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 163 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
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to your calculated Z score.
To find the probability that a single, randomly selected piece of material will be defective, we need to calculate the cumulative probability of the breaking strength being less than 163 pounds.
To do this, we will standardize the breaking strength using the formula:
Z = (X - μ) / σ
where:
X = breaking strength
μ = mean of breaking strength
σ = standard deviation of breaking strength
In this case, X = 163, μ = 172, and σ = √9 = 3.
Substituting these values into the formula, we get:
Z = (163 - 172) / 3
Z = -3
Next, we will use the standardized value (Z-score) to find the cumulative probability using a standard normal distribution table or calculator. The cumulative probability represents the area under the standard normal curve to the left of the Z-score.
Looking up the Z-score of -3 in a standard normal distribution table, we find that the cumulative probability is approximately 0.0013.
Therefore, the probability that a single, randomly selected piece of material will be defective is 0.0013, or 0.13% (rounded to two decimal places).