The breaking strength (in pounds) of a certain new synthetic is normally distributed, with a mean of 172 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? (Give the answer to two decimal places.)
Z = (score-mean)/SD
SD = √variance
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.
To find the probability that a single, randomly selected piece of material will be defective, we can use the concept of the standard normal distribution.
First, we need to standardize the breaking strength value of 166 pounds using the formula for standardization:
Z = (X - μ) / σ
Where:
X = breaking strength value (166 pounds)
μ = mean (172 pounds)
σ = standard deviation (square root of variance = square root of 9 = 3 pounds)
Substituting the values into the formula:
Z = (166 - 172) / 3
Z = -6 / 3
Z = -2
Now, we need to find the area under the standard normal curve to the left of Z = -2. We can consult a standard normal distribution table or use statistical software to calculate this probability.
Using a standard normal distribution table, the probability associated with Z = -2 is approximately 0.0228.
Therefore, the probability that a single, randomly selected piece of material will be defective is approximately 0.0228 (or 2.28% when rounded to two decimal places).