# Given a normal distribution with u=20 and the standard deviation =2.5, how do you find the value of x that has (a) 25% of the distribution's area to the left and (b) 45% of the distributions area to the right?

May 17, 2015

Firstly we must look at the z-score formula, which is $z = \frac{\overline{x} - \mu}{\sigma}$

now we can add what we got into our formula so far.
$z = \frac{\overline{x} - 20}{2 , 5}$

now in question a
it tells us that $\Phi \left(z\right) = 0.25$ (note that they said 25% to the left)

$\Phi$ is the symbol to say you using the CDF of the normal distribution

So to solve for $z$ we can just use the z-score table, which we will get. (The table gives us area, or probability to the left)

Thus $z \approx - 0.675$

then we plot into our formula.

$- 0.675 = \frac{\overline{x} - 20}{2.5}$

then we solve to get $\overline{x}$

$\overline{x} = 18.3125$

now to go to question b

note that in section b here they ask for 45% to the right of the point, and our tables gives us to the left of the point.

as our table is using a CDF we know the total area underneat the curve is going to equal $1$ which will leave us with the sum.

$1 - \Phi \left(z\right) = 0.45$ so we actually look for
$\Phi \left(z\right) = 0.55$

Thus we get that $z \approx 0.13$ by using our table.

put the value into our formula and we get.

$0.13 = \frac{\overline{x} - 20}{2.5}$

then we solve for $\overline{x}$

$\overline{x} = 20.325$