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 0.81 inch? (Give the answer to four decimal places.)
To find the probability that the diameter will exceed 0.81 inch, we need to calculate the area under the normal distribution curve beyond that value. Here's how you can do it:
Step 1: Standardize the value
We need to convert 0.81 inch to a standardized value using the formula:
Z = (X - μ) / σ
Where:
Z is the standardized value
X is the observed value
μ is the mean of the distribution
σ is the standard deviation of the distribution
In this case:
X = 0.81 inch
μ = 0.8 inch
σ = 0.01 inch
Plugging these values into the formula, we get:
Z = (0.81 - 0.8) / 0.01
Z = 0.01 / 0.01
Z = 1
Step 2: Find the area under the curve
Now that we have the standardized value, we can find the area under the normal distribution curve beyond that value. Since we want to find the probability that the diameter will exceed 0.81 inch, we need to calculate the area to the right of the standardized value.
Using a standard normal distribution table, we can look up the area corresponding to a Z-score of 1. The closest value we find is 0.8413, which represents the area to the left of the Z-score.
However, we want the area to the right of the Z-score. To find that, we subtract the area we found from 1:
Area to the right = 1 - 0.8413
Area to the right ≈ 0.1587
So the probability that the diameter will exceed 0.81 inch is approximately 0.1587, rounded to four decimal places.