The fracture strength of a certain type of manufactured glass is normally distributed with a mean of
579 MPa with a standard deviation of 14 MPa. (a) What is the probability that a randomly chosen
sample of glass will break at less than 579 MPa? (b) More than 590 MPa? (c) Less than 600 MPa?
(Data are from Science 283 [February 26, 1999], p. 1296.)
Use process indicated in the response to your previous post.
To solve these probability questions, we will use the standard normal distribution and convert the values using z-scores. The standard normal distribution has a mean of 0 and a standard deviation of 1.
To find the probability that the fracture strength is less than a given value, we need to find the area under the curve to the left of that value.
(a) Probability that a randomly chosen sample of glass will break at less than 579 MPa:
First, we calculate the z-score for 579 MPa using the formula:
z = (x - mean) / standard deviation
Substituting the values, we get:
z = (579 - 579) / 14
z = 0
Now, we find the probability using a standard normal distribution table or a calculator. The probability of getting a z-score of 0 or less is 0.5000.
Therefore, the probability that a randomly chosen sample of glass will break at less than 579 MPa is 0.5000 or 50%.
(b) Probability that a randomly chosen sample of glass will break at more than 590 MPa:
Again, we calculate the z-score for 590 MPa:
z = (590 - 579) / 14
z = 11 / 14
z ≈ 0.7857
To find the probability of getting a z-score greater than 0.7857, we subtract the area under the curve from 0 to the left of 0.7857. Using the standard normal distribution table or a calculator, the probability is approximately 0.2177.
Therefore, the probability that a randomly chosen sample of glass will break at more than 590 MPa is 0.2177 or 21.77%.
(c) Probability that a randomly chosen sample of glass will break at less than 600 MPa:
We calculate the z-score:
z = (600 - 579) / 14
z ≈ 1.5
Using the standard normal distribution table or a calculator, we find that the probability of getting a z-score greater than 1.5 is approximately 0.0668.
Therefore, the probability that a randomly chosen sample of glass will break at less than 600 MPa is 0.0668 or 6.68%.