A machine that cuts corks for wine bottles produces corks with diameters that ha

Question

A machine that cuts corks for wine bottles produces corks with diameters that have a distribution that is well approximated by a normal distribution with mean = 30 mm and standard deviation 0.8 mm. A cork that is too small leaks, and a cork that is too big will not fit in the bottle. Specifications call for corks with diameters between 29 mm and 31 mm Corks that do not meet the specifications are considered defective. 4. (a) What proportion of corks produced by this machine are defective? (b) Suppose three corks are selected at random and independently of each other. What is the probability that: i. all three corks are defective? ii. none of the corks are defective? iii. at least one of the corks is defective? (c) Ten percent of the corks produced by this machine have diameters of mm or es

Explanation / Answer

Solution

(a) Let X be the diameter in mm

P(29<X<31) = P((29-30)/0.8<z<(31-30)/0.8))

P(-1.25<z<1.25) = 2P(Z<1.25)-1 = 78.87%

1- 78.87% = 21.13% corks produced by this machine are defective

(b) Let y be the defective corks

(1) P(Y=3) = 3C3(0.2113)3 = 0.0094

P(Y=0) = 3C0(0.7887)3 = 0.4906

P(Y>=1) = 1-P(Y=0) = 0.5094

(c) P(Z<(X-30)/0.8) = 0.1

X = 1.2816*0.8+30 = 31.0253

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