Problem 1. Bernoulli Distribution When a certain glaze is applied to a ceramic s

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Problem 1. Bernoulli Distribution

When a certain glaze is applied to a ceramic surface, theprobability is 5% that there will be discoloration, 20% that therewill be a crack, and 23% that there will be either discoloration ora crack, or both. Let X=1 if there is discoloration,and let X=0 otherwise. Let Y=1 if there isa crack, and let Y=0 otherwise. Let Z=1 ifthere is either discoloration or a crack, or both, and letZ=0 otherwise.

a.     LetpX denote the success probability forX. Find pX.

b.     LetpY denote the success probability forY. Find pY.

c.     LetpX denote the success probability forZ. Find pZ.

d.     Is itpossible both X and Y to equal 1?

e.     DoespZ = pX +pY?

f.     Does Z = X + Y? Explain.

Problem 2. BinomialDistribution

A certain large shipment comes with a guarantee that it containsno more than 15% defective items. If the proportion ofdefective items in the shipment is greater than 15%, the shipmentmay be returned. You draw a random sample of 10 items. Let X be the number of defective items in the sample.

a.     If infact 15% of the items in the shipment are defective (so that theshipment is good, but just barely), what is P(X7)?

b.     Basedon the answer to part (a), if 15% of the items in the shipment aredefective, would 7 defectives in a sample size 10 be an unusuallylarge number?

c.     If youfound that 7 of the 10 sample items were defective, would this beconvincing evidence that the shipment should be returned? Explain.

d.     If infact 15% of the items in the shipment are defective, what isP(X2)?

e.     Basedon the answer to part (d), if 15% of the items in the shipment aredefective, would 2 defectives in a sample of size 10 be anunusually large number?

f.     If you found that 2 of the 10 sample items were defective, wouldthis be convincing evidence that the shipment should bereturned? Explain.

Problem 3. PoissonDistribution

The number of defective components produced by a certain processin one day has a Poisson distribution with mean 20. Eachdefective component has probability of 0.60 of beingrepairable.

a.     Findthe probability that exactly 15 defective components areproduced.

b.     Giventhat exactly 15 defective components are produced, find theprobability that exactly 10 of them are repairable.

c.     LetN be the number of defective components produced, and letX be the number of them that are repairable. Giventhe value of N, what is the distribution ofX?

d.     Findthe probability that exactly 15 defective components are produced,with exactly 10 of them being repairable.

Problem 4. OtherDistribution

In a lot of 10 microcircuits, 3 are defective. Fourmicrocircuits are chosen at random to be tested. LetX denote the number of tested circuits that aredefective.

a.     FindP(X=2).

b.     FindmX.

c.     FindsX.

Problem 5. NormalDistribution

Penicillin is produced by the Penicillium fungus, whichis grown in a broth whose sugar content must be carefullycontrolled. The optimum sugar concentration is 4.9mg/mL. If the concentration exceeds 6.0 mg/mL, the fungusdies and the process must be shut down for the day.

a.     Ifsugar concentration in batches of broth is normally distributedwith mean 4.9 mg/mL and standard deviation 0.6 mg/mL, on whatproportion of days will the process shut down?

b.     Thesupplier offers to sell broth with a sugar content that is normallydistributed with mean 5.2 mg/mL and standard deviation 0.4mg/mL. Will this broth result in fewer days of productionlost? Explain.

Problem 6. Central LimitTheorem

A certain electronic component manufacturing process producesparts, 205 of which are defective. Parts are shipped in unitsof 400. Shipments containing more than 90 defective parts maybe returned. You may assume that each shipment constitutes asimple random sample of parts.

a.     What isthe probability that a given shipment will be returned?

b.     On aparticular day, 500 shipments are made. What is theprobability that 60 or more of these shipments are returned?

c.     A newmanufacturing process is introduced that is supposed to reduce thepercentage of defective parts. The goal of the company is toreduce the probability that a shipment is returned to 0.01. What must the percentage of defective parts be in order to reachthis goal?

Explanation / Answer

X=1 if discolored
X=0 otherwise
px= .05
Y=1 if cracked
Y=0 otherwise
py= .2
Z=1 if either discolored or cracked
Z=0 otherwise
pz= .23

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