The breaking strength (in pounds) of a certain new synthetic is normally distributed, with a mean of 167 and a variance of 9. The material is considered defective if the breaking strength is less than 158 pounds. What is the probability that a single, randomly selected piece of material will be defective? (Give the answer to two decimal places.)
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Z = (score-mean)/SD
variance = SD^2
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 need to find the area under the normal distribution curve to the left of the breaking strength value of 158 pounds.
Step 1: Calculate the standard deviation (σ) by taking the square root of the variance:
√(variance) = √9 = 3
Step 2: Standardize the breaking strength of 158 pounds using the formula:
z = (x - mean) / standard deviation
where x is the breaking strength value
z = (158 - 167) / 3
z = -3
Step 3: Use a standard normal distribution table or calculator to find the cumulative probability associated with the z-score. The cumulative probability represents the area under the standard normal distribution curve to the left of the given 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 approximately 0.0013, or 0.13% (rounded to two decimal places).