Assigment Prob. n Stat.

8/2/2019 Assigment Prob. n Stat.

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REVIEW CHAPTER 1

1. How many five-digit even numbers can be formed from the digits 0,1,2,3,4,5,6 if eachdigits can only be used only once?

a)The last position is 0,

b)The last position is not o, 2. How many distinct arrangements are possible from the word CLICK>

3. How many different numbers can be formed using three of the digits 3,4,5,6,7 if the

repetition of the digit is not allowed? 4. Letter from the word JANGKAAN are to be arranged

a)find the numbers of different arrangement that can be made.


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b)find the probability of getting an arrangement where the two Ns are always

In the middle.

5. How many different numbers can be formed using three of the digits 3,4,5,6,7 if therepetition is not allowed?

6. 11 employees of LTK company are assigned identity codes. The code consists of the

first two letters of the employees name followed by four digits.

a)How many different identity codes are possible if repetitions are not allowed?

b)How many different identity codes are there for employees whose names start

with an S?


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REVIEW CHAPTER 2

1. A bacth of 200 boxes of frozen orange juice contains 5 boxes that are defective.Two boxes are selected at selected at random without replacement.

i) Draw a tree diagram for the above experiment.

D1 (5/200) D2 (4/199)D2 (195/199)

D1 (195/200) D2 (5/199)

D2 (194/199)

ii) Find the probability that the second box is defective given that the first box isdefective.

P(D2/D1)=

=

iii) Find the probability that both boxes are defective.P(D1 D2) = x

= 0.0005

iv) Find the probability that both boxes are non defective.

P(D1 D2) = x = 0.95


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2. In my bookcase there are four shelves and the number of books on each shelf is asshown in the table below :

Hardback Paperback

Shelf 1 11 9

Shelf 2 8 12

Shelf 3 16 4

Shelf 4 9 3

a) If I choose abook at random, irrespective of ts position in the cases, what isthe probability that it is paperback ?

P(P) = =

= 0.39

b) I am equally likely to choose any shelf.I choose a shelf at random thenchoose a book.

i) What is the probability that it is a hardback ?

S1 (1/4) H (11/20)

P (9/20)

S2 (1/4) H (8/20)

P (12/20)

S3 (1/4) H (16/20)

P (4/20)

S4 (1/4) H (9/12)

P (3/12)

=( + + + ) = 0.625

ii) If the book chosen is hard back, what is the probability that it is from shelf 3 ?P(P) =

= =


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3. An experiment consist of randomly selecting a card from a deck of 52 playing game cards

and then flipping a coin once,if the selected card is red.If the card selected is black,the coin

flipped twice.

1.Draw a tree diagram of the above experiment and list down the sample space.

HR T

H H T

B H T T

S={

11. Find the probability of getting one black card and two heads.

P(BHH)=

A=Red/Black

B=HH,HT,TH,TT

P(BHH) =


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4.If events A and B are independentand P(A)=3/5,P(AB)=2/5.Find

1. P(B)=

P(AB)=P(A).P(B/A)

=. P(B/A)P(B/A)=

P(B)=P(B/A)=

11.P(AB)

P(AB) = P(A).P(B/A)

b)Give that P(A)=1/7,P(A/B)=4/7 and P(AB)=4/9.Find

1.P(B)

P(A/B) = () P(B) = () = 11.P(AB)

P(AB) = P(B)-P(AB)

P(AB) = P(B)-P(AB)

111.P(A/B)

= P(AB)P(B)=

=


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5. An unbalanced die is such that an even number occurs twice as often as an odd number. If

A is the event that and odd number comes up on the first toss and B is the event thatan even number comes up on the second toss.

i. Draw the three diagram to show the above experiment.

ii. Find the probability of event A and B occurring.


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6. A housing development company employs three construction firms X, Y and Z with the

probability 0.45, 0.30, 0.25 to carry out its project, past experience has shown thatprobabilities of costs overruns for the firms are 0.03, 0.15, 0.05 respectively. Suppose that the

company is experiencing a cost overrun.

a) Find the probability that the construction firm involved is:

i. X = 0.0135

ii.Y = 0.045

iii. Z = 0.0125

R 0.03 0.0135

X 0.45

R 0.97 0.4365

R 0.15 0.045

Y 0.3

R 0.85 0.255

R 0.05 0.0125

Z 0.25

R 0.95 0.2375


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b) Which construction firm has the least involvement of the cost over run.

P|

|

|

7. The events A and B are such that P (A) =

,

| , and

. Find

i.


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ii.

8. A certain company uses customers to evaluate preliminary product designs. In the past,

98% of highly successful products received good review, 65% of moderately successful

products received good review, and 8% poor products received good reviews. Suppose this

company produced 40% of products have been highly successful, 35% have been moderately

successful, and 25% have been poor products.

a) Construct a tree diagram to portray the above events.

G 0.98

H 0.4

G 0.02

G 0.65

M 0.35

G 0.35

G 0.08

P 0.25

G 0.92


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b) Find the probability that a randomly chosen product attains a good review.

P (G) = 0.98(0.4) + 0.65(0.35) + 0.08(0.25)

= 0.392 + 0.2275 + 0.02

= 0.6395

c) If a product chosen does not attain a good review, what is the probability that it will behighly successful product?

| =

P (G) = 1-0.6395

= 0.3605

| = 0.022


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REVIEW CHAPTER 3

1. The time (in seconds) taken for a certain chemical to react and produce light at a given

temperature is a continuous random variable X with p.d.f as follows :

(a) Prove that Answer :

As is a continuous random variable X

*

+

* + Thus,


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(b) Find Answer :

* +

*

+


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(c) Using

obtained in (b), calculate the probability that the chemical will react and

produce light less than 4.5 seconds

Answer :

* +

2. Cumulative distribution function, F(x) for discrete random variable X is given by

{

(a) Compute Answer :


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(b) Find the probability distribution function, f(x)

Answer :

{

{

{

3. The total number of hours, measured in units of 10, that a student spends on reading books

over a period of one week is a continuous random variable X with density function

(a) Find the value of k

As f(x) is a continuous random variable X

Thus,


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* + * +

(b) Find the probability that over a period of one week, a student will spend less than 15

hours reading

Answer :

* + [ ] * + [ ] [ ]


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4. Below is the probability distribution of a discrete random variable X,

,

(a)Determine the cumulative distribution function for X , by using

(b)Use the result obtained in part (a) to find the probability(i) (ii)


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5. Suppose that the cumulative distribution of a continuous random variable X is

(a)Find (b)Find () (c)Find

{

6. If the probability distribution of a discrete variable X is given by

,0 , elsewhere.


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a) Findi) ii)

b) What is the cumulative distribution of X?X 0 1 2 3 4

P(X=x)


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0

1

7. Suppose that the probability density function of a continuous random variable X is given

by

, 0 , elsewhere.

a) Determine the cumulative distribution function F(x).

b) Findi)

= 0.9817


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ii) [ ] [ ] [ ] [ ]= 0.018

8. The following table shows the probability distribution of X, the number of defective bulbs

purchased by a shopkeeper.

X 0 1 2 3 4

P(X=x) 0.71 0.07 0.16 0.05 0.01

a) Construct the cumulative distribution of X.


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0

0.71 0.78 0.94 0.99

1

b) Find the probability of getting fewer than 3 defective bulbs.

=0.71+0.07+0.16

=0.94


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REVIEW CHAPTER 6

Exercise 1

1. For each of the following data, determine whether the data type is quantitative, qualitative,

discrete or continuous:

a. The daily production of palm oil in Jengka in the last 3 monthsquantitative,discrete

b. The type of car driven by the university staffqualitative

c. The number of contacts made to the lecturers by the Darul Hikmah bookseller

quantitative,discrete

d. The rating (excellent, good, fair, or poor) given to particular food vendor by a sample ofstudentsqualitative,discrete

e. The department in which each of a sample of university professors teachesqualitative

f. The number of rooms in an apartmentquantitative,discrete

g. The teaching evaluation rating of an instructor (1=poor, 5=excellent)qualitative,discrete

2. State whether each of the following variables is quantitative or qualitative :

a. Agequantitative

b. Annual salesquantitative

c. Soft drink sizes (small, medium, large)qualitative

d. Earningsquantitative

e. Methods of payment (cash, cheque, credit cards)qualitative


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3. Identify the following as discrete or continuous random variables

a. The number of bank failure in a given yeardiscrete

b. The floor-space area in a new office buildingdiscrete

c. The number of people waiting for treatment at a hospital emergency roomdiscrete

d. The total points in a football gamediscrete

e. The number of claims received by an insurance company during a day - discrete

4. A survey on television viewing habits of cables TV customers is conducted. One question

asks for the most frequently watched channel number. What type of variable is being reported

for that question?qualitative

Exercise 2

1. Construct a bar chart from the data given in the table below:

Malaysias Gross Imports by Commodity Sections in 1991

Commodity Sections $ MillionFood and live animals 5135.8

Beverages and tobacco 405.5

Crude materials inedible 2810.4

Mineral fuels 4214.1

Animal/vegetable oils and fats 395.3

Chemicals 7724.8

Manufactured goods 16009.7

Machinery and transport equipment 54352.8

Miscellaneous manufactures articles 5660.1

Other imports 4119.6

Totals 100831.1


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2. The Malaysian Labor Force Employment by Sector, 1995-2000 (thousand persons) is

potrayed in the following table. Draw a pie chart for the year 1995 and a bar chart that

represents both years.

Employment by Sector, 1995 and 2000 (thousand persons)

Industry 1995 2000Agriculture, forestry and fishing 1428.7 1187.2

Mining and quarrying 40.7 44.5

Manufacturing 2051.6 2616.3

Construction 659.4 845.4

Electricity, gas and water 69.1 84.0

Transport, storage and communications 395.2 506.9

Wholesale and retail trade, hotels and restaurant 1327.8 1469.6

Finance, Insurance, Real Estate & Business

Services

378.8 479.0

Government Services 872.2 894.2

Other Services 692.2 938.6

Total 7915.4 9066.2

0 10000 20000 30000 40000 50000 60000

Food and live animals

Beverages and tobacco

Crude materials inedible

Mineral fuels

Animal/vegetable oils and fats

Chemicals

Manufactured goods

Machinery and transport equipment

Miscellaneous manufactured articles

Other imports

Malaysias Gross Imports by Commodity Sections in 1991


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Agriculture, foresty

and fishing

18%

Mining and quarrying

0%

Manufacturing

26%

Construction

8%

Electricity,ga

s and water

1%

Transport,

storage and

communication

5%

Wholesale and retailtrade, hotels and

restaurant

17%

Finance, Insurance,

Real Estate &

Business Services

5%

Government Services

11%

Other

Services

9%

Employment by Sector, 1995 (thousand persons)

0

500

1000

1500

2000

2500

3000

Employment by Sector, 1995 and 2000

(thousand persons)

1995

2000


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Exercise 3

1. Each week the net amount of electric energy distributed by the electric utilities across the

country is compiled by the Edison Electric Institute. In one particular week, the electric

output (in thousands of kilowatt-hours) for the ten midsize cities reads as follow:

49,70,54,67,59,40,61,69,71,52

Find the mean, median, and mode

Answer:

i) Mean = =

= 59.2

ii) median

rearrange in ascending order

40,49,52,54,59,61,67,69,70,71

Median = =

= 60

iii) no mode


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2. During a particular month, the eight salesmen in an air-conditioning firm sold the

following number of central air-conditioning units : 8,11,5,14,8,11,16,11

a) Find the mean(average) number of units sold

b) Find the mode

Answer:

a) Mean = =

= 10.5

b) Mode = 11

3. A sample of teachers teaching in elementary schools in Rompin, Pahang revealed thefollowing income: (RM) 726, 499, 590, 687, 480, 439, 500 and 565.

a)

What is the mean monthly income?Sol:

439, 480, 499, 500, 565, 590, 687, 72

b) What is the median monthly income?

Sol:


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c) How many observations are below the median?

Ans: 4

439, 480, 499, 500

d) How many observations are above it?Ans: 4

565, 590, 687, 726

EXERCISE 4.

1. Each week the net amount of electric energy distributed by electric utilities across thecountry is compiled by the electricity company. In one particular week the electric

output (in thousands of kilowatt-hours) for ten cities reads as follow: 48, 70, 54, 67,

59, 40, 61, 69, 71, 52.

Find the first and third quartiles.

Sol:

40, 48, 52, 54, 59, 61, 67, 69, 70, 71


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2. A sample of salesgirls working in a textile store received the following monthly

income (in RM): 426, 299, 290, 687, 480, 439 and 565.

a) What is the third quartile of the monthly income?Sol:

290, 299, 426, 439, 480, 565, 687

b) How many observations are below the third quartile?Ans: 5

290, 299, 426, 439, 480

c) How many observations are above it?Ans: 1

687

3. A sample of eight companies in the motor bicycle industry was surveyed as to theirreturn on investment last year. The results are (in percent): 10.6, 12.6, 14.8, 18.2,

12.0, 14.8, 12.2 and 15.6. Find:

a) The first quartile.Sol:

10.6, 12.0, 12.2, 12.6, 14.8, 14.8, 15.6, 18.2


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b) The second quartile.

Sol:

EXERCISE 5.

1. Ten experts rated a newly developed pizza, PIZZA RIA on a scale of 1 to 5. Theratings were: 34, 35, 41, 28, 26, 29, 32, 36, 38 and 40.

a) Find the range.Sol:

26, 28, 29, 32, 34, 35, 36, 38, 40, 41

b) Find the arithmetic mean.

Sol:


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c) Find the standard deviation.Sol:

d) The rating of another pizza company, PIZZA JIA, was also studied. The mean

and standard deviation of this company is and s = 6.4. Compare themean and dispersion among the two firms.

Sol:

PIZZA IJA: PIZZA RIA: To compare dispersions use coefficient of variation:CV= CV for IJA CV for RIA


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2. The annual incomes of the five executives of Khaldun Industries are: RM 75 000, RM78 000, RM 72 000, RM 83 000 and RM 90 000. Consider this a population.

a) What is the range?Sol:

RM 72 000, RM 75 000, RM 78 000, RM 83 000 and RM 90 000 b) What is the arithmetic mean income?

Sol:

c) What is the population variance? The standard deviation?

Sol:

d) The annual incomes of officers of another firm similar to Khaldun Industries

were also studied. The mean and standard deviation of this firm is = RM 79

900 and = RM 8612. Compare the mean and dispersion among the twofirms.Sol:

Khaldun Industry: Another firm: To compare dispersions use coefficient of variation:

CV=


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CV for Khaldun Industry CV for the another firm

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