bm0011revision notes aug 2014

8/12/2019 Bm0011revision Notes Aug 2014

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BM0011 Final Exam Paper29 August 2014

3 sectionsTHEORY & CALCULATION(weightage = 60%)

Section A (20 marks Compulsory)

5 short questions-All Topics

Section B( 40 marks Compulsory)

2 Questions - 1 Topics A & B Data presentation and analysis; 1 -TOPIC ENORMAL DISTBN.

Section C(40 marks-Choose 2 out of 3 Questions)

1 Question on Topic CProbability

1 Question on Topic DBinomial and Poisson

1 Question on Topic F - Topic F - Sampling Distribution

STUDENTS CAN REFER TO lecture notes, TUTORIALS AND PAST EXAM PAPERS TO

GET AN IDEA OF THE QUESTIONS ASKED AND TOPICS COVERAGE IN VARIOUS

SECTIONS


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TOPIC A &B - Data Presentation & AnalysisCompulsory question

1. THEORY: ALL THE BASIC CONCEPTSPOPULATION,

SAMPLE, PARAMETER, STATISTIC, QUANTITATIVE,

QUALITATIVE, DISCRETE, CONTINUOUS

2. SYMMETRIC, RIGHT SKEWED AND LEFT SKEWED

DISTRIBUTION WITH REASONS

3. CALCULATION, MEAN, MEDIAN AND MODE

UNGROUPED/ GROUPED data

4. CALCULATION : RANGE, IQR, and standard deviation

5. Graphs with tablesHistogram and ogives


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TOPIC C - Probability

1. DEFINITIONS - THEORY QUESTIONSSample Space, MutuallyExclusive, Independent events, etc.

3. CALCULATION USING MULTIPLICATION AND ADDITION RULES

Refer fully worked examples in the lecture notes and tutorial questions


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TOPIC D - Discrete Probability distributions

1. CHARACTERISTICS OF BINOMIAL AND POISSON

application

2. CALCULATION OF MEAN AND STD.DEVIATION FOR

PROBABILITY DISTRIBUTIN, BINOMIAL DISTRIBUTION and

POISSION DISTRIBUTION

3. CALCULATION OF BINOMIAL/POISSON DISTRIBUTION.


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Binomial Distributionsrefer to tutorialquestions for revisionEach experiment/trial has 2 outcomessuccess

(p) or failure (q or 1-p)

Probability of r successes in n trials P(x=r)

P(X) =!

! !

OR P(X) =

Mean = = Standard Deviation =


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Poisson Probability Distribution - refer totutorial questions for revision

Number of events that occur in an interval

P(X) =

!

=

e= 2.71828

For Poisson distribution, the mean and standarddeviation is the same as the events occur at very

small intervals

THUS Variance, =

And standard Deviation = or


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TOPIC E Normal Distribution

Need to know Z table and expressing Probability equationsP(X>..) P(X


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TOPIC A & B Revision QuestionsWorked out EXAMPLE covered intutorial1. A sample of twenty five students marks in a Statistics ICA is given below:

65 70 59 65 67 68 63 62 63 70 72 66 63 66 66 62 70 58 60 61 59 59

67 71 65

FORMa frequency distribution consisting of 5 classes for the above data

tep 1 - arrange the data in ascendingrder

58 59 59 59 60 62 62 62 63 63 63

65 65 65 66 66 66 67 67 68

70 70 70 71 72

tep 2 - Given, No of classes, k

5

tep 3 - class width, W =( L-S)/K= (72-58)/5= 2.8, round up to 3

tep 4 & 5 draw frequency table

Student

marks Frequency (f)

58-60 5

61-63 6

64-66 6

67-69 3

70-72 5

TOTAL 25


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Student

marks Frequency (f)

Class

boundary

Class

mark

(midpt)

Relative

frequency

Cumulative

frequency

58-60 5 57.5-60.5 59 0.2000 0.2000

61-63 6 60.5-63.5 62 0.2400 0.4400

64-66 6 63.5-66.5 65 0.2400 0.680067-69 3 66.5-69.5 68 0.1200 0.8000

70-72 5 69.5-72.5 71 0.2000 1.0000

TOTAL 25 1.0000


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Frequency histogram

0.0000

0.0500

0.1000

0.1500

0.2000

0.2500

0.3000

57.5-61.5 61.5-63.5 63.5-66.5 66.5-69.5 69.5-72.5Frequencyofstudentsmarks

Student marks

frequency Histogram of student

marks

If polygon, just a line chart joining the midpoints


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Student

marks median> mode, the

data set is said to be skewed to the right. It

contains some extreme high / low outliers.

For a data set, mean < median< mode, the

data set is said to be skewed to the left. It

contains some extreme high / low outliers.


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Revision questions to try yourself1. Classify the following as quantitative or qualitative variables:

a) Colour of cars

b) Length of time to move through a maze

c) Classification of police administrations as city, country or state

d) The rating given to pizza in a taste test : poor, good or excellent

e) Number of married respondents in a survey

2. Classify the following as descriptive or inferential statistics:

AC Neilsen uses data from a sample of viewers to give estimates of average

television viewing time per week for all Singaporeans

a) The Singapore DEPT of statistics publish the leading causes of deaths in

year 2012. the estimates are based on a sample of death certificates.

b) The number of votes received by Candidate A in the last student union

office bearer election

c) The PSI index at 12noon on 25 June 2013.

3. Classify the following as sample or population:

a) All diabetics in Singapore

b) 374 individuals selected for a MediaCorp news poll

c) All owners of Toyota cars in Singapore

d) The registered voters in Ang Mo Kio GRC

e) 220 participants in a health study last month


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More Revision Practice Questions if you want to try or you can get other past

year papers on Blackboard

2. The following table shows the medicine sold by a pharmacy to 30 patients in

a week.

Sale of medicine ($) Number of patients (f)

21-27 228-34 4

35-41 7

42-48 8

49-55 5

59-62 4

Compute

(i) the class boundary

(ii) Class mark

(iii) Relative Frequency

(iv) Cumulative relative frequency


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TOPIC C PROBABILITY REVISION QUESTIONSin class practice


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TOPIC C PROBABILITY REVISION QUESTIONSin class

TT HH

TH

HT

ANS

(a) S = {HH, HT, TH, TT}

(b )

c) P( one head) = P(TH) + P(HT) =0.25 +0.25 = 0.5


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Revision in class

Answer next page


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ANS:(a) Empirical or experiment since it can be tested

(b) Classical, since probability is known in advance

(c) Classical, since probability is known in advance

(d) Subjective, based on someones opinion


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Revision questions for self practice:Determine if each of the following events are mutually exclusive

events?

Answer next page


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ANS:

(a) NOT mutually exclusive,

(b) Not mutually exclusive(c ) YES

(d) NOT


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Revision questions for self practice:AT a semiconductor plant, 60% of workers are skilled, 80% of the workers

are full time, 90% of the skilled workers are full time.

a) What is the probability that a an employee selected at random is a skilled full

me employee?

b) What is the probability that an employee selected at random is a skilled workeror a full time worker?

c ) What percentage of full time workers are skilled?


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Revision questions for self practice:AT a semiconductor plant, 60% of workers are skilled, 80% of the workers

are full time, 90% of the skilled workers are full time.

a) What is the probability that a an employee selected at random is a skilled full

me employee?

b) What is the probability that an employee selected at random is a skilled workeror a full time worker?

c ) What percentage of full time workers are skilled?

P (S) = 0.6, P (F)= 0.8, P(F I S) =0.9

P( S and F) = P(S) x P(F I S) = 0.6 x 0.9 = O.54

P(S or F) = P(S) + P(F)P (S and F) = 0.6 + 0.8 -0.54 = 0.86P(S I F) = P( S and F)/P(F) = 0.54/0.8 = 0.675


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Promoted No Yes

No 350 80

Yes 50 20

Minority

Revision questions for self practice:Question:

The table below classifies the 500 members of a police department according

to their status and their promotional status during the past year.

(a) Find the probability that a randomly selected individual from the police

dept is a minority

(b) Find the probability that a random selected individual was promoted last

year

(c )Find the probability that the selected officer is not a minority and was not

promoted

(d) Find the probability that the selected officer is a minority and was

promoted

(e) Is Being a Minority and getting promoted mutually exclusive?

Ans: (a) 0.2 (b) 0.14 (c ) 0.70 (d) 0.04


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Minority

Promoted No Yes Total

No 350 80 430

Yes 50 20 70

Total 400 100 500

(a) P(Minority) = 100/500 0.2

(b) P(Promoted)= 70/500 0.14

( c ) P (Not M and not P)= 350/500 0.7

(d) P( M and P) = 20/500 0.04

( e) Not mutually exclusive, as there are 20 minorities that are promoted or P(M and P) is not equal to 0

ANSWER:


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TOPIC DDiscrete , Binomial and Poisson Distributionrefer to Past year papers

On BB for revision questions

TOPIC E & Frefer following pages for revision


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Revisionshort topic D standard normal distribution (Z) questionsFind the value of c in each of the following probability statements.

Support your answers with appropriate diagrams.

(i) P (Z>c)=0.88

(ii) P (Z


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Normal Distribution Revisionshort questionsFind the value of c in each of the following probability statements. Support your answers with appropriate diagrams.

(i) P (Z>c)=0.88

(ii) P (Z


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Sampling Distribution Revision question

The net weight of cans of salmon is normally distributed with a mean of

60.5 grams and a standard deviation of 1.8 grams. An officer from

CASE selected 36 cansfor testing. Calculate the probability that themean net weightis less than 59.7 grams.

Solution: refer to next page


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Solution


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QUESTION 12 ( PAGE 33 lecture notes) - Binomial tutorial questions donein class previously

(a) X= random variable = no of cars with engine malfunction; follows a binomialdistribution with n= 10, p=0.02, q=0.98

(i) P(X=3) =103(0.02)3

(0.98)7

= 0.0008

(ii) P(X=0) = 10(0.02)(0.98)= 0.8171

(iii) Mean no of cars that will not have engine malfunction = np =10 x 0.98 =9.8

cars

S= = 100.020.98= 0.196 cars


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QUESTION 13 ( PAGE 33 lecture notes)Poisson tutorial questionsX= no of complaints in MacDonald restaurant with

=

= 1

For 3 days, , =

Using formula P(X) =

!

(i) P(X=0) =3(.7)

!= 0.0498

(ii) In a 5 day week, = 5

Expected number of complaints = mean x 4 weeks = x4 = 5 x 4 = 20

complaints


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Normal & Sampling Distribution -Revision Question 1

2011/12 Semester 1 Semestral Exam Q8

Question 8

The owner of the popular Koi Bubble Tea chain of shops has received complaints that

some cups of bubble tea contain less tea than usual. To maintain the reputation of his

shops, he conducts a check on all cups of bubble tea sold and finds that the average

amount of bubble tea per cup is 550ml. The standard deviation is 25ml. Given that the

amount of bubble tea per cup follows a Normal distribution,

(a) Calculatethe percentage of cups that contain more than 500ml of bubble tea.

(8 marks)

(b) Calculatethe amount of bubble tea per cup for the fullest 20% of the cups of bubbletea sold.

(5 marks)

(c) Computethe probability that a random sample of 25 cups has an average amount of

less than 545 ml.

(7 marks)


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Answer

Question 8

(a)

X = Amount of bubble tea per cup [1]

X ~ N (550, 252)Z ~ N (0, 1)

[1]

P(X > 500) =? [1]

For X = 500, Z =(500 - 550)/25 = -2 [1]

P(X >500) =P (Z>-2)

= 0.5 +0.4772 [1]

=0.9772

[1]

=97.72%

[1]

-2

[1]

500

0 Z

X

550


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(b) P(X >?) = 0.20

From the table, Z= 0.84 [2]

(X -550) /25 = 0.84 [1]

X = 571ml [1]

________________________________________________________________________________________

(c) X ~ N (550, 252) [1]

[1]

[1]

P (


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Revision questionsself practice

2011/12 Semester 1 Supplementary Exam Q8

Question 1

The Willy Wonky Candy Company has produced a new chocolate bar called Amazingly

Good. To ensure that quality is maintained and Amazingly Good chocolate bars meet the

minimum weight requirement, the company conducts a check on the chocolate bars

manufactured by the factory. The average weight of the bars is 115g and the standard

deviation is 12g. Given that the weight of the chocolate bars follows a Normal distribution,

(a) Calculatethe percentage of chocolate bars that weigh less than 100g.

(b) Calculatethe weight of the lightest 15% of the chocolate bars.

(c) Computethe probability that a random sample of 12 bars has an average weight of

more than 105g.

Answer

a Ans =10.56%

(b ) X = 102.52 grams

(c) Ans = 0.9981


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Revision question 2self practice

In recent years, students gave feedback that they did not have sufficient time tocomplete a university entrance examination. The duration of the examination was 2hours. The chief examiner wanted to investigate the validity of the feedback. It was

found that the average time taken by students to complete the examination was 1hour 30 minutes. The standard deviation was 12 minutes. Assume that the timetaken to complete the examination follows a normal distribution.

(a) Calculate the percentage of students who took more than 2 hours tocomplete the examination.

(7 marks)

(b) From your answer in part (a), explainif the feedback by students was valid.

(2 marks)

(c) Calculatethe minimum time required by the slowest 15% of the students.

(5 marks)

(d) Computethe probability that a random sample of 10 students took more thanan average of 85 minutes to complete the examination.

(6 marks)

Answers

(a) =0.62%(b) The claim is invalid since only 0.62% of the students have insufficient

time to complete the exam.(c) X = 102.48 minutes

(d) =0.9066


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Revision Question 3self practice

A city mayor wants to assess the effect of the economic slowdown on the averagehousehold income in the city in order to come up with appropriate unemployment

benefits for the households. It is known that the average annual household income inthe city is $50000 and the standard deviation is $5000. A random sample of 50households is selected.

(a) Computethe probability that the sample mean household income is between$48000 and $51000.

(9 marks)

(b) Calculatethe probability that the sample mean household income is less than$45000.

(5 marks)

(c) The city mayor will increase the unemployment benefits for households ifthere is a higher than 5% chance that the sample average household incomeis less than $45000. Explainif the mayor should increase the unemploymentbenefits.

(3 marks)

(d) The advisor of the mayor feels that a larger sample size is necessary for theabove computation. Explainif the sample size needs to be increased.

(3 marks)Answers

(a) Ans=0.9184

(b) ANs =0

(c) Should not increase unemployment benefits[1]since P(


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Revision Question 4

With the help of appropriate diagrams, findthe value of b in the following probabilitystatements.

(a) P ( Z < 0.64 ) = b

(b) P ( Z > b) = 0.64

(c) P(b < Z < 1)=0.64

Question 3 Answers

(a) P ( Z < 0.64 ) = bB= 0.7389

(b) P ( Z > b) = 0.64b= - 0.36

(c) P(b < Z < 1)=0.64

b= - 0.84

bm0011revision notes aug 2014

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