The diameter of an electric cable is normally distributed, with a mean of 0.9 inch and a standard deviation of 0.02 inch. What is the probability that the diameter will exceed 0.92 inch? (You may need to use the standard normal distribution table. Round your answer to three decimal places.)
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To solve this problem, we can use the concept of standard deviation and the standard normal distribution table.
First, we need to standardize the value of 0.92 inch using the formula:
Z = (X - μ) / σ
Where:
Z is the z-score,
X is the value we want to standardize (0.92 inch in this case),
μ is the mean of the distribution (0.9 inch),
σ is the standard deviation (0.02 inch).
Plugging in the values:
Z = (0.92 - 0.9) / 0.02
Z = 0.02 / 0.02
Z = 1
Now, we can use the standard normal distribution table to find the probability associated with a z-score of 1. The table provides the cumulative probability to the left of a given z-score.
Looking up a z-score of 1 in the table, we find that the cumulative probability is approximately 0.8413.
To find the probability that the diameter will exceed 0.92 inch, we subtract the cumulative probability from 1:
P(X > 0.92) = 1 - 0.8413 = 0.1587
Therefore, the probability that the diameter will exceed 0.92 inch is approximately 0.159 (rounded to three decimal places).