semester test 2: 30 april 2010 - varsity field · semester test 2: 30 april 2010 time: ......

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University of Pretoria

Faculty of Economic and Management Sciences Department of Statistics Statistics 110 (STK110)

Semester Test 2: 30 April 2010

Time: 75 minutes Total: 50 Marks Internal Examiners: Ms. J Coetsee Dr. G Crafford Ms. M Graham Ms. F Reyneke Mr. P van Staden External Examiner: Prof F Steffens Instructions:

• Answer all questions. • Complete the multi-choice answer sheet with pencil on Side 1. • Check that your student number is filled in correctly on the multi-choice

answer sheet.

• Do calculations on the counter page of the question paper if necessary.

• List of formulae and tables are attached.

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Outeursreg Voorbehou

Universiteit van Pretoria Fakulteit Ekonomiese en Bestuurswetenskappe

Departement Statistiek Statistiek 110 (STK110)

Semestertoets 2: 30 April 2010

Tyd: 75 minute Totaal: 50 Punte Internal Examiners: Me. J Coetsee Dr. G Crafford Me. M Graham Me. F Reyneke Mnr. P van Staden External Examiner: Prof F Steffens Instruksies:

• Beantwoord al die vrae. • Vul die meerkeuse antwoordblad op Kant 1 met potlood in.

• Kontroleer dat jou studentenommer korrek op die meerkeuse antwoordblad ingevul is.

• Doen berekeninge op die teenblad van die vraestel indien nodig. • Lys van formules en tabelle is aangeheg.

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Test Regulations

1. No candidate may enter the test hall later than half an hour after the test

session has commenced and no candidate may leave the test hall earlier than half an hour after the test session has commenced.

2. Candidates are obliged to execute immediately all instructions given by an

invigilator.

3. Candidates may not take into the test hall or have in their possession any unauthorized apparatus, books, notes or paper of whatever nature or size.

4. Once the invigilator has announced that the test has commenced, all conversation or any other form of communication between candidates must cease. During the course of the test there may be no communication of any nature between candidates.

5. No candidates may assist or attempt to assist another candidate, or obtain

assistance or obtain assistance from another candidate with regard to any information.

6. Candidates shall not act dishonestly in any respect.

7. Writing on any paper other than that supplied for test purposes is strictly

prohibited.

8. Rough work may be done on the counter page of the test paper.

9. Smoking is not permitted in the test hall and candidates may not leave the test hall during the test in order to smoke outside.

10. Only in exceptional circumstances will a candidate be given permission to

leave the test hall temporarily, and then only under the supervision of an invigilator.

11. Candidates may not remove used or unused answer sheets from the test hall.

12. When the invigilator announces that the time has expired, candidates must

stop writing immediately. PLEASE NOTE:

Candidates are seriously warned about contravening any of these instructions and that, if found guilty, a candidate could forfeit all credits for a whole year and also be suspended from the University (and consequently from all South African universities) for a period that could range from one year to permanent suspension.

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Toetsregulasies

1. Geen kandidaat sal later as ‘n halfuur na die aanvang van ‘n toetssessie tot die toetslokaal toegelaat word nie en geen kandidaat mag die toetslokaal vroeër as ‘n halfuur na die aanvang van die toetssessie verlaat nie.

2. Kandidate is verplig om alle opdragte van ‘n opsiener onmiddellik na te kom.

3. Kandidate mag geen ongemagtigde apparaat, boeke, aantekeninge van

welke aard ookal of enige papier hoe gering ookal, in die toetslokaal inneem nie of in hulle besit hê nie.

4. Nadat die opsiener aangekondig het dat die toets begin, moet alle gesprekke

of enige ander wyse van kommunikasie tussen studente gestaak word. Tydens die duur van toets mag daar geen gedagtewisseling van welke aard ookal tussen kandidate plaasvind nie.

5. Geen kandidaat mag ‘n ander kandidaat help of probeer help, of hulp verkry,

of probeer om hulp van ‘n ander kandidaat te verkry met betrekking tot enige inligting nie.

6. Kandidate mag hulle nie aan enige oneerlikhede skuldig maak nie.

7. Dit is ten strengste verbied om op enige ander papier te skryf as dié wat vir

toetsdoeleindes verskaf word.

8. Rofwerk mag op die teenblad van die vraestel gedoen word.

9. Binne die toetslokaal mag daar nie gerook word nie en kandidate mag nie gedurende die toets die lokaal verlaat met die doel om buitekant te rook nie.

10. Slegs in buitengewone omstandighede sal aan ‘n kandidaat toestemming

verleen word om die toets tydelik te verlaat en dan slegs onder toesig van ‘n opsiener.

11. Kandidate mag nie gebruikte of ongebruikte antwoordblaaie uit die toetslokaal

uit neem nie.

12. Sodra die opsiener aankondig dat die tyd verstreke is, moet kandidate onmiddellik ophou skryf.

LET WEL: Kandidate word ernstig gewaarsku teen oortreding van hierdie instruksies en daarop gewys dat by skuldigbevinding ‘n kandidaat alle krediet vir ‘n hele jaar se werk verbeur en daarbenewens uit die Universiteit (en gevolglik alle Suid-Afrikaanse universitêre inrigtings) geskors kan word vir ‘n tydperk wat kan wissel van een jaar tot permanente skorsing.

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Questions 1 to 13 are based on the following information:

It is known that 30% of all foreign visitors coming to South Africa during the World

Cup will hire a car while in South Africa. A random sample of 20 foreigners were

contacted and asked whether they will hire a car while in South Africa.

Let � �the number of foreign visitors who will hire a car.

�̅ �the sample proportion of foreign visitors who will hire a car.

� �the population proportion of foreign visitors who will hire a car.

Consider the following information provided in the Excel spreadsheet:

Formula worksheet:

Value worksheet:

Note: The formula of row 5 is copied up to row 25.

Rows 6-7 and 16-25 are hidden.

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Vrae 1 tot 13 is gebaseer op die volgende inligting:

Dit is bekend dat 30% van alle buitelandse besoekers wat na Suid-Afrika toe kom vir

die Wêreldbeker ‘n kar gaan huur terwyl hulle in Suid-Afrika is. ‘n Ewekansige

steekproef van 20 buitelandse besoekers is gekontak en gevra of hulle ‘n kar gaan

huur terwyl hulle in Suid-Afrika is.

Laat � �die aantal buitelandse besoekers wat ‘n kar gaan huur.

�̅ �die steekproefverhouding van buitelandse besoekers wat ‘n kar gaan huur.

� � die populasieverhouding van buitelandse besoekers wat ‘n kar gaan huur.

Beskou die volgende inligting wat beskikbaar is in die Excel spreiblad:

Formule werkblad:

Waarde werkblad:

Let wel: Die formule van ry 5 is gekopieer tot by ry 25.

Rye 6-7 en 16-25 is versteek.

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Question 1

The probability that at least seven of the 20 foreign visitors will hire a car is:

(A) 0.7723 (B) 0.2277

(C) 0.6080 (D) 0.3920

(E) 0.1916

(2)

Question 2

��3 < � < 9� �

(A) 0.8449 (B) 0.6492

(C) 0.0428 (D) 0.7145

(E) 0.7796

(2)

Question 3

The value of the Excel function

=SUMPRODUCT(A5:A25,B5:B25)

in the spreadsheet above is:

(A) 5.74 (B) 6.00

(C) 52.15 (D) 37.90

(E) 10.00

Hint: Make use of basic principles.

(1)

Question 4

The standard deviation of � is:

(A) 0.0105 (B) 2.0494

(C) 4.2000 (D) 0.1025

(E) 6.0000

(1)

Question 5

The probability that twelve foreign visitors will not hire a car is:

(A) 0.1144 (B) 0.0039

(C) 0.6000 (D) 0.4000

(E) 0.0308

(2)

Question 6

The probability distribution of � is:

(A) Symmetric (B) Skewed to the left

(C) Skewed to the right (D) Bi-modal

(E) Multi-modal

(1)

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Vraag1

Die waarskynlikheid dat ten minste sewe van die 20 buitelandse besoekers ‘n kar

gaan huur is:

(A) 0.7723 (B) 0.2277

(C) 0.6080 (D) 0.3920

(E) 0.1916

(2)

Vraag 2

��3 < � < 9� �

(A) 0.8449 (B) 0.6492

(C) 0.0428 (D) 0.7145

(E) 0.7796

(2)

Vraag 3

Die waarde van die Excel funksie

=SUMPRODUCT(A5:A25,B5:B25)

in die bogenoemde spreiblad is:

(A) 5.74 (B) 6.00

(C) 52.15 (D) 37.90

(E) 10.00

Wenk: Maak gebruik van basiese beginsels.

(1)

Vraag 4

Die standaardafwyking van � is:

(A) 0.0105 (B) 2.0494

(C) 4.2000 (D) 0.1025

(E) 6.0000

(1)

Vraag 5

Die waarskynlikheid dat twaalf besoekers nie ‘n kar gaan huur nie is:

(A) 0.1144 (B) 0.0039

(C) 0.6000 (D) 0.4000

(E) 0.0308

(2)

Vraag 6

Die waarskynlikheidsverdeling van � is:

(A) Simmetries (B) Skeef na links

(C) Skeef na regs (D) Bi-modaal

(E) Multi-modaal

(1)

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Question 7

The total number of experimental outcomes is:

(A) 400 (B) 184 756

(C) 190 (D) 1 048 576

(E) 380

(1)

Question 8

The total number of experimental outcomes with 2 out of 20 foreign visitors who will

hire a car is:

(A) 400 (B) 184 756

(C) 190 (D) 1 048 576

(E) 380

(1)

Question 9

In a random sample of five foreign tourists it was found that 2 out of the 5 foreign

tourists will hire a car. The sampling error of �̅ is:

(A) 0.3 (B) 0.4

(C) 0.2 (D) 0.5

(E) 0.1

(1)

Question 10

The sampling distribution of �̅ has an approximate normal distribution with mean �

and standard deviation . The values for � and are respectively:

(A)� � 0.30 and � 4.2 (B) � � 0.30and � 0.21

(C) � � 0.30and � 0.1025 (D) � � 0.30and � 0.0105

(E) � � 0.30and � 14.00

(1)

Question 11

��̅ < 0.36� �

(A) 0.7224 (B) 0.5854

(C) 0.2776 (D) 0.6406

(E) 0.3594

(2)

Question 12

20% of the sample proportions,�̅, are greater than________:

(A) 0.2139 (B) 0.3000

(C) 0.2514 (D) 0.3861

(E) 0.8157

(2)

Question 13

The probability that the sampling error of �̅ is more than 0.04 is:

(A) 0.3861 (B) 0.6966

(C) 0.3483 (D) 0.6517

(E) 0.3035

(2)

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Vraag 7

Die totale aantal eksperimentele uitkomste is:

(A) 400 (B) 184 756

(C) 190 (D) 1 048 576

(E) 380

(1)

Vraag 8

Die totale aantal eksperimentele uitkomste waar 2 van die 20 buitelandse besoekers

‘n kar gaan huur, is:

(A) 400 (B) 184 756

(C) 190 (D) 1 048 576

(E) 380

(1)

Vraag 9

In ‘n ewekansige steekproef van vyf buitelandse besoekers is gevind dat 2 uit die 5

buitelandse besoekers ‘n kar gaan huur. Die steekproeffout van �̅ is:

(A) 0.3 (B) 0.4

(C) 0.2 (D) 0.5

(E) 0.1

(1)

Vraag 10

Die steekproefverdeling van �̅ het ‘n benaderde normaalverdeling met ‘n gemiddelde

� en standaardafwyking . Die waardes vir � en is respektiewelik:

(A)� � 0.30 en � 4.2 (B) � � 0.30en � 0.21

(C) � � 0.30en � 0.1025 (D) � � 0.30en � 0.0105

(E) � � 0.30en � 14.00

(1)

Vraag 11

��̅ < 0.36� �

(A) 0.7224 (B) 0.5854

(C) 0.2776 (D) 0.6406

(E) 0.3594

(2)

Vraag 12

20% van die steekproefverhoudings,�� , is groter as________:

(A) 0.2139 (B) 0.3000

(C) 0.2514 (D) 0.3861

(E) 0.8157

(2)

Vraag 13

Die waarskynlikheid dat die steekproeffout van �̅ meer is as 0.04, is:

(A) 0.3861 (B) 0.6966

(C) 0.3483 (D) 0.6517

(E) 0.3035

(2)

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The time (in minutes) travelling from the ‘Park-and-Ride’ facility to Soccer City in

Soweto is uniformly distributed between 20 and 35 minutes.

Let � �time (in minutes) travelling from the ‘Park-and-Ride’ facility.

�̅ �average amount of time (in minutes) travelling from the ‘Park-and-Ride’

facility of 36 randomly selected bus trips.

Given: �� 18.75

Question 14

The probability density function of � is:

(A) ��

�� where 20 ≤ � ≤ 35

0 elsewhere

(B) �� 0 where 20 ≤ � ≤ 35

�

�� elsewhere

(C) ��

�� where 20 ≤ � ≤ 35

0 elsewhere

(D) �� 0 where 20 ≤ � ≤ 35

�

�� elsewhere

(E) ��

�� where 20 ≤ � ≤ 35

0 elsewhere

(1)

Question 15

The probability that the time (in minutes) spent travelling to Soccer City is less than

23 minutes is:

(A) 0.15 (B) 0.80

(C) 0.20 (D) 0.09

(E) 0.66

(1)

Question 16

The probability that the time (in minutes) spent travelling to Soccer City is exactly 25

minutes is:

(A) 0.3333 (B)1.0000

(C) 0.0000 (D) 0.6667

(E) 0.1389

(1)

Question 17

The probability that the time (in minutes) spent travelling to Soccer City is between

23 and 38 minutes is:

(A) 1.00 (B) 0.80

(C) 0.15 (D) 0.20

(E) 0.42

(2)

Question 18

The expected value of � is:

(A) 35.0 (B) 7.5

(C) 27.5 (D) 20.0

(E) 36.0

(1)

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Die tyd (in minute) om vanaf die ‘Parkeer-en-Ry’ fasiliteit na Soccer City in Soweto te

ry is uniform verdeel tussen 20 en 35 minute.

Laat � �tyd (in minute) om vanaf die ‘Parkeer-en-Ry’ fasiliteit te ry.

�̅ �gemiddelde tyd (in minute) van 36 ewekansiggekose busritte wat vanaf

die ‘Parkeer-en-ry’ fasiliteit ry.

Gegee: �� 18.75

Vraag 14

Die waarskynlikheidsdigtheidsfunksie van � is:

(A) ��

�� waar 20 ≤ � ≤ 35

0 andersins

(B) �� 0 waar 20 ≤ � ≤ 35

�

�� andersins

(C) ��

�� waar 20 ≤ � ≤ 35

0 andersins

(D) �� 0 waar 20 ≤ � ≤ 35

�

�� andersins

(E) ��

�� waar 20 ≤ � ≤ 35

0 andersins

(1)

Vraag 15

Die waarskynlikheid dat die tyd (in minute) spandeer om na Soccer City te ry minder

as 23 minute is, is:

(A) 0.15 (B) 0.80

(C) 0.20 (D) 0.09

(E) 0.66

(1)

Vraag 16

Die waarskynlikheid dat die tyd (in minute) spandeer om na Soccer City te ry presies

25 minute is, is:

(A) 0.3333 (B)1.0000

(C) 0.0000 (D) 0.6667

(E) 0.1389

(1)

Vraag 17

Die waarskynlikheid dat die tyd (in minute) spandeer om na Soccer City te ry tussen

23 en 38 minute is, is:

(A) 1.00 (B) 0.80

(C) 0.15 (D) 0.20

(E) 0.42

(2)

Vraag 18

Die verwagte waarde van � is:

(A) 35.0 (B) 7.5

(C) 27.5 (D) 20.0

(E) 36.0

(1)

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Question 19

According to the central limit theorem, the sample mean �̅, will have an

approximated

(A) binomial distribution.

(B) normal distribution.

(C) bi-modal distribution.

(D) uniform distribution.

(E) standard normal distribution.

(1)

Question 20

The standard error of �̅ is:

(A) 3.13 (B) 0.52

(C) 18.75 (D) 0.72

(E) 0.12

(2)

Question 21

��̅ > 26� �

(A) 0.0188 (B) 0.6000

(C) 0.6368 (D) 0.9812

(E) 0.3632

(2)

Question 22

The 95th percentile of �̅ is:

(A) 28.68 (B) 1.645

(C) 1.96 (D) 28.91

(E) 34.62

(2)


Given:! is a standard normal random variable.

Question 23

��−2.45 < ! < 1.45� �

(A) 0.0071 (B) 0.9265

(C) 0.9336 (D) 0.9194

(E) 0.0735

(1)

Question 24

��! < �� 0.35. The value of � is:

(A) 0.64 (B) -0.39

(C) -1.81 (D) 0.36

(E) -2.70

(2)

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Vraag 19

Volgens die sentrale limietstelling het die steekproefgemiddelde �̅, ‘n benaderde

(A) binomiaalverdeling.

(B) normaalverdeling.

(C) bi-modale verdeling.

(D) uniforme verdeling.

(E) standaard normaalverdeling.

(1)

Vraag 20

Die standaardfout van �̅ is:

(A) 3.13 (B) 0.52

(C) 18.75 (D) 0.72

(E) 0.12

(2)

Vraag 21

��̅ > 26� �

(A) 0.0188 (B) 0.6000

(C) 0.6368 (D) 0.9812

(E) 0.3632

(2)

Vraag 22

Die 95ste persentiel van �̅ is:

(A) 28.68 (B) 1.645

(C) 1.96 (D) 28.91

(E) 34.62

(2)


Gegee:! is ‘n standaard normaal stogastiese veranderlike.

Vraag 23

��−2.45 < ! < 1.45� �

(A) 0.0071 (B) 0.9265

(C) 0.9336 (D) 0.9194

(E) 0.0735

(1)

Vraag 24

��! < �� 0.35. Die waarde van � is:

(A) 0.64 (B) -0.39

(C) -1.81 (D) 0.36

(E) -2.70

(2)

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Question 25

In the following probability statement, ��! > #� � 0.55, the value # represents:

(A) the 55th percentile of the standard normal distribution.

(B) the 55th value of the standard normal distribution.

(C) the 45th value of the standard normal distribution.

(D) the 45th percentile of the standard normal distribution.

(E) the 55th !-value of the standard normal distribution.

(1)


The time (in minutes) spent in the queue of a kiosk at a soccer stadium is normally

distributed with a mean of 25 minutes and a standard deviation of 5 minutes.

Let � �time (in minutes) waiting in the queue.

Let �̅ �the average amount of time (in minutes) of 25 randomly selected soccer

spectators waiting in the queue.

Question 26

The probability that a randomly selected spectator will wait more than 35 minutes in

the queue is:

(A) 0.0228 (B) 0.9772

(C) 0.7143 (D) 0.2857

(E) 0.2000

(2)

Question 27

The probability that a randomly selected spectator will wait between 18 and 28

minutes is:

(A) 0.7257 (B) 0.0808

(C) 0.8065 (D) 0.6449

(E) 0.2743

(2)

Question 28

25% of the spectators wait less than __________minutes in queues.

(A) 27.99 (B) 28.35

(C) 27.01 (D) 22.55

(E) 21.65

(2)

Question 29

The expected value and standard deviation of �̅ are:

(A) �$̅ � 25 and $̅ � 1 (B) �$̅ � 25 and $̅ � 5

(C) �$̅ � 5 and $̅ � 1 (D) �$̅ � 5 and $̅ � 0.2

(E) �$̅ � 25 and $̅ � 0.2

(2)

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Vraag 25

In die volgende waarskynlikheidstelling, ��! > #� � 0.55, stel die waarde #:

(A) die 55ste persentiel van die standaard normaalverdeling voor.

(B) die 55ste waarde van die standaard normaalverdeling voor.

(C) die 45ste waarde van die standaard nomraalverdeling voor.

(D) die 45ste persentiel van die standaard normaalverdeling voor.

(E) die 55ste !-waarde van die standaard normaalverdeling voor.

(1)


Die tyd (in minute) spandeer in toue van ‘n kiosk by ‘n sokkerstadion is normaal

verdeel met ‘n gemiddelde van 25 minute en ‘n standaardafwyking van 5 minute.

Laat � �tyd (in minute) in toue gewag.

Laat �̅ �die gemiddelde tyd (in minute) wat 25 ewekansiggekose sokkertoeskouers

in toue wag.

Vraag 26

Die waarskynlikheid dat ‘n ewekansiggekose toeskouer meer as 35 minute in ‘n tou

wag is:

(A) 0.0228 (B) 0.9772

(C) 0.7143 (D) 0.2857

(E) 0.2000

(2)

Vraag 27

Die waarskynlikheid dat ‘n ewekansiggekose toeskouer tussen 18 en 28 minute wag

is:

(A) 0.7257 (B) 0.0808

(C) 0.8065 (D) 0.6449

(E) 0.2743

(2)

Vraag 28

25% van die toeskouers wag minder as __________minute in toue.

(A) 27.99 (B) 28.35

(C) 27.01 (D) 22.55

(E) 21.65

(2)

Vraag 29

Die verwagte waarde en standaardafwyking van �̅ is:

(A) �$̅ � 25 en $̅ � 1 (B) �$̅ � 25 en $̅ � 5

(C) �$̅ � 5 en $̅ � 1 (D) �$̅ � 5 en $̅ � 0.2

(E) �$̅ � 25 en $̅ � 0.2

(2)

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Question 30

The probability that the average amount of time (in minutes) waiting in queue, by a

random sample of 25 spectators, is less than 23 minutes, is:

(A) 0.0228 (B) 0.9772

(C) 0.3446 (D) 0.6554

(E) 0.4000

(2)


According to a magazine article 40% of all foreign visitors who will visit South Africa

during the World Cup will visit the Kruger National Park during their stay in South

Africa. A random sample of 50 foreign visitors were contacted and asked their age

and which game park (Kruger National, Pilanesberg, Other, None) they will visit

while staying in South Africa. The answers were recorded in the following Excel

Spreadsheet:

Let � �the population proportion of foreign visitors who will visit the Kruger National

Park.

Let � � the age of the foreign visitors who will visit a game park.

Formula worksheet:

Value worksheet:

Note: Only the first 11 rows are shown, the rest are hidden.

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Vraag 30

Die waarskynlikheid dat die gemiddelde tyd (in minute) wat 25 ewekansiggekose

toeskouers in toue wag minder as 23 minute is, is:

(A) 0.0228 (B) 0.9772

(C) 0.3446 (D) 0.6554

(E) 0.4000

(2)


Volgens ‘n tydskrif gaan 40% van alle buitelandse toeskouers wat Suid-Afrika

besoek gedurende die Wêreldbeker die Nasionale Krugerwildtuin besoek terwyl hulle

in Suid-Afrika is. ‘n Ewekansige steekproef van 50 buitelandse besoekers is

gekontak en gevra wat hulle ouderdom is en watter wildtuin hulle gaan besoek

(Nasionale Krugerwildtuin, Pilanesberg, Ander, Geen) terwyl hulle in Suid-Afrika is.

Die antwoorde word gegee in die volgende Excel Spreiblad:

Laat � �die populasieproporsie van buitelandse besoekers wat die Nasionale

Krugerwildtuin gaan besoek.

Laat � � die ouderdom van die buitelandse besoekers wat ‘n wildtuin gaan besoek.

Formule werkblad:

Waarde werkblad:

Let wel: Slegs die eerste 11 rye word gewys, die res is versteek.

19

Question 31

The point estimate of the population proportion, �, is:

(A) 0.16 (B) 0.44

(C) 0.24 (D) 0.40

(E) 0.26

(2)

Question 32

The sample proportion is an unbiased estimator since:

(A) &��̅� � � (B) &�� ̅

(C) &��̅� � ��1 − �� (D) &�� ̅�1 − �̅�

(E) &�� '��1 − ��

(1)

Question 33

The point estimate of the population mean of ages, �, is:

(A) 29.0 (B) 31.9

(C) 95.6 (D) 45.8

(E) 38.0

(1)

20

Vraag 31

Die puntberaming van die populasieverhouding, �, is:

(A) 0.16 (B) 0.44

(C) 0.24 (D) 0.40

(E) 0.26

(2)

Vraag 32

Die steekproefverhouding is ‘n onsydige beramer want:

(A) &��̅� � � (B) &�� ̅

(C) &��̅� � ��1 − �� (D) &�� ̅�1 − �̅�

(E) &�� '��1 − ��

(1)

Vraag 33

Die puntberaming van die populasiegemiddelde van ouderdomme , �, is:

(A) 29.0 (B) 31.9

(C) 95.6 (D) 45.8

(E) 38.0

(1)

21

Cumulative probabilities for the standard normal

distribution

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

-3.0 .0013 .0013 .0013 .0012 .0012 .0011 .0011 .0011 .0010 .0010

-2.9 .0019 .0018 .0018 .0017 .0016 .0016 .0015 .0015 .0014 .0014

-2.8 .0026 .0025 .0024 .0023 .0023 .0022 .0021 .0021 .0020 .0019

-2.7 .0035 .0034 .0033 .0032 .0031 .0030 .0029 .0028 .0027 .0026

-2.6 .0047 .0045 .0044 .0043 .0041 .0040 .0039 .0038 .0037 .0036

-2.5 .0062 .0060 .0059 .0057 .0055 .0054 .0052 .0051 .0049 .0048

-2.4 .0082 .0080 .0078 .0075 .0073 .0071 .0069 .0068 .0066 .0064

-2.3 .0107 .0104 .0102 .0099 .0096 .0094 .0091 .0089 .0087 .0084

-2.2 .0139 .0136 .0132 .0129 .0125 .0122 .0119 .0116 .0113 .0110

-2.1 .0179 .0174 .0170 .0166 .0162 .0158 .0154 .0150 .0146 .0143

-2.0 .0228 .0222 .0217 .0212 .0207 .0202 .0197 .0192 .0188 .0183

-1.9 .0287 .0281 .0274 .0268 .0262 .0256 .0250 .0244 .0239 .0233

-1.8 .0359 .0351 .0344 .0336 .0329 .0322 .0314 .0307 .0301 .0294

-1.7 .0446 .0436 .0427 .0418 .0409 .0401 .0392 .0384 .0375 .0367

-1.6 .0548 .0537 .0526 .0516 .0505 .0495 .0485 .0475 .0465 .0455

-1.5 .0668 .0655 .0643 .0630 .0618 .0606 .0594 .0582 .0571 .0559

-1.4 .0808 .0793 .0778 .0764 .0749 .0735 .0721 .0708 .0694 .0681

-1.3 .0968 .0951 .0934 .0918 .0901 .0885 .0869 .0853 .0838 .0823

-1.2 .1151 .1131 .1112 .1093 .1075 .1056 .1038 .1020 .1003 .0985

-1.1 .1357 .1335 .1314 .1292 .1271 .1251 .1230 .1210 .1190 .1170

-1.0 .1587 .1562 .1539 .1515 .1492 .1469 .1446 .1423 .1401 .1379

-0.9 .1841 .1814 .1788 .1762 .1736 .1711 .1685 .1660 .1635 .1611

-0.8 .2119 .2090 .2061 .2033 .2005 .1977 .1949 .1922 .1894 .1867

-0.7 .2420 .2389 .2358 .2327 .2296 .2266 .2236 .2206 .2177 .2148

-0.6 .2743 .2709 .2676 .2643 .2611 .2578 .2546 .2514 .2483 .2451

-0.5 .3085 .3050 .3015 .2981 .2946 .2912 .2877 .2843 .2810 .2776

-0.4 .3446 .3409 .3372 .3336 .3300 .3264 .3228 .3192 .3156 .3121

-0.3 .3821 .3783 .3745 .3707 .3669 .3632 .3594 .3557 .3520 .3483

-0.2 .4207 .4168 .4129 .4090 .4052 .4013 .3974 .3936 .3897 .3859

-0.1 .4602 .4562 .4522 .4483 .4443 .4404 .4364 .4325 .4286 .4247

-0.0 .5000 .4960 .4920 .4880 .4840 .4801 .4761 .4721 .4681 .4641

0 z

Cumulative probability

22

Cumulative probabilities for the standard normal distribution

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359

.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753

.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141

.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517

.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879

.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224

.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549

.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852

.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133

.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389

1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621

1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830

1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015

1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177

1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319

1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441

1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545

1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633

1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706

1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767

2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817

2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857

2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890

2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916

2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936

2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952

2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964

2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974

2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981

2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986

3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990

0 z

Cumulative probability

23

Formulae STK110 (Page 1 of 2)

n

xx

i∑=

N

xi∑=µ

1

)(2

2

−

∑ −=

n

xxs

i

N

xi∑ −=

22

)( µσ

2ss =

2σσ =

np

i

=

100 IQR = Q3 – Q1

100×=x

scv

s

xxz ii

−=

1

))((

−

∑ −−=

n

yyxxs iixy

N

yx yixixy

∑ −−=

))(( µµσ

yx

xyxy

ss

sr =

yx

xyxy

σσ

σρ =

∑

∑=

i

ii

w

xwx

n

Mfx

ii∑=

N

Mf ii∑=µ

1

)( 22

−

∑ −=

n

xMfs

ii

N

Mf ii∑ −=

2)(2 µσ

)1)(2)...(2)(1(! −−= NNNN 1!0 =

)!(!

!

nNn

N

n

NC

Nn

−=

=

)!(

!!

nN

N

n

NnP

Nn

−=

=

∑== )()( xfxxE µ ∑ −== )()()( 22 xfxxVar µσ

xnxpp

x

nxf

−−

= )1()(

npxE == µ)( )1()( 2 pnpxVar −== σ

24

Formulae STK110 (Page 2 of 2)

≤≤

= −

elsewhere0

for)(

1 bxaxf ab

2)(

baxE

+=

12

)()(

2abxVar

−=

µ=)(xE n

xσ

σ =

ppE =)(

n

ppp

)1( −=σ

x

xz

σ

µ−=

p

ppz

σ

−=

±x margin of error µ−x

±p margin of error pp −

nzx

σα 2±

n

stx 2α±

n

ppzp

)1(2/

−± α

n

xz

σ

µ0−=

ns

xt 0µ−

=

n

pp

ppz

)1( 00

0

−

−=

2

2

2

1

2

1

21

nn

xxz

σσ+

−=

2

2

2

1

2

1

21

n

s

n

s

xxt

+

− where

2

2

2

2

2

2

1

2

1

1

2

2

2

2

1

2

1

1

1

1

1

−+

−

+

=

n

s

nn

s

n

n

s

n

s

df

semester test 2: 30 april 2010 - varsity field · semester test 2: 30 april 2010 time: ......

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