The diameter of an electric cable is normally distributed, with a mean of 0.8 inch and a standard deviation of 0.01 inch. What is the probability that the diameter will exceed 0.815 inch? (Give the answer to four decimal places.)
To find the probability that the diameter will exceed 0.815 inch, we need to calculate the area under the normal distribution curve to the right of 0.815.
First, we need to convert the given diameter into a standard score (z-score) using the formula:
z = (x - μ) / σ
where x is the given diameter, μ is the mean, and σ is the standard deviation.
Substituting the values into the formula, we get:
z = (0.815 - 0.8) / 0.01
z = 0.015 / 0.01
z = 1.5
Next, we can use the standard normal distribution table or a calculator to find the cumulative probability associated with the z-score of 1.5. This represents the probability of getting a value less than or equal to the given z-score.
Looking up the z-score of 1.5 in the table or using a calculator, we find that the cumulative probability is approximately 0.9332.
Since we want the probability of the diameter exceeding 0.815, we subtract the cumulative probability from 1:
P(diameter > 0.815) = 1 - 0.9332
P(diameter > 0.815) ≈ 0.0668
Therefore, the probability that the diameter will exceed 0.815 inch is approximately 0.0668 or 6.68% (rounded to four decimal places).