The U.S. Army is working with a major defense contractor 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. Flechettes are about the size of an average nail, with small fins at one end to stabilize them in flight. Since World War I, when France dropped them in large quantities from aircraft on to masses of ground troops, munitions experts have experimented with using flechettes in a variety of guns. The problem with using flechettes as ammunition is accuracy: Current weapons that fire large quantities of flechettes have unsatisfactory hit ratios when fired at long distances.

The defense contractor (not named here for both confidentiality and security reasons) 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 a distance of 500 meters (approximately 1,500 feet) from the weapon. With a number line used as a reference, the centers of the three targets were at 0, 5, and 10 feet, respectively, as shown in Figure SIA5.1. The prototype gun was aimed at the middle target (center at 5 feet) and fired once. The point where each of the 1,100 flechettes landed at the 500-meter distance was measured with the use of a horizontal and vertical grid. For the purposes of this application, only the horizontal measurements are considered. These 1,100 measurements are saved in the MOAGUN file. (The data are simulated for confidentiality reasons.) For example, a flechette with a value of $x=5.5$ hit the middle target, but a flechette with a value of $x=2.0$ did not hit any of the three targets. (See Figure SIA5.1.)

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 hits. The weapon is designed to have a mean horizontal value equal to the aim point (e.g., $\mu =5$ feet when aimed at the center target). By changing specifications, the contractor can vary the standard deviation $\sigma$. The MOAGUN file contains flechette measurements for three different range tests: one with a standard deviation of $\sigma =1$ foot, one with $\sigma =2$ feet, and one with $\sigma =4$ feet.

In this chapter, two Statistics in Action Revisited examples demonstrate how we can use one of the probability models discussed in the chapter—the normal probability distribution—to aid the defense contractor in developing its “super” weapon.