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?
Z = (x - mean)/Standard Deviation
Calculate the z scores for the various values. Then look them up in the back of your stats text called something like "areas under normal distribution" to find the probabilities.
I hope this helps.
To answer these questions, we need to use the standard normal distribution, also known as the z-distribution, since we are given the mean and standard deviation of the fracture strength in the population.
(a) Probability of breaking at less than 579 MPa:
First, we need to calculate the z-score, which represents the number of standard deviations a given value is from the mean:
z = (x - μ) / σ
In this case, x is 579 MPa, μ is the mean of 579 MPa, and σ is the standard deviation of 14 MPa.
z = (579 - 579) / 14 = 0
Next, we need to find the probability associated with the z-score using a standard normal distribution table or a calculator. Since the z-score is 0, the probability is 0.5, or 50%.
Therefore, the probability that a randomly chosen sample of glass will break at less than 579 MPa is 50%.
(b) Probability of breaking at more than 590 MPa:
Using the same steps as above, we calculate the z-score:
z = (590 - 579) / 14 = 0.79
To find the probability associated with this z-score, we need to subtract the cumulative probability up to the z-score from 1:
P(x > 590) = 1 - P(x ≤ 590)
Using a standard normal distribution table or a calculator, we can find the cumulative probability of a z-score of 0.79 to be approximately 0.786.
P(x > 590) = 1 - 0.786 ≈ 0.214
Therefore, the probability that a randomly chosen sample of glass will break at more than 590 MPa is approximately 0.214, or 21.4%.
(c) Probability of breaking at less than 600 MPa:
Again, we calculate the z-score:
z = (600 - 579) / 14 ≈ 1.50
Using a standard normal distribution table or a calculator, we find the cumulative probability of a z-score of 1.50 to be approximately 0.933.
Therefore, the probability that a randomly chosen sample of glass will break at less than 600 MPa is approximately 0.933, or 93.3%.