Question

For a normal distribution, what is the probability that the data falls somewhere .5 standard deviations from the mean?

Answer 1

`P(-0.5 <= Z <= +0.5) = 0.383`

Explanation:

For a normal distribution, if we are using the standard normal distribution `N(0,1^2)`, a z-score of +0.5 represents a half of a standard deviation above the mean `mu`. A z-score of -0.5 represents a half of a standard deviation below the mean `mu`.

This question is asking about the area under the normal distribution curve between the z-scores of -0.5 and +0.5. This can be done one of two ways. EIther you can find a z-score table which provides things known as cumulative from the mean areas, or you can use a regular z-score table (cumulative).

In the latter case, you have to be clever; you look up the +0.5 z-score, find that probability, and then subtract the -0.5 z-score value from the same table. This is because we are only interested in the amount between those z-score values, while cumulative z-score tables gives us the probability from `-oo` to the searched z-score value.

z-score +0.5: 0.6915
z-score -0.5: 0.3085

`P(-0.5 <= Z <= +0.5) = 0.6915 - 0.3085 = 0.383`