If X is distributed as , then the standardized variable has the standard normal distribution. This property of the normal distribution allows us to cast a probability problem concerning X into one concerning Z. To find the probability that X lies in a given interval, convert the interval to the z-scale and then calculate the probability by using the standard normal table.

If X is distributed as , then

where Z has the standard normal distribution.

The number of calories in a salad on the lunch menu is normally distributed with mean=200 and sd=5. Find the probability that the salad you select will contain
(a)
More than 208 calories.
(b)
Between 190 and 200 calories.

Letting X denote the number of calories in the salad, we have the standardized variable

(a)
The z-value corresponding to x=208 is . Therefore,
(b)
The z-values corresponding to x=190 and x=200 are and respectively. We calculate