MBA First Semester Assignment

7/24/2019 MBA First Semester Assignment

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MBA 1st Sem SFM/RMS notes

STATISTICS FOR MANAGEMENT (SFM)/RESEARCH METHODOLOGY & STATISTICS

(RMS) NOTES

Statistis !o" Mana#ement $ S%a's as e" O*+

Unit – 1 Introduction to statistics and probability

Nature and Scope of StatisticsGeneral concept of probability

T,eo"ems o! -"o'a'iit%

1. Additional theorem2. Multiplication theorem3. ayes! theorem

Unit – 2 "robability #istributions

inomial #istribution"oisson #istribution Normal #istribution

Unit – 3 Samplin$ %heory

asics of samplin$Standard error %estin$ of &ypothesis – lar$e sample test '( test)

Unit – * Small sample tests

%estin$ of &ypothesis – small sample tests 't test)+hi,s-uare test

AN/A

Unit – 0 +orrelation Analysis

+orrelation Analysise$ression analysis%ime series analysis



+NIT $ I

.e i0e in a o"2 in ,i, e a"e na'e to !o"east t,e !t"e it, omete

e"taint%3

-"o'a'iit% De!inition4%he probability of a $ien eent is an epression of li4elihood or chance of occurrence of an eent. A probability is a number 5hich ran$es from 6 to 1 – 7ero for an eent 5hichcan not occur and 1 for an eent certain to occur.Cassia o" a "io"i "o'a'iit% 2e!inition4

%he classical approach to probability is the oldest and simplest. It ori$inated in ei$hteenthcentury in problems pertainin$ to $ames of chance8 such as thro5in$ of coins8 dice or dec4 of cards8 etc. the basic assumption underlyin$ the classical theory is that theoutcomes of a random eperiment are 9e-ually li4ely:.

%he definition of probability $ien by ;rench Mathematician <aplace is 9probability is

the ratio of the number of =faorable! cases 'eents) to the total number of e-ually li4elycases 'eents). If probability of occurrence of A is denoted by p 'A)8 then by thisdefinition 5e hae> Nm'e" o! !a0o"a'e ases (e0ents)

- (A) 5666666666666666666666666666666666666666666666666666666666

Tota nm'e" o! e7a% i8e% ases (e0ents)

;or calculatin$ probability 5e hae to find out t5o thin$s>1. Number of faorable cases or eents2. %otal number of e-ually li4ely cases or eents

Caation o! "o'a'iit%

efore discussin$ the procedure for calculatin$ probability it is necessary to definecertain terms as $ien belo5>

1. E9e"iment4 %he term eperiment refers to describe an act 5hich can be repeatedunder some $ien conditions. andom eperiments are those eperiments 5hoseresults depend on chance such as tossin$ of a coin8 thro5in$ of dice.

2. E0ent4 An eent is the outcome or result of the eperiment. ?ents are $enerallydenoted by capital letters A8 8 +. etc.

An eent 5hose occurrence is ineitable 5hen a certain random eperiment is performedis called a e"tain o" s"e e0ent.An eent 5hich can neer occur 5hen a certain random eperiment is performed is calledan imossi'e e0ent.An eent 5hich may or may not occur 5hile performin$ a certain random eperiment is4no5n as a "an2om e0ent*



3. Mta% E9si0e E0ents4 t5o eents are said to be mutually eclusie or incompatible 5hen both cannot happen simultaneously in a sin$le trial or8 in other 5ords8 the occurrence of any one of them precludes the occurrence of the other.;or eample8 if a sin$le coin is tossed either head can be up or tail can be up8 bothcannot be up at the same time.

*. In2een2ent an2 Deen2ent E0ents4 %5o or more eents are said to beindependent 5hen the outcome of one does not affect8 and is not affected by theother. ;or eample8 if a coin is tossed t5ice8 the results of the second thro5.Similarly8 the results obtained by thro5in$ a dice are independent of the resultsobtained by dra5in$ an ace from a pac4 of cards.

Deen2ent e0ents are those in 5hich the occurrence or non occurrence of one eent inany one trial affects the probability of other eents in other trials. ;or eample8 if a card isdra5n from a pac4 of playin$ cards and is not replaced8 this 5ill alter the probability thatthe second card dra5n.

0. E7a% i8e% e0ents4 ?ents are said to be e-ually li4ely 5hen one does not

occur more often than the others. ;or eample8 if an unbiased coin or dice isthro5n8 each face may be epected to be obsered approimately the samenumber of times in the lon$ run.

@. Sime an2 Comon2 E0ents4 In case of simple eents 5e consider the probability of the happenin$ or not happenin$ of sin$le eents. ;or eample8dra5in$ a red ball from a ba$ containin$ 16 5hite and @ red balls. n the other hand8 in case of compound eents 5e consider the oint occurrence of t5o or more eents. ;or eample8 if a ba$ contains 16 5hite and @ red balls and if t5osuccessie dra5s of 3 balls are made8 probability of $ettin$ 3 5hite balls in thefirst dra5 and 3 blac4 balls in the second dra5.

B. E9,asti0e E0ents4 eents are said to be ehaustie 5hen their totality includes

all the possible outcomes of a random eperiments. ;or eample8 5hile tossin$ adice8 the possible 5hile tossin$ a dice8 the possible outcomes are 18 28 38 *8 0 and@ and hence the ehaustie number of cases is @. If t5o dice are thro5n once8 theehaustie eents are 3@ '@C).

-"o'ems on #ene"a "o'a'iit%

1. Dhat is the probability of $ettin$ a =&ead! in a sin$le toss of a fair coin.2. Dhat is the probability of obtainin$ 'i) an een number 'ii) a number less than 0

'iii) a =0! in a sin$le thro5 of an unbiased dice.3. Dhat is the chance that a non leap year should hae 03 SundaysE*. Dhat is the chance that a leap year selected at random 5ill contain 03 SundaysE0. In a thro5 of t5o unbiased dice find the probability of 'i) total B points 'ii) total F

points.@. %5o dice are thro5n simultaneously and the points are multiplied to$ether. ;ind

the probability that the product is *.B. Dhat is the probability of $ettin$ 3 5hite balls in a dra5 of 3 balls from a bo

containin$ 0 5hite and * blac4 balls.



F. A ba$ contains @ 5hite8 * red and 16 blac4 balls. %5o balls are dra5n at random.;ind the probability that they both 5ill be blac4.

. A ba$ contains 0 5hite and F red balls. %5o dra5in$s of 3 balls are made suchthat 'a) the balls are replaced before the second trial8 and 'b) the balls are notreplaced before the second trial. ;ind the probability that the first dra5in$ 5ill

$ie 3 5hite and the second 3 red balls in each case.16. ne ba$ contains * 5hite and 2 blac4 balls. Another contains 3 5hite and 0 blac4 balls. If one ball is dra5n from each ba$8 find the probability that 'a) both are5hite8 'b) both are blac48 and 'c) one is 5hite and one is blac4.

T,eo"ems o! -"o'a'iit%%here are t5o important theorems of probability8 namely>1. %he Addition %heoremH and2. %he Multiplication %heorem.

A22ition T,eo"em

%he addition theorem states that if t5o eents A and are mutually eclusie the probability of the occurrence of either A or is the sum of the indiidual probability of Aand . symbolically8

- (A o" B) 5 - (A) : - (B) o" - (A+B) 5 - (A) : (B)

In ase o! t,"ee mta% e9si0e e0ents8- (A o" B o" C) 5 - (A) : - (B) : - (C) o" - (A+B+C) 5 - (A) : - (B) : -(C),en e0ents a"e not mta% e9si0e4 5hen eents are not mutually eclusie or8 inother 5ords8 it is possible for both eents to occur8 the addition rule must be modified.

%he modified form of the addition theorem is as follo5s.- (A+B) 5 - (A) : - (B)6- (A;B)

In t,e ase o! t,"ee e0ents<- (A+B+C)5- (A) :- (B) : - (C) $ - (A;B) $ - (B;C) $ - (A;C) : - (A;B;C)

-"o'ems on A22itiona T,eo"em

1. ne card is dra5n from a standard pac4 of 02. Dhat is the probability that it iseither a 4in$ or a -ueenE

2. +alculate the probability of pic4in$ a card that 5as a heart or a spadeE3. Dhat is the probability of pic4in$ a card that 5ad red or blac4E*. A card is dra5n from a standard pac4 of 02 cards. 5hat is the probability that it is

an Ace or in$ or a JueenE0. Dhat is the probability of $ettin$ a Kac4 or a Spade from a pac4 of 02 cardsE@. A person can hit a tar$et in 0 out of B shots8 5hereas another person can hit the

tar$et in 16 out of 11 shots. Dhat is the probability of the tar$et bein$ hit 5henthey ma4e an attempt to hitE



B.

Mtiiation T,eo"em

%his theorem states that if t5o eents A and are independent8 the probability that they

both 5ill occur is e-ual to the product of their indiidual probability. Symbolically8 if Aand are independent8 then- (A an2 B) 5 - (A) = - (B) o" - (A;B) 5 - (A) = - (B)

In ase o! t,"ee in2een2ent e0ents

- (A< B an2 C) 5 - (A) = - (B) = -(C) o" - (ABC) 5 - (A) = - (B) = - (C)In ase o! Deen2ent E0ents (Con2itiona -"o'a'iit%)

%he multiplication theorem eplained aboe is not applicable in case of dependenteents. %5o eents A and are said to be dependent 5hen can occur only 5hen A is4no5n to hae occurred or ice ersa. %he probability attached to such an eent is calledthe conditional probability and is denoted by " 'AL) or8 in other 5ords8 probability of A$ien that has occurred.

- (A;B) 5 - (A) = - (B/A)- (A;B) 5 - (B) = - (A/B)

Fo" t,"ee e0ents A< B an2 C< e ,a0e

- (A;B;C) 5 - (A) = - (B/A) = - (C/A;B) i.e.8 the probability of occurrence of A8 and + is e-ual to the probability of A8 times the probability of $ien that A hasoccurred8 times the probability of + $ien that both A and has occurred.

BAYES> THEOREM OR -OSTERIOR -ROBABILITY

%he concept of conditional probability ta4es in to account information about theoccurrence of one eent to predict the probability of another eent. %his concept can beetended to 9reise: probabilities based on ne5 information and to determine the

probability that a particular effect 5as due to a specific cause. %he procedure of reisin$these probabilities is 4no5n as Bayes’ theorem,

%he ayes! theorem named ater the british mathematician re. %homas ayes '1B62,@1)and published in 1B@3 in a short paper has become one of the most famous memories inthe history of science and one of the most controersial. &is contribution consists primarily of a uni-ue method for calculatin$ conditional probabilities.

An eent 9?: can occur only if any one of the set of ehaustie and mutually eclusieeents A18 A28 A3..An occurs. %he probabilities "'A1)8 "'A2)8 "'A3)"'An) andthe conditional probabilities "'?LAi) 5here 'i 182838.n) for ? to occur are 4no5n.%hen the conditional probability " 'AiL?) 5hen 9?: has actually occurred is $ien by " 'Ai) " '?LAi) " 'AiL?) ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 5here i 18 28 3..n O " 'Ai) " '?LAi)

CORRELATION ANALYSIS



1. +alculate 4arl pearsons correlation coefficient from the follo5in$ data andinterpret its alue.

Ro No* o!

St2ents 1 2 3 * 0

Ma"8s inAonts *F 30 1B 23 *BMa"8s in

Statistis *0 26 *6 20 *0

2. Ma4in$ the use of data summari7ed belo5 calculate coefficient of correlation. Also calculate probable error.

+ase A + # ? ; G &P 16 @ 16 12 13 11

Q * @ 11 13 F *3. +alculate 4arl pearsons coefficient of correlation from the follo5in$ data ta4in$

166 and 06 as the assumed aera$es of P and Q respectiely.

P 16* 111 16* 11* 11F 11B 160 16F 16@ 166 16* 160Q 0B 00 *B *0 *0 06 @* @3 @@ @2 @ @1

*. %he follo5in$ data $ies the sales and adertisement alues. +alculate thecoefficient of correlation from the data by ta4in$ ** and 2@ as assumed means and

interpret its alues.

Sales 0B *2 *6 3F *2 *0 *2 *0 *2 ** *6 *@Adert. 16 2@ 36 *1 2 2B 2B 1 1F 1 31 2

0. %he follo5in$ table sho5s the amount of adertisin$ epenses'P) and the sales'Q) of 16 firms in s. 666!s. ;ind the 4arl pearsons coeeficient of correlation andcomment.

P 1.1 1.3 1.* 1.@ 1.@ 1.0 1.0 1.* 1.3 1.@ Q @6 F6 B6 06 B6 @0 0F @6 B0 F6

@. +alculate coefficient of correlation from the follo5in$ data.

P 066 1666 1066 2666 2066 3666 3066Q 166 266 366 *66 066 @66 B66



B. ;rom the follo5in$ data compute the coefficient of correlation bet5een P and Q.

= Se"ies Y Se"ies No. of items 10 10 Arithmetic mean 20 1F

S-uares of deiations from mean 13@ 13F Summation of product of deiations of P and Q series from their respectieArithmetic mean is 122.

F. A computer 5hile calculatin$ correlation coefficient bet5een t5o ariables P andQ from 20 pairs of obserations obtained the follo5in$ results.

N 208 O 1208 Oy 1668 Oy 06F8 OC @068 OyC *@6.It 5as ho5eer discoered at the time of chec4in$ that t5o pairs of obserations 5ere notcorrectly copied. %hey 5ere ta4en as '@8*) and 'F8@) 5hile the correct alues are 'F812)and '@8F). "roe that the correct alue of the correlation coefficient should be 2L3.

. +alculate 4arl pearsons coefficient of correlation and find out probable error.

P> 13,1*8 1*,108 10,1@8 1@,1B8 1B,1F8 1F,18 1,268 26,218 21,22Q> 3 *6 *3 *3 3@ 3 *F ** 0@

16. %he ran4s of 16 students in t5o subects of accountancy and auditin$ are asfollo5s.

Accountancy> 3 0 F * B 16 2 1 @ Auditin$> @ * F 1 2 3 16 0 B

Dhat is the coefficient of ran4 correlationE11. 16 competitors in a beauty contest are ran4ed by 3 ud$es in the follo5in$ order.

1st Kud$e> 1 @ 0 16 3 2 * B F2nd Kud$e> 3 0 F * B 16 2 1 @ 3rd Kud$e> @ * F 1 2 3 16 0 BUse the ran4 correlation coefficient to determine 5hich pair of ud$es has the nearestapproach to common taste in beauty.

12. +alculate the correlation coefficient from the follo5in$ data by the spearman!sran4 correlation method.

"rice of tea> B0 FF 0 B6 @6 F6 F1 06"rice of coffee> 126 13* 106 110 116 1*6 1*2 166

13. Juotations of inde no. of security prices of a certain oint stoc4 company are$ien belo5.



Qear> 1 2 3 * 0 @ B#ebenture "rice> B.F .2 F.F F.3 F.* @.B B.1Share price> B3.2 F0.F BF. B0.F BB.2 FB.2 F3.FUsin$ ran4 correlation method determine the relationship bet5een debenture price andshare price.

1*. +alculate the spearman!s ran4 correlation coefficient in maths and statistics.

Maths> BF 20 3@ BF @2 @0 @2 B F @3Statistics> 01 F* 1 @6 @2 @F @6 *B 03 01

10. +alculate ran4 correlation coefficient of the follo5in$ dataE

Series A> 110 16 112 FB F 126 F 166 F 11FSeries > B0 B3 F0 B6 B@ F2 @0 B3 @F F6

1@. ;ollo5in$ are the scores of 16 students in a class and their IJ.

Students> 1 2 3 * 0 @ B F 16Scores > 30 *6 20 00 F0 6 @0 00 *0 06IJ > 166 166 116 1*6 106 136 166 126 1*6116

1B. +oefficient of ran4 correlation bet5een mar4s in statistics and maths obtained bya $roup of students is 2L3 and the sum of s-uares of differences in ran4s is 00.+alculate the number of obserations or alue of nE

1F. In a biariate samples the sum of s-uares of difference bet5een the ran4s of

obsered alues of t5o ariables is 33. %he coefficient of correlation bet5eenthem is 6.F. calculate no of obserations.

REGRESSION ANALYSIS

1. Qou are $ien data relatin$ to purchase and sales. obtain the t5o re$ressione-uations and estimate li4ely sales 5hen purchase is e-ual to 166.

"urchase> @2 B2 F B@ F1 0@ B@ 2 FF *Sales > 112 12* 131 11B 132 @ 126 13@ B F0

2. %he follo5in$ table sho5s the a$es 'P) and " 'Q) of F persons obtain t5ore$ression e-uations and find the epected " of a person 5ho is * years old.

P> 02 @@ *0 3@ B2 @0 *B 20Q> @2 00 01 20 B *3 @6 33

3. In a correlation study the follo5in$ alues are obtained.



"articulars P QMean @0 @BS.# 2.0 3.0+oefficient of+orrelation 'r) 6.F

;ind the t5o re$ression e-uations that are associated 5ith the aboe alues.*. De are $ien the follo5in$ data.

"articulars P QArithmetic mean 3@ F0S.# 11 F+oefficient of +orrelation 'r) 6.@@;ind the t5o re$ression e-uations and estimate the alue of P 5hen Q is B0.

0. In a partially established laboratory record of an analysis of correlation data8 the

follo5in$ results are only aailable./ariance of P FP – 16Q R @@ 6*6P – 1FQ 21*;ind on the basis of aboe information

'a) %he mean alues of P and Q.'b) +oefficient of correlation bet5een P and Q.'c) Standard deiation of Q

@. ;or certain P and Q series 5hich are correlated the t5o lines of re$ression are 0P – @Q R 6 68 10P – FQ – 136 6. ;ind the means of t5o series and the

correlation coefficient.

B. %5o random ariables hae the re$ression e-uations 3P R 2Q – 2@ 68 @P R Q – 31 6. ;ind the mean alues and correlation coefficient bet5een P and Q. if theariance of P 208 ;ind the standard deiation of Q from the date $ien aboe.

TIME SERIES ANALYSIS

1. +alculate 3 yearly moin$ aera$es of the production fi$ures $ien belo5. Alsocalculate short term fluctuations.

Qear> 1F0 F@ FB FF F 6 1 2 3 * 0 @ B F "roduction> 10 21 36 3@ *2 *@ 06 0@ @3 B6 B* F2 6 0 162

2. Usin$ 3 yearly moin$ aera$es determine the trend and short term fluctuations.



Qear> 1FB FF F 6 1 2 3 * 0 @"roductionIn tons > 21 22 23 20 2* 22 20 2@ 2B 2@

3. ;ind the trend for the follo5in$ series usin$ * yearly moin$ aera$e.

Qear> 1* 0 @ B F 66 61 62 63 6*"roduction> @1* @10 @02 @BF @F1 @00 B1B B1 B6F BB B0B

*. ;ind the fie years 5ei$hted moin$ aera$e 5ith 5ei$hts 182828281 respectielyfor measurin$ the trend of the follo5in$ time series.

Qear> 1B* B0 B@ BB BF B F6 F1 F2Sales> 2 @ 1 0 3 B 2 @ *

0. ;or the belo5 data calculate * yearly 5ei$hted moin$ aera$e 5ith 5ei$hts

1828281.Qear> 1B* B0 B@ BB BF B F6 F1 F2Slaes> 2 @ 1 0 3 B 2 @ *

@. +alculate the e-uation of a strai$ht line trend for the follo5in$ data.

Qear> 1B* B0 B@ BB BFSales> 30 0@ B F6 *6

B. ;it a strai$ht line by the method of least s-uares and estimate the alues for 1F*

and [email protected] 'P)> 1BF B F6 F1 F2Q > F6 F0 FB 3 166

F. ;it a strai$ht line trend by the method of least s-uares and estimate the productionin 1F1.

Qear> 1F0 F@ FB FF F 6"roduction> 10 1F 1@ 1F 26 1

-ROBABILITY

1. Dhat is the probability of $ettin$ a head in a sin$le toss of a fair coin.



2. Dhat is the probability of obtainin$ 'i) an een number 'ii) a number less than 0 'iii) anumber =0! in a sin$le thro5 of a unbiased die.

3. ;ind the probability that a thro5 of an unbiased die results in 'i) an ace 'number 1) 'ii) aneen number 'iii) a multiple of B.

*. A ba$ contains 3 5hite8 * red and 2 $reen balls. ne ball is selected at random from the

ba$. ;ind the probability that the selected ball is 'i) 5hite 'ii) non – 5hite 'iii) 5hite or $reen.0. Dhat is the chance that a non leap year should hae 03 Sundays.@. Dhat is the chance that a leap year selected at random 5ill contain =03! Sundays.B. In a thro5 of an unbiased dice find the probability of 'i) total B points 'ii) total F pointsF. %5o unbiased dice are thro5n. ;ind the probability that the sum of the faces is not less

than 16.. %5o dice are thro5n are simultaneously and the points are multiplied to$ether8 find the

probability that the product is *.16. Dhat is the probability of $ettin$ 3 5hite balls in a dra5 of 3 balls from a bo containin$

0 5hite balls from a bo containin$ 0 5hite and * blac4 balls.

A22itiona T,eo"em

1. ne card is dra5n from a standard pac4 of 02 cards. Dhat is the probability that either 4in$ or -ueen.

2. In dra5in$ a card from 5ell shuffled pac4 find the probability that the card dra5n 5ill beeither spade or Arc.

3. In a colle$e8 there are fie lecturers. Amon$ them8 three are doctorates. If a committeeconsistin$ three lecturers is formed8 5hat is the probability that at least t5o of them aredoctoratesE

*. A ba$ contains 36 balls numbered from 1 to 36. ne ball is dra5n at random. ;ind the probability that the no of the ball dra5n 5ill be a multiple of a) 0 or B b) 3 or B

Mtiiation T,eo"em

1. A man 5ants to marry a $irl hain$ -ualities8 5hite compleion – the probability of $ettin$ such a $irl is 1 in 26. &and some do5ry – the probability of $ettin$ this is 1 in 06.Desterni7ed manner – the probability here is 1 in 166. ;ind out the probability of his$ettin$ married to such a $irl 5here the possession of these three attributes areindependent.

2. %he personnel department of a company has records sho5 the follo5in$ of 266en$ineers.

A$e achelors Masters %otal #e$ree de$reeUnder 36 6 16 16636 – *6 26 36 06Aboe *6 *6 16 06If an en$ineer is selected at random from the company8 find the probability thata) &e has only a bachelor!s de$ree



b) &e has a master!s de$ree $ien that he is oer *6c) &e is under 36 $ien that he has only a bachelor!s de$ree.

3. %he data for the promotion status and academic -ualification re$ardin$ 166

?mployees of a company is as follo5s. "romotional status MA non , MA %otal

"romoted 12 *F @6

Not – promoted 1F 22 *6 %otal 36 B6 166

At random one employee is pic4ed up. Dhat is the probability thatA) &e is an MA) &e is promoted+) &e is promoted $ien that he is an MA

#)

&e is an MA promoted $ien that he is promoted.3. %he records of *66 eaminees are $ien belo5.

Score A .S+ .+M %otal

elo5 06 6 36 @6 1F606 – @6 26 B6 B6 1@6Aboe @6 16 36 26 @6%otal 126 136 106 *66If an eaminee is selected at the random from this $roup of eaminees8 find the probability that

a) &e is a commerce $raduate b) &e is a science $raduate $ien that his score is aboe @6.c) &is score is belo5 06 $ien that he is a A*. In a bolt factory 3 machines M18M28M3 manufactures respectiely 208 30 and *6

of the total of the output. 08 *8 2 respectiely are defectie bolts. ne bolt is dra5nat random from the products and is found to be defectie. Dhat is the probability that it ismanufactured in Machine 2E

0. %hree A8 8 + produce 068 36 and 26 respectiely of the total no of items in afactory. %he percenta$e of defectie output of these machines are 38 *8 0. An itemfrom production of the factory is selected at random and is found to be defectie. Dhat isthe probability it 5as produced by machine AE

@. Suppose that there is a chance for a ne5ly constructed buildin$ to collapse 5hether thedesi$n is faulty or not. %he chance that the desi$n is faulty is 16. %he chance that the buildin$ collapse is 08 if the desi$n is faulty and other5ise it is *0. It is seen that the buildin$ collapsed. Dhat is the probability that it is due to faulty desi$nE

B. %here are t5o identical boes containin$ respectiely * 5hite8 3 red and 3 5hite8 B red balls. A bo is choosen at random and a ball is dra5n from it. If the ball is 5hite8 5hat isthe probability it is from 1st boE



Binomia Dist"i'tion

1. @ coins are tossed in succession. ;ind the probability of $ettin$ more than * heads.2. %he probability that an eenin$ colle$e student 5ill $raduate is 6.*. determine the

probability that out of fie students8 one student8 t5o students and at least one student5ill be $raduate.

3. A ba$ containin$ 06 items has fie defecties. Indicate the probability of the follo5in$ if * items are dra5n 5ith replacement

'a) ?actly t5o defectie items'b) At least t5o defectie items'c) At the most t5o defectie items

*.

A random ariable =! is binomially distributed 5ith mean T standard deiation 2.*.find the probability that more than half of the trials are success.0. F coins are tossed at a time 20@ times. No of heads obsered at each thro5 is recorded

and the results are $ien belo5. ;ind the epected fre-uency. Dhat is the theoreticalalue of mean and standard deiation.

@. %he scre5s produced by a certain machine 5ere chec4ed by eaminin$ centers of B. %hefollo5in$ sho5s the distribution of 12F samples accordin$ to the no of defectie itemsthey contained.

No. of #efecties> 6 1 2 3 * 0 @ B No. of Samples> B @ 1 30 36 23 B 1;it a binomial distribution and find the epected fre-uencies if the chance of machine bein$ defectie is V. ;ind the mean and ariance.

-oisson 2ist"i'tion

1. Assumin$ that on an aera$e 2 of the output in a factory manufacturin$ certain bolts isdefectie and that 266 units are in pac4a$e. Dhat is the probability that

'i) None is defectie'ii) At most 3 defecties bolts may be found in that pac4a$e.

2. A boo4 contains 120 misprints distributed at random throu$hout its 120 pa$es. Dhat isthe probability that a pa$e obsered at random contains at least 2 misprints.

3. A random =! follo5s a poisson distribution 5ith parameter 3. ;ind the probability that=! assumes the alues

'i) 68 18 2'ii) <ess than 3'iii) At least 3

*. In an office bet5een *T0 "M the aera$e no of incomin$ phone calls per minute in thes5itch board is 1.F. find the probability that durin$ one particular minute there 5ill be

'i) No phone call at all'ii) ?actly 2 phone calls



0. Suppose that a manufactured product has t5o defecties per unit of product inspected.Usin$ "oisson distribution calculate the probabilities of findin$ a product 5ithout anydefect8 3 defects and * defects.

@. ;it a poisson distribution to the follo5in$ data and calculate the theoretical fre-uencies. No of deaths> 6 1 2 3 *

;re-uencies> 122 @6 10 2 1B. %he distribution of typin$ mista4es committed by a typist is as follo5s. Assumin$ a poisson model8 find out the epected fre-uencies.Mista4esLpa$e> 6 1 2 3 * 0 No of pa$es> 1*2 10@ @ 2B 0 1

No"ma Dist"i'tion1. A sample of 166 dry battery cells are tested to find out the len$th of life produced the

follo5in$ results. W12 hrs. S.#. 3 hrs. assumin$ that the data are normally distributed5hat percenta$e of battery cells are epected to hae

'a) More than 10 hrs

'b)

<ess than @ hrs'c) et5een 16 and 1* hrs.2. %he mean and standard deiation of the 5a$es of @666 5or4ers en$a$ed in a factory are

s. 1266 and *66 respectiely. Assumin$ the distribution to be normal estimate'a) "ercenta$e of 5or4ers $ettin$ 5a$es aboe s.1@66'b) No of 5or4ers $ettin$ 5a$es bet5een s.@66 and s.66.'c) No of 5or4ers $ettin$ 5a$es bet5een s.1166 and s.1066.3. %he daily 5a$es of 1666 5or4men are normally distributed around a mean of s.B6 and

5ith a standard deiation of s.0. estimate the no of 5or4ers 5here daily 5a$es 5ill be'a) More than B0'b) More than F6'c) <ess than @3'd) et5een B6TB2'e) et5een @TB2

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MBA First Semester Assignment