The probability will be the area under the normal distribution [tex]N(27,3^2)[/tex] from 24 to 32.
We normalize these so we can use the standard normal distribution
[tex]z_1 = \dfrac{24 - 27}{3} = -1[/tex]
[tex]z_2 = \dfrac{32-27}{3}=\dfrac 5 3[/tex]
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, 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 0.45254.
So our total probability, integral from -1 to 5/3, is
[tex]p =0.34134+0.45254=0.79388[/tex]
That's really from z=-1 to z=+1.67 so likely a bit off in the last couple digits.
