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 use the concept of the standard normal distribution.
Step 1: Standardizing the value
First, we need to standardize the value of 0.81 inch using the formula:
Z = (X - μ) / σ
where:
Z = the standardized value (also known as the z-score)
X = the value we want to standardize (0.81 inch in this case)
μ = the mean of the distribution (0.8 inch)
σ = the standard deviation of the distribution (0.01 inch)
Using the values given, we can calculate the z-score:
Z = (0.81 - 0.8) / 0.01
Z = 0.01 / 0.01
Z = 1
Step 2: Finding the probability
Next, we need to find the probability that the standardized value (Z) is greater than 1. We can do this by referring to the standard normal distribution table or by using statistical software.
Using the standard normal distribution table, we find that the probability associated with Z = 1 is approximately 0.8413.
Since the question asks for the probability that the diameter will exceed 0.81 inch, we need to find the area beyond Z = 1. This is equal to 1 - the area up to Z = 1.
P(Z > 1) = 1 - P(Z < 1)
P(Z > 1) = 1 - 0.8413
P(Z > 1) = 0.1587
Therefore, the probability that the diameter will exceed 0.81 inch is approximately 0.1587 (or 15.87% when expressed as a percentage), rounded to four decimal places.