The fracture strength of a certain type of manufactured glass is normally distributed with a mean of 575 MPa with a standard deviation of 17 MPa.
a) What is the probability that a randomly chosen sample of glass will break at less than 575 MPa?(Round your answer to 4 decimal places.)
b) What is the probability that a randomly chosen sample of glass will break at more than 586 Mpa? (Round your answer to 4 decimal places.)
c) What is the probability that a randomly chosen sample of glass will break at less than 596 MPa? (Round your answer to 4 decimal places.)
To solve these probability questions, we need to use the standard normal distribution and the z-score.
a) To find the probability that a randomly chosen sample of glass will break at less than 575 MPa, we need to find the area to the left of 575 MPa on the standard normal distribution.
We can calculate the z-score using the formula: z = (x - μ) / σ, where x is the value we are interested in, μ is the mean, and σ is the standard deviation.
In this case, x = 575 MPa, μ = 575 MPa, and σ = 17 MPa.
Plugging in the values, we have z = (575 - 575) / 17 = 0.
Using a standard normal distribution table or calculator, we can find the probability corresponding to a z-score of 0 is 0.5000.
So, the probability that a randomly chosen sample of glass will break at less than 575 MPa is 0.5000.
b) To find the probability that a randomly chosen sample of glass will break at more than 586 Mpa, we need to find the area to the right of 586 Mpa on the standard normal distribution.
Again, we are looking for the z-score. Using the same formula as before, we have z = (586 - 575) / 17 = 0.6471.
Using a standard normal distribution table or calculator, we can find the probability corresponding to a z-score of 0.6471 is 0.7412.
So, the probability that a randomly chosen sample of glass will break at more than 586 Mpa is 0.7412.
c) To find the probability that a randomly chosen sample of glass will break at less than 596 Mpa, we need to find the area to the left of 596 Mpa on the standard normal distribution.
Again, we calculate the z-score: z = (596 - 575) / 17 = 1.2353.
Using a standard normal distribution table or calculator, we can find the probability corresponding to a z-score of 1.2353 is 0.8907.
So, the probability that a randomly chosen sample of glass will break at less than 596 Mpa is 0.8907.
To answer these questions, we need to use the concept of Z-scores and the standard normal distribution.
a) To find the probability that a randomly chosen sample of glass will break at less than 575 MPa, we need to calculate the area under the normal distribution curve to the left of 575 MPa.
Step 1: Convert the given value of 575 MPa to a Z-score using the formula: Z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.
Z = (575 - 575) / 17 = 0
Step 2: Look up the Z-score in the standard normal distribution table or use a Z-score calculator to find the corresponding probability.
The Z-score of 0 corresponds to a probability of 0.5000.
Therefore, the probability that a randomly chosen sample of glass will break at less than 575 MPa is 0.5000.
b) To find the probability that a randomly chosen sample of glass will break at more than 586 MPa, we need to calculate the area under the normal distribution curve to the right of 586 MPa.
Step 1: Convert the given value of 586 MPa to a Z-score.
Z = (586 - 575) / 17 = 0.6471
Step 2: Find the probability corresponding to the Z-score.
The Z-score of 0.6471 corresponds to a probability of 0.7404.
Therefore, the probability that a randomly chosen sample of glass will break at more than 586 MPa is 0.7404.
c) To find the probability that a randomly chosen sample of glass will break at less than 596 MPa, we need to calculate the area under the normal distribution curve to the left of 596 MPa.
Step 1: Convert the given value of 596 MPa to a Z-score.
Z = (596 - 575) / 17 = 1.2353
Step 2: Find the probability corresponding to the Z-score.
The Z-score of 1.2353 corresponds to a probability of 0.8907.
Therefore, the probability that a randomly chosen sample of glass will break at less than 596 MPa is 0.8907.