A scale drawing of a rectangular park is 5 inches wide and 7 inches long. The actual park length is 280 yards. What is the width
2 answers:
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The probability will be the area under the normal distribution 
from 24 to 32.
We normalize these so we can use the standard normal distribution
Typically we'd just use a computer to tell us the integral of the standard normal from -1 to 5/3. In the old days we'd use the z table. It tells us the normal integral from 0 to positive z.
We know the integral of the Gaussian from -1 to 1 standard deviation is 68%; from the z table we see the probability from 0 to 1 is half that, <span>0.34134. That will also be the integral from -1 to 0.
The z table says the integral from 0 to 5/3, 1.67, is </span><span>0.45254.
So our total probability, integral from -1 to 5/3, is
</span>
That's really from z=-1 to z=+1.67 so likely a bit off in the last couple digits.
We normalize these so we can use the standard normal distribution
Typically we'd just use a computer to tell us the integral of the standard normal from -1 to 5/3. In the old days we'd use the z table. It tells us the normal integral from 0 to positive z.
We know the integral of the Gaussian from -1 to 1 standard deviation is 68%; from the z table we see the probability from 0 to 1 is half that, <span>0.34134. That will also be the integral from -1 to 0.
The z table says the integral from 0 to 5/3, 1.67, is </span><span>0.45254.
So our total probability, integral from -1 to 5/3, is
</span>
That's really from z=-1 to z=+1.67 so likely a bit off in the last couple digits.