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1. The sports offered at a retirement village are Golf (G), Tennis (T) and Swimming (S). The Venndiagram shows the numbers of people involved in each activity.

G T

S

1 1

1

!"

#"

$

(a) %ow many people

(i) only play golf&

(ii) play both tennis and golf&

(iii) do not play golf&

(b) Shade the part of the Venn diagram that represents the set G S.

Working:

Answers'

(a) (i) ..

(ii) .

(iii) .

(Total 4 marks)

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2. The table below shows the relative freuencies of the ages of the students atIngham HighSchool.

*ge(in years)

+elativefreuency

1! .11

1" .!

1- .!

1$ .1

1 .1-

Total 1

(a) /f a student is randomly selected from this school, find the probability that

(i) the student is 1- years old0

(ii) the student is 1$ years of age or older.

There are 1 students atIngham High School.

(b) alculate the number of 1- year old students.

Working:

Answers'

(a) (i) ...(ii) ..

(b) ..

(Total 4 marks)

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3. The table below shows the number of left and right handed tennis players in a sample of -males and females.

2eft handed +ight handed Total

3ale ! 4 !

5emale 1$ 1#

Total - "- -

/f a tennis player was selected at random from the group, find the probability that the player is

(a) male and left handed0

(b) right handed0

(c) right handed, given that the player selected is female.

Working:

Answers'

(a) ..

(b) ..

(c) ..

(Total 4 marks)

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4. 2et 6 7x' 1 8x9 1,x :.

P, QandRare the subsets of such that

P6 7multiples of four:0

Q6 7factors of !$:0

R6 7suare numbers:.

(a) 2ist the elements of

(i)

(ii) PQR.(2)

(b) ;escribe in words the setPQ.(1)

(c) (i) ;raw a Venn diagram to show the relationship between setsP, QandR.(2)

(ii)

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(iii) (a) =se a truth table to determine the values of (pr) q.

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* second light bulb is selected at random and put into the light fitting in room @.

(b) actly two defective batteries0(3)

(b) at least one defective battery.(2)

(Total 5 marks)

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7. /n each of the Venn diagrams, shade the region indicated.

(a) A%

A %

(b) The complement of (A%)

A %

(c) The complement of (A%)

A %

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(d) *(@ )

A %

&

Working:

(Total 4 marks)

8. * bag contains red, ! yellow and - green sweets.

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9. Three propositionsp, qand rare defined as follows'

p' the water is cold. q' the water is boiling. r' the water is warm.

(a)

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10. /n a school, 1# pupils are asAed which is their favourite outdoor sport in winter. The pie chartshows the result of the survey. The diagram is notaccurately drawn.

5 o o t b a l l( - $ )

% o c A e y( " ! )

+ u g b y( ! # )

D e t b a l l( ! " )

? t h e r

(a) alculate the angle of the sector representing rugby.

(b) Estimate the probability that a pupilFs favourite outdoor sport in winter will be hocAey.

Working:

Answers'

(a) ..

(b) ..

(Total 4 marks)

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11. 2et

6 7positive integers less than 1-:0

*6 7multiples of :0

6 7multiples of !:.

(a) Show, in a Venn diagram, the relationship between the sets ,*and +.(1)

(b) 2ist the elements of'

(i) * +(1)

(ii) * +.

(2)

(c) 5ind the number of elementsin the complement of (*+).(2)

(Total 6 marks)

12. Three propositions are defined as follows'

p' The oven is worAing.

q' The food supply is adeuate.

r' The visitors are hungry.

(a)

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(c) * tautology is a compound statement which is always true. =se a truth table to determinewhether or not your answer to part (b) is a tautology.

Hint' @egin by writing the first two columns of your truth table in the following format'

p q

T T

T 5

5 T

5 5

(3)

(Total 9 marks)

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13. onsider the following statements'

p' Good mathematics students go to good universities

q' Good music students are good mathematics students

r' Students who go to good universities get good Hobs

(a) 5rom these statements, write two valiconclusions.

(b)

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14. * group of - females were asAed how many children they each had. The results are shown inthe histogram below.

1 ,

4

#

$

-

"

!

1

,

, 1 ! "D u m b e r o f h i l d r e n

5 r e u e n c y

D u m b e r o f h i l d r e n p e r 5 e m a l e

(a) Show that the mean number of children per female is 1.".(2)

(b) Show clearly that the standard deviation for this data is appro>imately 1.$.(3)

(c) *nother group of - females was surveyed and it was found that the mean number ofchildren per female was ." and the standard deviation was . =se the results from parts(a) and (b) to describe the differences between the number of children the two groups offemales have.

(2)

(d) * female is selected at random from the first group.

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(e) Two females are selected at random from the first group.

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16. 2et the propositionsp, qand rbe defined as'

p' 3atthew arrives home before si> oFclocAq' 3atthew cooAs dinnerr' Iill washes the dishes

(a) (i) E>press the following statement in logical form.

/f 3atthew arrives home before si> oFclocA then he will cooA dinner.(1)

(ii) plain the significance of the truth table above.(2)

(Total 9 marks)

17. Dene and ;eAa both play netball. The probability that Dene will score a goal on her first attemptis .-. The probability that ;eAa will score a goal on her first attempt is .#.

alculate the probability that

(a) Dene and ;eAa will both score a goal on their first attempts0

(b) neither Dene nor ;eAa will score a goal on their first attempts.

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Working:

Answers'

(a) ..

(b) ..

(Total 4 marks)

18. The propositionspand qare defined as follows'

p: #o! ha,e !n$ersoo$ his opic

q: #o! will )e a)le o $o his q!esion

(a)

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(b) E>plain, in words only, what the following symbolic proposition represents'

(p q) p.

Working:

Answers'

(a) ..

(b) ..

(Total 4 marks)

19. /n a club with $ members, everyone attends either on Tuesday for ;rama (0) or on Thursdayfor Sports (S) or on both days for ;rama and Sports.

?ne weeA it is found that "# members attend for ;rama and "" members attend for Sports and xmembers attend for both ;rama and Sports.

(a) (i) ;raw and label full!a Venn diagram to illustrate this information.(3)

(ii) 5ind the number of members who attend for both ;rama and Sports.(2)

(iii) ;escribe, in words, the set represented by (0S)L.(2)

(iv)

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The club has # female members, # of whom attend for both ;rama and Sports.

(b)

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21. Two Hars contain a number of coloured balls as indicated in the diagrams below.

@ l a c A! < h i t e

@ l a c A1 < h i t e

"ar #ne "ar T$o

Two e>periments are carried out.

Firs 1xperimen' * Har is first chosen at random and then a ball is drawn from that Har.

(a) ;raw, an label full!, a tree diagram to show allpossible outcomes of this e>periment.(2)

(b)

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follows'

@illiards SnooAer ;arts 5encing

3ale " 1- # 1

5emale 1 1 1 !

(b) /f the test were applied to this new set of data,

(i) why would it be necessary to combine billiards with another game&

(ii) which other game would you combine with billiards and why&(2)

* club member is to be selected at random.

(c)

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(b) Clace the elements ofAand%in the appropriate region in the Venn diagram below.

2

A %

Working:

Answers'

(a) (i) (ii) ............................................................

(Total 4 marks)

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24. * bag contains two red sweets and three green sweets. Iacues taAes one sweet from the bag,notes its colour, then eats it. %e then taAes another sweet from the bag.

omplete the tree diagram below to show all probabilities.

+ e d

G r e e n

+ e d

G r e e n

+ e d

G r e e n

-

"

Working:

(Total 4 marks)

25. /f each of the following compound propositions is true, what conclusions can be made&

(a) x9 orx!, andx9

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(b) p6 ! if and only if q6 -, and if q- then r1.

Working:

Answers'

(a) ..

(b) ..................................................................

(Total 4 marks)

26. /n a survey of - students it was found that ! study Spanish and 1- have computers. Seven ofthe students who study Spanish also have computers.

(a) %o&!and complete this table.

Study

Spanish

;o not study

Spanish Total

%ave computers

;o not have computers

Total -

(3)

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(b) ;raw and label full!a Venn diagram to illustrate this information. =se 2to represent theset of all students surveyed, Sthe set of students who study Spanish and &the set ofstudents who have computers.

(2)

(c) ;escribe, in words, the set represented by &S .(2)

(d) 5ind n(&S ).(1)

* student is selected at random to attend a computer worAshop given in Spanish.

(e)

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27. J(p q) pK q

(a) omplete the truth table below for the compound statement above.

p q p

q (p

q) p J(p

q) pKqT T

T '

' T

' '

(b) E>plain the significance of your result.

Working:

Answers'

(b) ....................................................................................................................................

(Total 4 marks)

28. Two propositionspand qare defined as follows'

p: hen!m)eren$sin3ero

q' hen!m)eris$i,isi)le)#4

(a)

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(b)

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(ii)

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(b) omplete the following truth table for the logic statement p q.

p q p pq

T T

T 5

5 T

5 5

Working:

Answers'

(a) (i) ...

(ii) ...

(Total 4 marks)

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31. * committee 2has three subMcommittees' researchR, financeFand purchasingP. Do memberbelongs to both finance and purchasing subMcommittees. Some members belong to both researchand purchasing committees. *ll members of the finance subMcommittee also belong to theresearch subMcommittee.

;raw a Venn diagram, showing the relationship between the sets 2,R,FandP.

Answer:

(Total 4 marks)

32. * teacher has a bo> containing si> type * calculators and four type @ calculators.

The probability that a type * calculator is faulty is .1 and the probability that a type @calculator is faulty is .1.

(a) omplete the tree diagram given below, showing all the probabilities.

5 * = 2 T

D ? T 5 * = 2 T

5 * = 2 T

D ? T 5 * = 2 T

, . 1

t y p e *

t y p e @

, . $

, . "

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(b) * calculator is selected at random from the bo>. 5ind the probability that the calculator is

(i) a faulty type *0

(ii) not faulty.

Working:

Answers'

(b) (i) ...

(ii) ...

(Total 4 marks)

33. 2etFbe the set of all families that have e>actly children.

(a) *ssuming C(boy) 6 C(girl), copy and complete the following tree diagram, for familieswith children.

@ o y

G i r l

@ o y

G i r l@ o y

G i r l

1

(2)

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(b) actly

(i) boys&

(ii) boys, if it is Anown that the first child is a boy&

(iii) boys, if it is Anown that there is a boy in the family&(3)

(Total 5 marks)

34. 1 students were asAed which television channel (3TV, DD or @@) they had watched theprevious evening. The results are shown in the Venn diagram below.

2

3 T V D D

@ @

! -

$ -

!

! 1 4

5rom the information in the Venn diagram, write down the number of students who watched

(a) both 3TV and @@0

(b) 3TV or @@0

(c) DD and @@ but not 3TV0

(d) 3TV or DD but not @@.(Total 4 marks)

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35. 2etpand qbe the statements

p: #o! wach he m!sic T- channel

q: #o! like m!sic

(a) onsider the following logic statement.

I( #o! wach he m!sic T- channel hen #o! like m!sic.

(i)

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(b) ?nly one of the statements in part(a) is true.

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(a) (i) %ow many students belong to the region labelledA&

(ii) ;escribe in words the region labelledA.

(iii) %ow many students do not study any of the three subHects&

(5)

(b) ;raw a sAetch of the Venn diagram above and shade the region which represents S H.(1)

(c) alculate n(S H).(2)

This group of students is to compete in an annual uiO evening which tests Anowledge of3athematics, Science and %istory. The names of the twenty students are written on pieces of

paper and then put into a bag.

(d) ?ne name is randomly selected from the bag. alculate the probability that the studentselected studies

(i) all three subHects0

(ii) %istory or Science.(2)

(e) * team of two students is to be randomly selected to compete in the uiO evening. Thefirst student selected will be the captain of the team. alculate the probability that

(i) the captain studies all three subHects and the other team member does not study anyof the three subHects0

(ii) one student studies Science only and the other student studies %istory only0

(iii) the second student selected studies %istory, given that the captain studies %istoryand 3athematics.

(5)

(Total 15 marks)

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38. * poll was taAen of the leisure time activities of 4 students.

$ students watch TV (T), $ students read (R), students go to the cinema (&).$ students watch TV, read ango to the cinema. students watch TV and go to the cinema only.

1# students read and go to the cinema only.1 students read and watch TV only.

(a) ;raw a Venn diagram to illustrate the above information.

(b) alculate how many students

(i) only watch TV0

(ii) only go to the cinema.

0iagram:

Working:

Answers'

(b) (i) ..

(ii) ..

(Total 8 marks)

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39. onsider the following statements.

p: s!$ens work har$

q: s!$ens will s!ccee$

(a)

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(b)

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41. /n the Venn diagram below,A,%and &are subsets of a universal set 26 71,,!,",$,,#,4:.

2 %

A &

!

"1

#

4

$

2ist the elements in each of the following sets.

(a) A%

(b) A%&

(c) (AF&) %

Working:

Answers'

(a) ..................................................................

(b) ..................................................................

(c) ..................................................................

(Total 8 marks)

42. The sets 2,P,Rand Sare defined as follows'

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26 7all uadrilaterals:P6 7all parallelograms:R6 7all rectangles:S6 7all suares:

(a) ;raw a Venn ;iagram illustrating the relationships of the above sets.(4)

(b) ;raw a separate Venn ;iagram for each of the e>amples below./ndicate by shading each of the following'

(i) (P S)

(ii) (R S) P(4)

(Total 8 marks)

43. onsider each of the following statements'

p' *le> is from =ruguayq' *le> is a scientistr' *le> plays the flute

(a)

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(c) onstruct a truth table for the argument in part (b) using the values below forp, q, rand

r. Test whether or not the argument is logically valid.

p q r r

T T T 5

T T 5 T

T 5 T 5

T 5 5 T

5 T T 5

5 T 5 T

5 5 T 5

5 5 5 T

(4)(Total 10 marks)

44. 5ifty students at 2ayton %igh School recorded how much money each student in their classspent on video rentals this month (to the nearest dollar). The results are shown in the freuencytable below'

%lass interval in ounaries in 're*uen+!

1P1 .-P1.- 1

11P 1.-P.-

1P! .-P!.- 1

!1P" !.-P".-

"1P- ".-P-.- "

-1P$ -.-P$.-

$1P $.-P.- "

(a) ?n graph paper using a scale of cm to represent each interval (Q1.) on the horiOontala>is and 1 cm to represent - people on the vertical a>is, draw and clearly label afreuency histogram which displays the above information.

(5)

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(b) *nswer the following uestions'

(i) t month a student will spend an amount in the class interval'

(i) 5rom Q1 to Q! inclusive on video rentals.

(ii) Q! or less on video rentals.

(iii) 5rom Q"1 to Q$ on video rentals, given that they spent more than Q on videorentals.

(iv) Dot more than Q$ on video rentals, given that they spent over Q1 on videorentals.

(6)

(Total 13 marks)

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45. onsider two propositionspand q. omplete the truth table below for the compoundproposition.

(p q) (p q)

p q p q p q p q (p q) (p q)

T T 5 5 5 () T

T 5 5 T (b) 5 (f)

5 T (a) 5 (+) T (,)

5 5 T T 5 (e) (-)

Working:

(Total 8 marks)

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46. The following Venn ;iagram shows the sets 2,A,%and &.

2

A %

&

State whether the following statements are true or false for the information illustrated in theVenn ;iagram.

(a) A&6

(b) &% 7 &

(c) &(A%)

(d) A &

Working:

Answers'

(a) ..................................................................

(b) ..................................................................(c) ..................................................................

(d) ...................................................................

(Total 8 marks)

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47. Today Chilip intends to go walAing. The probability of good weather (G) is "

!

. /f the weather is

good, the probability he will go walAing (

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(b) onsider the logic statements.

p' xR ! 6 - q'x6x

The compound proposition xR ! 6 - x6xis given.

/s this compound proposition true&

(c) ample to show that the converse is false.

Working:

Answers'

(a) ..................................................................

(b) ..................................................................

(c) ..................................................................

..................................................................

(d) ..................................................................

..................................................................

(Total 8 marks)

49. ?n a particular day 1 children are asAed to maAe a note of what they dranA that day.

They are given three choices' water (

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(c) * child is chosen at random. 5ind the probability that the child dranA

(i) coffee0

(ii) water or fruit Huice but not coffee0

(iii) no fruit Huice, given that the child did drinA water.

(4)

(d) Two children are chosen at random. 5ind the probability that both children dranA all threechoices.

(3)

(Total 13 marks)

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50. 2et 26 7P", !

P

, 1, , 1!, $., $4, 1!!

:.

Ais the set of all the integers in 2.

%is the set of all the rational numbers in 2.

(a) 2ist all the prime numbers contained in 2.

(b) 2ist all the members ofA.

(c) 2ist all the members of%.

(d) 2ist all the members of the setA%.

Working:

Answers'

(a) ..................................................................

(b) ..................................................................(c) ..................................................................

(d) ..................................................................

(Total 8 marks)

51. 2etpand qbe the statements'

p: Sarah eas los o( carros.

q: Sarah can see well in he $ark.

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I( Sarah canno see well in he $ark" hen she $oes no ea los o( carros.

(d) /s the statement in part (c) the in,erse, the con,erseor the conraposii,eof the statementin part (a)&

Working:

Answers'

(a) ....................................................................................................................................

(b) ..................................................................

..................................................................

(c) ..................................................................

(d) ..................................................................

(Total 8 marks)

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52. Two identical dice have sides numbered one to si>. T-e i+e are $ei,-te. *ll the numberse>cept the four have eual probability of appearing on top. The four is three times as liAely aseach of the other numbers to appear on top.

The tree diagram below shows some of the probabilities.

a#

1#

1#

)#

1#1

#

o n e

t w o

t h r e e

f o u r

f i v e

s i >

(a) 5ind the values of aand ) in the diagram.

(b) @oth dice are thrown. alculate the probability that two fours appear on top.

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(c) ?ne of the dice is thrown once. The result is not a two or a three. &

Working:

Answers'

(a) ..................................................................

(b) ..................................................................

(c) ..................................................................

(Total 8 marks)

53. There are two biscuit tins on a shelf. The retin contains three chocolate biscuits and sevenplain biscuits. The bluetin contains one chocolate biscuit and nine plain biscuits.

(a) * child reaches into the retin and randomly selects a biscuit. The child returns that

biscuit to the tin, shaAes the tin, and then selects another biscuit.

5ind the probability that

(i) both biscuits chosen are chocolate.(2)

(ii) one of the biscuits is plain and the other biscuit is chocolate.(3)

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(b) * second child chooses a biscuit from the bluetin. The child eats the biscuit and choosesanother one from the bluetin. The tree diagram below represents the possible outcomesfor this event.

11 ,

41 ,

a

b

C

C

C

(i)

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54. Shade the given region on the corresponding Venn ;iagram.

(a) A %

2

* @

(b) & %

2

* @

(c) (A % &)

2

* @

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(d) A &

2

* @

(Total 8 marks)

55. onsider the following logic statements'

p' the train arrives on timeq' / am late for school

(a) pressionp q as a logic statement.

(b)

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(d) *re the two compound propositions (pq) and (p q) logically euivalent&

Working:

Answers'

(a) ..................................................................

..................................................................

(b) ..................................................................(d) ..................................................................

(Total 8 marks)

56. The probability,p, that Iames gets up before . is .4-./f Iames gets up before ., the probability, , that he arrives at school on time is .4#./f Iames gets up later than ., the probability that he arrives at school on time is .--.

The above information is represented by the following tree diagram.

n o t p

n o t

n o t

p

(a) omplete the tree diagram.

(b) alculate the probability that Iames gets up before . and is on time for school.

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(c) alculate the probability that Iames does notarrive at school on time.

Working:

Answers'

(b) ..................................................................

(c) ..................................................................

(Total 8 marks)

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57. The figure below shows the set 2of all functions ofx.

Ais the set of all sine and cosine functions of x(measured in degrees).

%is the set of all functions ofxwith period 1.

&is the subset ofA, containing functions whose amplitude is !.

2

A

&

%

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W o r k i n g :

(Total 6 marks)

58. Deil has three dogs. Two are brown and one is grey.

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59. There are "4 mice in a pet shop.

! mice are white.

mice are male.

1# mice have short tails.

# mice are white and have short tails.

11 mice are male and have short tails.

mice are male but neither white nor shortMtailed.

- mice have all three characteristics and

have none.

opy the diagram below to your e>amination script.

W r e p r e s e n t s w h i t e m i c e .

6 r e p r e s e n t s m a l e m i c e .

S r e p r e s e n t s s h o r t M t a i l e d m i c e .

S W

6

2

!

-

(a) omplete the diagram, using the information given in the uestion.(4)

(b) 5ind (i) n(6W)

(ii) n(6U S)(3)

Two mice are chosen without replacement.

(c) 5ind C (both mice are shortMtailed).(2)

(Total 9 marks)

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60. The universal set 2is the set of integers from 1 to inclusive.

Aand%are subsets of 2where'

Ais the set of even numbers between and 1.

%is the set of multiples of !.

2ist the elements of the following sets'

(a) A0(1)

(b) %0(1)

(c) A%0

(2)

(d) A%U.(2)

W o r k i n g :

A n s w e r s :

( a ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

( b ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

( c ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

( d ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(Total 6 marks)

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61. * school offers three activities, basAetball (%), choir (&) and drama (0). Every student mustparticipate in at least one activity.

1$ students play basAetball only.

1# students play basAetball and sing in the choir but do not do drama.

!" students play basAetball and do drama but do not sing in the choir.

students are in the choir and do drama but do not play basAetball.

(a) Enter the above information on the Venn diagram below.(2)

x

@

;

=

44 of the students play basAetball, ## sing in the choir and 11 do drama.

(b) alculate the number of studentsxparticipating in all three activities.

(1)

(c) alculate the total number of students in the school.(3)

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W o r k i n g :

A n s w e r s :

( c ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

( b ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(Total 6 marks)

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62. The table below shows the number of words in the e>tended essays of an /@ class.

umber of

$ors!w

5

r

e

)

u

e

n

c

y

D u m b e r o f w o r d s

,

(3)

(b) imum word count is " words.

(c)

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63. The following histogram shows the weights of a number of froOen chicAens in a supermarAet.

The weights are grouped such that 1 weight

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A n s w e r s :

W o r k i n g :

( a ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

( b ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

( c ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(Total 6 marks)

64. (a) Shade (A%) &U on the diagram below.

A % &2

(2)

(b) /n the Venn diagram below, the number of elements in each region is given.

5ind n ((PQ) R).

P ! 1 Q

"

$

-

4

R

2

(2)

(c) = is the set of positive integers,R.

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1is the set of even numbers.

6is the set of multiples of !.

(i) 2ist the first si> elements of the set6.

(ii) 2ist the first si> elements of the set1U 6.

(2)

W o r k i n g :

A n s w e r s :

( b ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

( c ) ( i ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

( i i ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(Total 6 marks)

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65. laire and ate both wish to go to the cinema but one of them has to stay at home to babyMsit.

The probability that ate goes to the cinema is .. /f ate does not go laire goes.

/f ate goes to the cinema the probability that she is late home is .!.

/f laire goes to the cinema the probability that she is late home is .$.

(a) opy and complete the probability tree diagram below.

, .

, . !

2 a t e

2 a t e

D o t 2 a t e

D o t 2 a t e

a t e

l a i r e. . .

. . .

. . .

. . .

(3)

(b) alculate the probability that

(i) ate goes to the cinema and is not late0

(2)

(ii) the person who goes to the cinema arrives home late.(3)

(Total 8 marks)

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66. * survey of " people is carried out by a marAet research organiOation in two different cities,@uenos *ires and 3ontevideo. The people are asAed which brand of cereal they prefer out ofhocos, Wucos or 5ruti. The table below summariOes their responses.

hocos Wucos 5ruti Total

@uenos *ires "! #- $ 14

3ontevideo - !- 11# 1

Total 1 1 1# "

(a) ?ne person is chosen at random from those surveyed. 5ind the probability that this person

(i) does not prefer Wucos0

(ii) prefers hocos, given that they live in 3ontevideo.(4)

(b) Two people are chosen at random from those surveyed. 5ind the probability that they bothprefer 5ruti.

(3)

The marAet research organiOation tests the survey data to determine whether the brand of cereal

preferred is associated with a city. * chiMsuared test at the -%level of significance is

performed.

(c) State the null hypothesis.(1)

(d) State the number of degrees of freedom.(1)

(e) Show that the e>pected freuency for the number of people who live in 3ontevideo andprefer Wucos is $!.

(2)

(f)