Problem: The observed error "E" in a series of measurements is normally distributed with mean of 0. Approximately 2% of error are -10 or less. Approximately what fraction of the measurements have errors between 0 and 5?
How do you solve this without using the z score table?
Like Andre Nicolas suggested, you can check in a table: you are told that the value -10 represents the 2 percentile. A table would give you an approximate match between percentile and standard deviation. Without a table, a rule-of-thumb you can use is the $68-95-99.7$% rule, and the symmetry of the normal deviation. By symmetry, around $47.5$% of the values will be 2 standard deviations below the mean. This implies that the value -10 is around 2 deviations below the mean $0$, so that the standard deviation is around $5$. This implies that the number of values between $0$ and $5$ is around $34.3$, following the same $68-95-99.7$% rule, since the interval $[0,5]$ covers around 1 standard deviation from the mean.