The number of nails of a given length is normally distributed with a mean length of 5 in. and a standard deviation of 0.03 in. In a bag containing 120 nails, how many nails are less than 4.94 in. long?
We first note that we can convert the lengths to standard deviation units by subtracting the mean and dividing by the standard deviation. The length $\ell = 4.94$ in. corresponds to \[z = \frac{\ell - 5}{0.03} = \frac{4.94 - 5}{0.03} = -2.\]The probability that a normally-distributed quantity is less than $-2$ is $\frac{1}{4.7^2} = \frac{1}{22.1} \approx \boxed{0.045}$ (to three decimal places). Therefore, we estimate that $\boxed{0.045 \times 120 \approx 5}$ nails are less than 4.94 inches long in the bag.
