Question

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?

Answer 1

Firstly we must look at the z-score formula, which is `z= (barx - mu)/sigma`

now we can add what we got into our formula so far.
`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`