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? O the first machine O the second machine You may need to use the appropriate table in the Appendix of Tables to answer this question. Need Help? Read It Talk to a Tutor

Answer 1

Solution :

Given that ,

mean = $\mu$ = 3 cm ( first machine )

standard deviation = $\sigma$ = 0.08 cm

P(2.9 < x < 3.1 ) = P[(2.9 - 3 )/ 0.08) < (x - $\mu$ ) /$\sigma$  < (3.1 - 3) / 0.08) ]

= P(-1.25 < z < 1.25)

= P(z < 1.25) - P(z < -1.25)

Using z table,

= 0.8944 - 0.1056

= 0.7888

mean = $\mu$ = 3.04 cm ( second machine )

standard deviation = $\sigma$ = 0.04 cm

P(2.9 < x < 3.1 ) = P[(2.9 - 3.04 )/ 0.04) < (x - $\mu$ ) /$\sigma$  < (3.1 - 3.04) / 0.04) ]

= P(-3.5 < z < 1.5)

= P(z < 1.5) - P(z < -3.5)

Using z table,

= 0.9332 - 0.0002

= 0.9330

The second machine more likely