Chapter 4, Problem 4.205CTC

Super weapons development. The U.S. Army is working with a major defense contractor (not named here for both confidentiality and security reasons) to develop a “super” weapon. The weapon is designed to fire a large number of sharp tungsten bullets— called flechettes—with a single shot that will destroy a large number of enemy soldiers. (Fiechettes are about the size of an average nail, with small fins at one end to stabilize them in flight.) The defense contractor has developed a prototype gun that fires 1,100 flechettes with a single round. In range tests, three 2-feet-wide targets were set up at a distance of 500 meters (approximately 1500 feet) from the weapon. Using a number line as a reference, the centers of the three targets were at 0, 5, and 10 feet, respectively, as shown in the accompanying figure. The prototype gun was aimed at the middle target (center at 5 feet) and fired once. The point x where each of the 1,100 flechettes landed at the 500-meter distance was measured using a horizontal grid. For example, a flechette with a horizontal value of x = 5.5 (shown in the figure) hit the middle target, but a flechette with a horizontal value of x = 2.0 (also shown in the figure) did not hit any of the three targets. The 1,100 measurements on the random variable x are saved in the accompanying file. (The data are simulated for confidentiality reasons.)

The defense contractor is interested in the likelihood of any one of the targets being hit by a flechette, and in particular, wants to set the gun specifications to maximize the number of target hits. The weapon is designed to have a mean horizontal value, E(x), equal to the aim point (e.g., μ = 5 feet when aimed at the center target). By changing specifications, the contractor can vary the standard deviation,  . The data file contains flechette measurements for three different range tests- one with a standard deviation of σ = 1 foot, one with σ = 2 feet, and one with σ = 4 feet. Let x₁, x₂, and x₄ represent the random variables for horizontal measurements with σ = 1, σ = 2, and σ = 4, respectively. From past experience, the defense con-tractor has found that the distribution of the horizontal flechette measurements is closely approximated by a normal distribution.

1. a. For each of the three values of σ, use the normal distribution to find the approximate probability that a single flechette shot from the weapon will hit any one of the three targets. [Hint: Note that the three targets range from - 1 to 1, 4 to 6, and 9 to 11 feet on the horizontal grid.]
2. b. The actual results of the three range tests are saved in the data file. Use this information to calculate the proportion of the 1,100 flechettes that actually hit each target—called the hit ratio—for each value of σ. How do these actual hit ratios compare with the estimated probabilities of a hit using the normal distribution?
3. c. If the U.S. Army wants to maximize the chance of hitting the target that the prototype gun is

    aimed at, what setting should be used for a? If the Army wants to hit multiple targets with a single shot of the weapon, what setting should be used for σ?