The number of nails of a given length is normally distributed with a mean length of 5.00 in. and a standard deviation of 0.03 in. Find the number of nails in a bag of 120 that are between 4.97 and 5.03 in. Long.

Accepted Solution

4.97 is 1 standard deviation below the mean, since 5.00 - 0.03 = 4.97. Similarly, 5.03 is 1 standard deviation above the mean. The 68-95-99.7 rule (sometimes called "empirical rule") says that approximately 68% of any normally distributed population lies within 1 standard deviation of the mean, so[tex]P(4.97<X<5.03)\approx0.68[/tex]So out of 120 nails, we can expect [tex]0.68\cdot120=81.6\approx82[/tex] nails to be within the prescribed length.