&"" !" "# !8. Assuming that this sample came from an underlying normal population,test, at the "\$ signicance le%el, the hypothesis that the population mean length is "# cm.

3. A random sample of 12 steel ingots was taken from a production line. The masses, inkilograms, of these ingots are gi%en below. 2!.8 .8 28.1 2!.8 2+.! 22.1 2!.+ 2+.&2+." 2+.8 2&.9 2&.2 Assuming that this sample came from an underlying normalpopulation, in%estigate the claim that its mean e-ceeds 2".# kg. / 1\$

4. A car manufacturer introduces a new method of assembling a particular component. Theold method had a mean assembly time of !2 minutes. The manufacturer would like theassembly time to be as short as possible, and so he e-pects the new method to ha%e asmaller mean. A random sample of assembly times 0minutes taken after the new methodhad become established was 2+ &9 28 !1 !+ !2 &" &2 &8 tating any necessarydistributional assumptions, in%estigate the manufacturer's e-pectation./ 2 \$

5. A random sample of 1" workers from a %acuum 3ask assembly line was selected from alarge number of such workers. 4%or topwatch, a work5study engineer, asked each of theseworkers to assemble a one5litre %acuum 3ask at their normal working speed. The timestaken, in seconds, to complete these tasks are gi%en below: 1#9.2 1!.2 12+.9 92.#1#8." 91.1 1#9.8 11!.9 11".& 99.# 112.8 1.+ 1!1.+ 122. 119.9 Assuming thatthis sample came from an underlying normal population, in%estigate the claim that thepopulation mean assembly time is less than 2 minutes./ "\$

6. A pharmacist claims that more than #\$ of all customers simply collect a prescription. 6neof her assistants notes that, in a random sample of 12 customers, 1# simply collected aprescription. (oes this pro%ide su7cient e%idence, at the "\$ le%el, to support thepharmacist's claim

7. 4n a sur%ey carried out in un%ille, 1! children out of a random sample of said that theybought the opper comic regularly. Test, at the 1#\$ le%el of signicance, the hypothesisthat the true proportion of all children who buy this comic regularly is #.&".

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8. A random sample of & coee drinkers were each asked to taste5test a new brand ofcoee. The responses are listed below with ; representing 'like', 4 representing 'indierent',and ( representing 'dislike'.; ( ; ; ( ; ; ; ; ; ( (; ; ; ; ; ( ; ; ; ; ( (; ; ; ; ( ; ; ; ; ; ; ((o these data support the claim that more than half of all coee drinkers like this new

brand of coeeackets issued were of the following si?es.2 & & 1 & & 2 ! & 2 " ! 1 2 & & 2 ! " & 2 ! ! 1 " & & 2 & & 1 & ! & & 2" 1 ! !Assuming that the !# employees may be regarded as a random sample of all employees,test the hypothesis, at the "\$ signicance le%el, that !#\$ of all employees re@uire si?e &.

Test the claim that si?e & is the median si?e.

11. The Acme ompany has de%eloped a new battery. The engineer in charge claimsthat the new battery will operate continuously for at least+ minutes longer than the old

battery. To test the claim, the company selects a simple random sample of 1## new

batteries and 1## old batteries. The old batteries run continuously for 19# minutes with astandard de%iation of 2# minutesB the new batteries, 2## minutes with a standard

de%iation of !# minutes.

Test the engineer's claim that the new batteries run at least + minutes longer than the old.

Cse a #.#" le%el of signicance. 0Assume that there are no outliers in either sample.

12. orty5four si-th graders were randomly selected from a school district. Then, theywere di%ided into 22 matched pairs, each pair ha%ing e@ual 4D's. 6ne member of each pair

was randomly selected to recei%e special training. Then, all of the students were gi%en an

4D test. Test results are summari?ed below.

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)air Training Eo training

1 9" 9#

2 89 8"

& + +&

! 92 9#

" 91 9#

"& "&

+ + 8

8 88 9#

9 +" +8

1# 8" 89

11 9# 9"

)airTrainin

g

Eo

training

12 8" 8&

1& 8+ 8&

1! 8" 8&1" 8" 82

1 8 "

1+ 81 +9

18 8! 8&

19 +1 #

2# ! !+

21 +" ++

22 8# 8&

(o these results pro%ide e%idence that the special training helped or hurt studentperformance< Cse an #.#" le%el of signicance. Assume that the mean dierences are

appro-imately normally distributed.

13. Fithin a school district, students were randomly assigned to one of two Gathteachers 5 Grs. mith and Grs. Hones. After the assignment, Grs. mith had students,

and Grs. Hones had 2" students. At the end of the year, each class took the same

standardi?ed test. Grs. mith's students had an a%erage test score of +8, with a standard

de%iation of 1#B and Grs. Hones' students had an a%erage test score of 8", with a standard

de%iation of 1".

Test the hypothesis that Grs. mith and Grs. Hones are e@ually eecti%e teachers. Cse a

#.1# le%el of signicance. 0Assume that student performance is appro-imately normal.

14. The =6 of a large electric utility claims that 8# percent of his 1,###,### customersare %ery satised with the ser%ice they recei%e. To test this claim, the local newspaper

sur%eyed 1## customers, using simple random sampling. Among the sampled customers,

+& percent say they are %ery satisied. ased on these ndings, can we re>ect the =6's

hypothesis that 8#\$ of the customers are %ery satised< Cse a #.#" le%el of signicance.

15. The =6 of a large electric utility claims that 8# percent of his 1,###,### customersare %ery satised with the ser%ice they recei%e. To test this claim, the local newspaper

sur%eyed 1## customers, using simple random sampling. Among the sampled customers,

+& percent say they are %ery satised. ased on these ndings, can we re>ect the =6's

hypothesis that 8#\$ of the customers are %ery satised< Cse a #.#" le%el of signicance.

16. uppose the Acme (rug ompany de%elops a new drug, designed to pre%ent colds.The company states that the drug is e@ually eecti%e for men and women. To test this

claim, they choose a simple random sample of 1## women and 2## men from a population

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of 1##,### %olunteers. At the end of the study, &8\$ of the women caught a coldB and "1\$

of the men caught a cold. ased on these ndings, can we re>ect the company's claim that

the drug is e@ually eecti%e for men and women< Cse a #.#! le%el of signicance.

17. uppose the pre%ious e-ample is stated a little bit dierently. uppose the Acme(rug ompany de%elops a new drug, designed to pre%ent colds. The company states that

the drug is more eecti%e for women than for men. To test this claim, they choose a simplerandom sample of 1## women and 2## men from a population of 1##,### %olunteers. At

the end of the study, &8\$ of the women caught a coldB and "1\$ of the men caught a cold.

ased on these ndings, can we conclude that the drug is more eecti%e for women than

for men< Cse a #.#1 le%el of signicance.

18. Cn diseIador de productos estJ interesado en reducir el tiempo de secado de unapintura. e prueban dos fKrmulas de pinturaB la fKrmula 1 tiene el contenido @uLmico

estJndar y la fKrmula 2 tiene un nue%o ingrediente secante @ue tiende a reducir el tiempo

de secado. (e la e-periencia se sabe @ue la des%iaciKn estJndar del tiempo de secado esocho minutos y esta %ariabilidad inherente no debe %erse afectada por adiciKn del nue%o

ingrediente. e pintan &" placas con la fKrmula 1 y otras &" con la fKrmula 2. ;os dos

tiempos promedio de secado muestrales son 11 minutos para la fKrmula 1 y 112 minutos

para la fKrmula 2. MA @uN conclusiKn puede llegar el diseIador del producto sobre la

ecacia del nue%o ingrediente, al ni%el de signicancia #,#1oint replacement surgery. Thetrial compares the new pain relie%er to the pain relie%er currently in use 0called thestandard of care. A total of 1## patients undergoing >oint replacement surgery agreed toparticipate in the trial. )atients were randomly assigned to recei%e either the new painrelie%er or the standard pain relie%er following surgery and were blind to the treatmentassignment. efore recei%ing the assigned treatment, patients were asked to rate their

pain on a scale of #51# with higher scores indicati%e of more pain. =ach patient was thengi%en the assigned treatment and after minutes was again asked to rate their pain onthe same scale. The primary outcome was a reduction in pain of & or more scale points0dened by clinicians as a clinically meaningful reduction. The following data wereobser%ed in the trial.

T&e,t*e(t /&o\$p

(

'\$*+e&ith

Re\$#tio(

o 3 Poi(ts

'\$*+e&ith

Re\$#tio(

o 3

Poi(ts

'e P,i( Re%ie!e& "# 2&

t,(,& P,i(Re%ie!e&

"# 11

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Test whether there was a signicant dierence in the proportions of patients reporting ameaningful reduction 0i.e., a reduction of & or more scale points using a W statistic, asfollows.et up hypotheses and determine le%el of signicance "\$

30. =l director de una escuela clasica a los padres en tres categorLas socio5econKmicas

segPn su Jrea de residencia y en tres ni%eles de participaciKn en acti%idades escolares.)robar la hipKtesis de @ue no e-iste relaciKn entre el ni%el socio5econKmico y la

participaciKn en acti%idades escolares, con un ni%el de signicaciKn del !\$

)articipaciKn

Ei%el de ingreso

a>o Gedio Alto

Eunca 28 !8 1

6casional 22 " 1!Qegularme

nte 1+ +! &

31. Cna agencia de publicidad intenta determinar la composiciKn demogrJca delmercado para un nue%o producto. eleccionaron al a?ar +" personas de cada uno de "grupos de edad y les presentaron el producto. ;os resultados de la encuesta son lossiguientes:

Actitud frenteal producto

*rupo de edad18 529 5 &9 !# 5 !9 "# 5 "9 # 5 9

omprafrecuente 12 18 1+ 22 &2

ompra rara%e? 18 2" 29 2!

Eunca compra !" &2 29 29 1&

, (esarrolle una tabla de frecuencias obser%adas y esperadas para este problema+ alcule el %alor V2de la muestra.# =stable?ca las hipKtesis nula y alternati%a. i el ni%el de signicancia es #.#1, Mdebe recha?arse la hipKtesis nulaar en una estaciKn de pesado para camiones, He impsonsiente @ue el peso por camiKn 0en miles de libras sigue una distribuciKn normal. on elob>eto de probar esta suposiciKn, He recolecta los siguientes datos un lunes y registra el

peso de cada camiKn @ue llega a su bJscula.85 57 60 81 89 6352 65 77 64

89 86 90 60 57 6195 78 66 92

50 56 95 60 82 5561 81 61 53

63 75 50 98 63 77

50 62 79 6976 66 97 67 54 9370 80 67 73

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i He usa la prueba de bondad de a>uste de >i5cuadrada para estos datos, M@uN concluyeacerca de la distribuciKn del peso de los camiones< 0Cse un ni%el de signicancia de #.1# yasegPrese de establecer la hipKtesis de interNs. 0Sugerencia: use cinco inter%alosigualmente probables.

33. =l coordinador de computaciKn en la escuela de administraciKn cree @ue el tiempo@ue un estudiante de posgrado dedica a leer y escribir correos electrKnicos cada dLa de lasemana tiene una distribuciKn normal. )ara e-aminar esta opiniKn, el coordinadorrecolecta datos un miNrcoles y registra el tiempo en minutos @ue cada estudiante gasta essus correos electrKnicos. Cse la prueba de bondad de a>uste de >i5cuadrada con estosdatos, M@uN concluye acerca de la distribuciKn del tiempo dedicado al correo electrKnicoo Gedio Alto

Eunca 28 !8 1

6casional 22 " 1!recuentemente 1+ +! &

Wonas

A ( ==scuchK

elprograma

12 1+ 8 21 2"

EoescuchK

elprograma

&+ !1 2+ 1

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RE/REI' CRREACI'

1. ;as notas de 12 alumnos de una claseen GatemJticas y Lsicason las siguientes:

GatemJticas 2 & ! ! " + + 8 1# 1#

Lsica 1 & 2 ! ! ! ! + 9 1#

Rallar el coeciente de correlaciKn de la distribuciKn e interpretarlo.

2. Cna compaILa de segurosconsidera @ue el nPmero de %ehLculos @ue circulan por unadeterminada autopista a mJs de 12# kmZh, puede ponerse en funciKn del nPmerode accidentes @ue ocurren en ella. (urante " dLas obtu%o los siguientes resultados:

A##ie(tes " + 2 1 9':*e&o e !eh"#\$%os 1" 18 1# 8 2#

a alcula el coeciente de correlaciKn lineal.b i ayer se produ>eron accidentes, cuJntos %ehLculos podemos suponer @ue circulaban

por la autopista a mJs de 12# km Z h