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?

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Answer 1 · 2015-05-17T08:16:17

Firstly we must look at the z-score formula, which is $z = \frac{¯ x - μ}{σ}$ $z= (barx - mu)/sigma$

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

now in question a
it tells us that $Phi(z)= 0.25$ (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 $z ~~ -0.675$

then we plot into our formula.

$-0.675 = (barx - 20)/2.5$

then we solve to get $barx$

$barx = 18.3125$

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.

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

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

put the value into our formula and we get.

$0.13 = (barx-20)/2.5$

then we solve for $barx$

$barx = 20.325$