The U.S. Department of
Defense makes use of statistics in designing weapon systems
for tanks and other forms of armor. By measuring
the results of the firing of a weapons system under simulated combat conditions, ballistics researchers can estimate the accuracy of the shells fired. If the design
specifications require that 99% of the shells fired fall within 1.5 meters of the target when the system is operating
properly, what is the maximum allowable standard deviation for the system?
± 1.5 m = ± 2 SD = .99%
To determine the maximum allowable standard deviation for the system, we need to use some statistical principles. First, let's understand the context of the problem.
The U.S. Department of Defense wants to design a weapon system for tanks and armor, and they want to ensure that the shells fired from the system are accurate. The design specifications state that 99% of the shells fired should fall within 1.5 meters of the target under simulated combat conditions.
Now, we need to consider the normal distribution and use the concept of empirical rule or the 68-95-99.7 rule. According to this rule, for a normal distribution, approximately 68% of the data falls within one standard deviation, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.
In this case, since the desired accuracy is 99%, we can consider it falls within the three standard deviations. So, we need to find the standard deviation for which 99% of the shells fall within 1.5 meters.
To solve this, we can use a standard normal distribution table or a statistical calculator. The standard normal distribution table provides the area under the curve (probability) up to a given number of standard deviations.
Since we want to find the maximum allowable standard deviation, we need to find the value of z for which 99% of the area lies to the left. Using the standard normal distribution table, this value is approximately 2.33.
Now, we can use the formula for standard deviation to find the maximum allowable value:
Maximum allowable standard deviation = (Desired accuracy / Number of standard deviations)
Maximum allowable standard deviation = 1.5 meters / 2.33
Calculating this, the maximum allowable standard deviation for the system is approximately 0.6431 meters.
Therefore, to meet the design specification of having 99% of the shells fired fall within 1.5 meters of the target, the maximum allowable standard deviation for the system is approximately 0.6431 meters.