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 and standard deviation , the z-score of a measure X is given by:
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
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
has a p-value of 0.8944
X = 2.9
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, . Now to find the probability, same procedure.
X = 3.1
has a p-value of 0.9772
X = 2.9
has a p-value of 0
0.9772 - 0 = 0.9772
0.9772 = 97.72% probability that the second machine produces an acceptable cork.