The breaking strength (in pounds) of a certain new synthetic is normally distributed, with a mean of 175 and a variance of 9. The material is considered defective if the breaking strength is less than 166 pounds. What is the probability that a single, randomly selected piece of material will be defective? (
z=(166-175)/9=-1
Look up the probability for z=-1.
To solve this problem, we can use the properties of a normal distribution. We will calculate the probability using the z-score formula.
The formula for the z-score is:
z = (x - μ) / σ
Where:
- x is the value we want to find the probability for (166 pounds)
- μ is the mean of the distribution (175 pounds)
- σ is the standard deviation of the distribution (square root of the variance, which is 3 pounds)
First, we need to calculate the z-score:
z = (166 - 175) / 3
z = -3
Next, we can use a z-table or calculator to find the probability associated with the z-score of -3. The probability of a z-score below -3 is approximately 0.0013.
So, the probability that a single, randomly selected piece of material will be defective is approximately 0.0013 or 0.13%.