The breaking strength (in pounds) of a certain new synthetic is normally distributed, with a mean of 174 and a variance of 9. The material is considered defective if the breaking strength is less than 168 pounds. What is the probability that a single, randomly selected piece of material will be defective? (Give the answer to two decimal places.)
To find the probability that a single, randomly selected piece of material will be defective, we need to calculate the probability that the breaking strength is less than 168 pounds.
Given that the breaking strength of the synthetic material is normally distributed with a mean of 174 and a variance of 9, we can use the standard normal distribution to calculate the probability.
Step 1: Standardize the threshold value 168 using the formula: z = (x - μ) / σ
where x is the threshold value, μ is the mean, and σ is the standard deviation.
In this case, x = 168, μ = 174, and σ = √9 = 3.
So, z = (168 - 174) / 3 = -2/3.
Step 2: Look up the standardized value in the standard normal distribution table or use a calculator to find the cumulative probability associated with this z-score.
The cumulative probability that corresponds to z = -2/3 is approximately 0.2520.
Step 3: Subtract the cumulative probability from 1 to get the probability that the breaking strength is less than 168 pounds.
P(X < 168) = 1 - 0.2520 = 0.7480
Therefore, the probability that a single, randomly selected piece of material will be defective is approximately 0.75 or 75% (rounded to two decimal places).