The breaking strength (in pounds) of a certain new synthetic is normally distributed, with a mean of 160 pounds and a variance of 9. The material is considered defective if the breaking strength is less than 154 pounds. What is the probability a randomly selected sample will have a breaking strength greater than 166 pounds? Please round the answer to the nearest thousandth, if necessary.
A) 0.977
B) 0.023
C) 0.5
D) 0.841
E) 0.159

Question

Quizplus · Accepted Answer

We need to standardize the value of 166 pounds using the formula, Z = (X - μ)/σ where X is the breaking strength, μ is the mean, and σ is the standard deviation. Thus,

Z = (166 - 160)/3 = 2/3 = 0.6667

Now, we need to find the probability that the breaking strength is greater than 166 pounds, which is the same as finding the area to the right of Z = 0.6667 on a standard normal distribution table.

Using the table, we get the probability as 0.253. However, since the probability of the breaking strength being less than 154 pounds is also given, we need to subtract this from 1 to get the final probability as:

P(X > 166) = 1 - P(X < 154) -  P(Z < 0.6667)
            = 1 - 0.3085 - 0.253
            = 0.4385

Rounding this to three decimal places, we get the final answer as 0.023, which is option B.

The Breaking Strength (In Pounds) of a Certain New Synthetic