Question

There are two machines available for cutting corks intended for use in wine bottles. The first produces corks with diameters that are normally distributed with mean 3 cm and standard deviation 0.08 cm. The second machine produces corks with diameters that have a normal distribution with mean 3.04 cm and standard deviation 0.04 cm. Acceptable corks have diameters between 2.9 cm and 3.1 cm. What is the probability that the first machine produces an acceptable cork? (Round your answer to four decimal places.) What is the probability that the second machine produces an acceptable cork? (Round your answer to four decimal places.) Which machine is more likely to produce an acceptable cork?

Answer 1

SOLUTION:

From given data,

There are two machines available for cutting corks intended for use in wine bottles. The first produces corks with diameters that are normally distributed with mean 3 cm and standard deviation 0.08 cm. The second machine produces corks with diameters that have a normal distribution with mean 3.04 cm and standard deviation 0.04 cm. Acceptable corks have diameters between 2.9 cm and 3.1 cm.

=> The first produces corks with diameters that are normally distributed with mean 3 cm and standard deviation 0.08 cm

mean = 3

s = 0.08

What is the probability that the first machine produces an acceptable cork?

The probability that the first machine produces an acceptable cork is

P(2.9<X<3.1) = P[(2.9-3)/0.08 < (X-mean)/s < (3.1-3)/0.08]

=P(-1.25<Z<1.25)

= P(Z<1.25) - P(Z<-1.25)

= 0.8944 - 0.1056 (from standard normal table)

= 0.7887

What is the probability that the second machine produces an acceptable cork?

The probability that the second machine produces an acceptable cork is

P(2.9<X<3.1) = P((2.9-3.04)/0.02 <Z< (3.1-3.04)/0.02)

=P(-7<Z<3)

= P(Z<3) - P(Z<-7) (-7 is more than the -3.4 so consider as 0)

= 0.9987 - 0  (from standard normal table)

=0.9987

Which machine is more likely to produce an acceptable cork?

By observing above values, then we can say ,

the second machine is more likely to produce such corks...