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?

Answer 1

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.