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 0.03 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.)
Please explain the math behind your answer so I am able to understand!(:

Answer 1

Answer:

0.7888 = 78.88% probability that the first machine produces an acceptable cork.

0.9772 = 97.72% probability that the second machine produces an acceptable cork.

Step-by-step explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

First machine:

Mean 3 cm and standard deviation 0.08 cm, which means that $\mu = 3, \sigma = 0.08$

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

This is the p-value of Z when X = 3.1 subtracted by the p-value of Z when X = 2.9. So

X = 3.1

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{3.1 - 3}{0.08}$

$Z = 1.25$

$Z = 1.25$ has a p-value of 0.8944

X = 2.9

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{2.9 - 3}{0.08}$

$Z = -1.25$

$Z = -1.25$ has a p-value of 0.1056

0.8944 - 0.1056 = 0.7888

0.7888 = 78.88% probability that the first machine produces an acceptable cork.

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

For the second machine, $\mu = 3.04, \sigma = 0.03$. Now to find the probability, same procedure.

X = 3.1

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{3.1 - 3.04}{0.03}$

$Z = 2$

$Z = 2$ has a p-value of 0.9772

X = 2.9

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{2.9 - 3.04}{0.03}$

$Z = -4.67$

$Z = -4.67$ has a p-value of 0

0.9772 - 0 = 0.9772

0.9772 = 97.72% probability that the second machine produces an acceptable cork.