X is a normally distributed random variable with mean 57 and standard deviation 17. What is the probability that X is between 40 and 74?

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Actually, the probability density of the normally distributed random variable is known. The probability in question is the integral of this probability density over the given interval.

Because this integral is not an elementary function, it is common to use approximate computations, and a z-table is the most used tool. Such a table is for the standard normal distribution. To use it, we need to transform our data.

The general formula is `L' = (L-mu)/sigma,` `U' = (U-mu)/sigma,` where `L` is the lower value (`40` in our case), `U` is the upper value `(74),`  `mu` is the mean `(57)` and `sigma` is the standard deviation `(17).` `L'` and `U'` are the equivalent endpoints for the standard normal distribution which may be used with a z-table.

Our case is somewhat special (very well-known), because

`L-mu = 40-57 = -17 = -sigma,` and  `U-mu = 74-57 = 17 = sigma,`

the limits are one standard deviation across the mean. The corresponding probability is about 68%, as we know from the mnemonic "68-95-99.7 Rule".

So the answer is about 0.68.

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