1. 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.

Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores calculated.

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To find the probabilities for the given questions, we will use the standard normal distribution.

(a) To find the probability that a randomly chosen sample of glass will break at less than 579 MPa, we need to find the area to the left of the value 579 in the standard normal distribution.

To calculate this, we will use the standard normal table (also known as the Z-table):

1. First, we need to standardize the value 579 using the formula:
Z = (X - μ) / σ

Where:
X is the value we want to standardize (579 MPa)
μ is the mean of the distribution (579 MPa)
σ is the standard deviation of the distribution (14 MPa)

In this case, Z = (579 - 579) / 14 = 0

2. Find the corresponding area in the standard normal table for Z = 0.

Looking up the Z-table, the area corresponding to Z = 0 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) To find the probability that a randomly chosen sample of glass will break at more than 590 MPa, we need to find the area to the right of the value 590 in the standard normal distribution.

1. Standardize the value 590:
Z = (590 - 579) / 14 = 0.7857 (approximately)

2. Find the corresponding area in the standard normal table for Z = 0.7857.

Looking up the Z-table, the area corresponding to Z = 0.7857 is 0.7832 (approximately).

The area to the left of Z = 0.7857 is 0.7832, so the area to the right of Z = 0.7857 is equal to 1 - 0.7832 = 0.2168 (approximately).

Therefore, the probability that a randomly chosen sample of glass will break at more than 590 MPa is 0.2168 or 21.68%.

(c) To find the probability that a randomly chosen sample of glass will break at less than 600 MPa, we need to find the area to the left of the value 600 in the standard normal distribution.

1. Standardize the value 600:
Z = (600 - 579) / 14 = 1.5

2. Find the corresponding area in the standard normal table for Z = 1.5.

Looking up the Z-table, the area corresponding to Z = 1.5 is 0.9332 (approximately).

Therefore, the probability that a randomly chosen sample of glass will break at less than 600 MPa is 0.9332 or 93.32%.