# The mean of a normally distributed population is 400 pounds.The standard deviation is 10 pounds.What is the probability of selecting a value between the mean and 415 pounds?

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### 1 Answer

We have X ~ N(400,10)

The probability of selecting x between the mean and 415 pounds is

Pr(400 < X < 415) = Pr(X < 415) - Pr(X ` `< 400)

Converting to Z scores so that we can use Normal tables for the cdf

`z_1 = (415-400)/10 = 1.5` and `z_2 = 0`

Now Pr(400 < X < 415) = Pr(Z < `z_1` ) - Pr(Z < `z_2` ) where Z ~ N(0,1)

For `z_1` = 1.5 the table gives a probability in the lower tail of 0.560

For `z_2` = 0 we know that the probability in the lower tail is 0.5 (z=0 is exactly in the middle of the distribution)

**So the probability we want is 0.560 - 0.5 = 0.060**