The 68-95-99.7 rule only works at whole numbers of standard deviations. To find the proportion below any value, convert it to a z-score and look up the area under the standard normal curve (mean 0, SD 1). Tables and calculators give that area; the problems here print the table values you need.
Key idea
$P(X < x) = P\!\left(Z < \frac{x - \mu}{\sigma}\right)$, read from a table of left areas. Then: below = the left area; above = $1 -$ the left area; between = the larger left area minus the smaller one.
Why it works
Standardizing does not change which values are below which: $x$ is below the cutoff exactly when its z-score is below the cutoff's z-score. So the proportion below $x$ on any normal curve equals the area to the left of $z$ on the standard one. The total area is 1, which is why "above" is the complement of "below." For a smooth curve the single point $x$ has no area, so "below" and "at most" are the same.
Worked example: Below a value
Battery lifetimes are normal with mean 500 hours and SD 50. What proportion last less than 560 hours? (A table gives $P(Z < 1.20) = 0.8849$.)
- $z = \dfrac{560 - 500}{50} = 1.20$.
- The table gives the left area directly: $0.8849$.
Answer: About $0.8849$, or 88.5%.
Worked example: Above a value
What proportion last more than 560 hours?
- Same z-score: $1.20$, with left area $0.8849$.
- Above is the rest: $1 - 0.8849 = 0.1151$.
Answer: About $0.1151$.
Worked example: Between two values
What proportion last between 450 and 560 hours? ($P(Z < -1.00) = 0.1587$.)
- Lower: $z = (450 - 500)/50 = -1.00$, left area $0.1587$.
- Upper: $z = 1.20$, left area $0.8849$.
- Between: $0.8849 - 0.1587 = 0.7262$.
Answer: About $0.7262$.
Common mistake
Reporting the table's left area when the question asks "more than." The table always gives the area to the left. Check with a quick sketch: shade the region the question asks about. If you shaded the right side, subtract from 1. Also check the size: a value above the mean must have less than half the data above it.
By symmetry, $P(Z > 1.2) = P(Z < -1.2)$. Either way the answer is $0.1151$.