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
0.81 inch is exactly one standard deviation (s.d.) above the mean of 0.80.
Assuming a normal distribution, you can look up tables, such as:
http://www.math.unb.ca/~knight/utility/NormTble.htm
to find that the probability of not exceeding 1 s.d. above the mean is 0.8413.
Therefore the probability of exceeding 1 s.d. is 1-0.8413.
To find the probability that the diameter will exceed 0.81 inch, we need to calculate the area under the normal distribution curve to the right of 0.81 inch.
In this case, we can use the standard normal distribution table or a statistical calculator that provides the probability associated with a given z-score.
The first step is to standardize the value of 0.81 inch into a z-score. The z-score can be calculated using the formula:
z = (x - μ) / σ
where:
- x is the value of interest (0.81 inch in this case)
- μ is the mean of the normal distribution (0.8 inch in this case)
- σ is the standard deviation of the normal distribution (0.01 inch in this case)
Substituting the values into the formula:
z = (0.81 - 0.8) / 0.01
z = 0.01 / 0.01
z = 1
Now that we have the z-score, we can find the probability associated with it. Using a standard normal distribution table, we can look up the probability corresponding to a z-score of 1. In this case, the probability will be the area under the curve to the right of the z-score.
Looking up the z-score of 1 in the table, we find that the corresponding probability is approximately 0.1587.
Therefore, the probability that the diameter will exceed 0.81 inch is approximately 0.1587, or 15.87%.