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Q1. An insurance salesman meets with 8 prospective customers each week. Fromhistorical data, the proportion of these customers who take out a policy is 18%.

• What is the probability that the salesman will sell 3 or more policies in a ivenweek!

• What is the probability that the salesman will sell 1" or more policies in aiven month #assumin a month is e$actly weeks&!

• 'an you comment on the probability that the salesman sells 1() or morepolicies in a year #assumin a year is e$actly (" weeks&!

• What is the mean and standard deviation of the number of policies he sells ina week, a month, a year! What do you notice about these three distributions!

*". +ou survey ( customers on whether or not they reconi-e your brand. n thebasis of historical norms you e$pect 3(% or 1/( to reconi-e your brand. 0ut youhave been spendin more money than usual on advertisin. How many more than1/( would start to convince you that the e$tra money had been worth the e$pense!

*3.

+2 mutual fund has outperformed the inde$ in 3/ of the past (" weeks. n thisbasis they are claimin that they can systematically outperform the market.

• What do you think of this claim!• Assumin that +2 are not outperformin the market, what is the chance

they would et to make this claim durin a 1 year period!• ow many years would +2 have to wait before they can make such a claim!

*. +ou are determined that the underlyin proportion of customers who ive yourproducts the lowest possible ratin for meetin e$pectations on a ( point scale is tobe no hiher than 1%. 4uppose it is e$actly 1%.

• From a survey of 1 customers describe in words the likely number who will

ive your product the lowest possible ratin.

• What is the chance that, by bad luck, 1( or more out of the sample report the

lowest level of satisfaction!

*(.5merency patients arrive at a lare hospital at the rate of .33 per minute. naverae, ""% of emerency patients are triaed into the most serious cateory.

• What is the probability of ) or more arrivals durin the ne$t 3 minutes!• What is the probability of ) or more arrivals durin the ne$t 3 minutes that

are into the most serious cateory!• 6f there are "( patients that arrive in the ne$t hour, what is that chance that

more than half of them will be into the most serious cateory!

*).• 7he number of people arrivin at a fast food drive thru #or throuh& in any

iven " minute interval obeys a oisson process with mean 1. 4uppose thatthe waiters can only process 3 orders in any iven minute interval #also,assume the waiters can process an order instantaneously, but are limited inhow many they can process in iven time intervals, as indicated&.

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• What is the e$pected number of people that leave the drive thru with theirorders 9lled in any iven minute interval!

*/.An aptitude test, when applied to the eneral population, ives scores that arenormally distributed with mean (/ and standard deviation ).

• What proportion of people sittin the test will score less than (!• What is the probability that a person scores between (/ and )!• For the sub:population of people who are already successful manaers, the

mean score is )( with standard deviation 8. What are the answers to parts#a& and #b& for this sub:population!

*8. 7he level of cholesterol in the blood s important because hih cholesterol mayincrease the risk of heart disease. 7he distribution of blood cholesterol levels in alare population of people of the same ae and se$ is rouhly normal. For 1:yearold boys, the mean is 1/ m;dl and s.d<3 m;dl. =evels above " may re>uiremedical attention.

• What percent of 1:year old boys have more than " m;dl of cholesterol!

*?.4cores on the 4A7 test in "" followed appro$imately normal distribution @#(,111& .

• ow hih must a student score in order to place in the top 1% of all studentstakin the 4A7!

*1.Are attitudes towards shoppin chanin! 4ample surveys show that fewer peopleenoy shoppin than in the past. A survey asked a nationwide random sample of"( adults if they areed or disareed that B 6 like buyin new clothes, butshoppin is often frustratin and time:consuminC. 4uppose )% of all adultresidents of 6ndia say BAreeC if asked the same >uestion.

• What is the probability that 1(" or more of the sample aree!

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*11.4ome people complain about the lenth of time it takes for a pharmacist to 9ll aprescription. A careful audit at Deliare has shown that the time it takes to 9ll arandomly selected prescription has a uniform distribution between ( and "( mins.4uppose a prescription is selected at random.

• Find the probability it takes at most 1 mins to 9ll the prescription• Find the probability it takes between 1 and " mins to 9ll the prescription• Find the mean time it takes to 9ll a prescription.

*1".

@ovartis harma claims that dru A relives many symptoms due to the commoncold. After the prescribed dose is taken, suppose the lenth of time #in hrs& untilsymptoms return is a random variable, , that has an e$ponential distribution withparameter .1.

• What is the probability the lenth of time until symptoms return is less thanmean!

• What is the probability the lenth of time until symptoms return is at least 1"hrs

• What is the probability the lenth of time until symptoms return is between 8and 1) hrs!

*13.

6n a factory, there are 3 machines producin respectively (%, "% and 3% of the

total output. 7he life distribution of a typical item produced by the machines are

e$ponential with mean 8 #in hrs&, 1 and 1" for 9rst, second and third machine

respectively. An item is drawn at random from the production lineE it is seen to

survive beyond 1( hrs.

• what is the probability that it came from machine 1!

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*1.

A carrier heat pump, desined to heat a home in the winter and cool the home

in summer, lasts, on averae , 1) years. 7he life time #in yrs& of this system can be

modeled by an e$ponential distribution with parameter .)"(. suppose a heat

pump is selected at random

• What is the probability the heat pump will last for at least ( yrs• 4uppose the heat pump lasts for 9ve years. What is the probability it will last

for at least another ( years.

*1(.

7he total weiht of a 9lled tire can dramatically aect the performance and safety of

an automobile. 4ome transportation oGcials arue that mechanics should check the

tire weihts of every vehicle as part of the annual inspection. 4uppose the weiht of

a 9lled tire is normally distributed with standard deviation 1."( pounds. 6n a random

sample of 1( 9lled tires, the sample mean weiht was 18./( pounds.

•

Find a ?(% con9dence interval for the true mean weiht of the tires.

*1).

=ihthouses are constructed to uide ships travellin in rocky waters and to allow

sailin at niht. 7here are many ways to report the si-e of a lihthouse. owever,

the heiht is usually measured from the base of the tower to the top of the

ventilator ball. A random sample of 18 lihthouses in France and 5nland was

obtained, and the heiht of each was recorded. 7he sample mean was 33./(

meters. Assume the distribution of lihthouse heihts is normal and population s.d.

is (. meters.

• Find ?(% '6 for the mean heiht

• ow lare a sample is necessary to ensure that the width of the resultin

?(% '6 is " meters!

*1/.

Histillation is a process for separatin and collectin substances accordin to their

reaction to heat. When heat is applied to a mi$ture, the substance that evaporates

and is collected as it cools is distillate. 7he unevaporated portion of the mi$ture is

the residue. il obtained from orane blossoms throuh distillation is used in

perfume. 4uppose the oil yield is normally distributed. 6n a random sample of

eleven distillations, the sample mean oil yield was ?8." rams with standard

deviation "/.) rams.• Find ?(% con9dence interval for the true mean oil yield per batch.

*18.

ih blood pressure, or hypertension, occurs when the force of blood aainst your

artery walls is too stron. 7his added pressure can cause a stroke. 6n a recent

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survey, 11 adult are randomly selected and e$amined for hih blood pressure.

7he number of patients classi9ed with hypertension is 31?.

• Find ?(% '6 for the true proportion of adult with hypertension

• 7he department of ealth and uman 4ervices has a taret of 1)% for the

hypertension prevalence by the year "1. 6s there any evidence to suest

the prevalnece of hypertension is dierent from the taret value!

*1?.

5ach year a reional 0ell telephone company inspects telephone poles and replaces

those that are defective. 6n order to eectively allocate resources, the company

plans to estimate the proportion of defective poles. A ?(% con9dence interval for p

with bound on the error of estimation ." is needed. ow lare a sample si-e is

necessary in each of the followin cases

• rior e$perience suest p<.1

• 7here is no prior information reardin the proportion of defective poles.

*".

5arthenware dishes are made from clay and are 9red, or e$posed to heat, in a lare

kiln. =are Iuctuations in the kiln temperature can cause cracks, bumps or other

Iaws. With kiln set at 8 deree, a random sample of 1? temperature

measurements was obtained. 7he sample variance was 1/.((

• Find the ?(% '6 for the true population variance

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