Velocity of Winchester bullets. The American Rifleman (June 1993) reported on the velocity of ammunition fired from the FEG P9R pistol, a 9 mm gun manufactured in Hungary. Field tests revealed that Winchester bullets fired from the pistol had a mean velocity (at 15 feet) of 936 feet per second and a standard deviation of 10 feet per second. Tests were also conducted with Uzi and the Black Hills ammunition.

1. Describe the velocity distribution of Winchester bullets fired from the FEG P9R pistol.
2. A brand unknown bullet is fired from the FEG P9R pistol. Suppose the bullet's velocity (at 15 feet) is 1,000 feet per second. Is the bullet likely to be manufactured by Winchester? Explain.

It is given that the mean velocity is 936 feet per second, and the standard deviation is 10 feet per second.

i.e.,$\overline{\mathrm{x}}=936\mathrm{and}\mathrm{s}=10$

Here, Chebyshev’s rule can be used to describe the distribution.

Since Chebyshev’s rule states that at least $\frac{3}{4}$of the data will fall within 2 standard deviations of the mean.

Therefore,

$$\begin{gathered} \overline{x}+2s=936\pm \left(2\right)\left(10\right) \\ =936\pm 20 \\ =\left(936-20,936+20\right) \\ =916,956 \end{gathered}$$

Thus, at least $\frac{3}{4}$of the data will fall in the interval (916,956).

Also, Chebyshev’s rule states that at least $\frac{8}{9}$of the data will fall within 3 standard deviations of the mean.

Therefore,

$$\begin{gathered} \overline{x}+3s=936\pm \left(3\right)\left(10\right) \\ =936\pm 30 \\ =\left(936-30,936+30\right) \\ =\left(906,966\right) \end{gathered}$$

Thus, at least $\frac{8}{9}$of the data will fall in the interval (906,966).

If an unknown bullet’s velocity is measured as 1,000 feet per second, it would be improbable that a well-known company manufactures this bullet. Its velocity would be almost 6 standard deviations above the mean of 936 if it were from our manufacturer, which is an improbable event.