# If X is a normal variable with mean 10 and SD 2, then what's the area under the normal curve of X bounded by the interval 7 and 13?

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• Many ways to do this. Here are two:

A) Transform your values to z space via z = (x - avg)/sd.

So the 7 becomes ?

And the 13 becomes ?

Now look up in your z table for the area between these two.

This usually requires to look up the individual areas & subtracting them.

What do you get?

B) Note that 7 & 13 are each 1.5 SD from the avg. You can apply

an extension of the 68–95–99.7_rule. Wiki it. You will see that

μ ± 1.5σ is 0.8664.

Done!

• You can determine this with a z-score table.

The area under a normal curve is 1.  So the area of a portion of that will be less than one.

First we need the z-scores of your two endpoints:

n = m + sz

Where n is the data point in question (7 and 13)

m is the mean (10)

s is the standard deviation (2)

z is the z-score (unknown)

7 = 10 + 2z and 13 = 10 + 2z

-3 = 2z and 3 = 2z

-1.5z and 1.5 = z

Using these points in a z-score table gives you the probability of a random data point being less than that point (aka, the area under the curve from -inf to that point).

So if you find the area under z = 1.5 and subtract it from the area under z = -1.5 that overlaps, the result is the area between the points.

P(z = 1.5) - P(z = -1.5)

0.9332 - 0.0668

0.8664