00:01
In this question, we are looking at a normal distribution.
00:05
So let's start by drawing it.
00:09
Now, we don't know the mean, and we don't know the standard deviation.
00:13
But we do know a couple of facts.
00:15
We know, if i put this here, this is 2 ,500.
00:20
We know that the area to the left is 8%.
00:24
I put on 1 ,500, the area to the left is 1 .4%.
00:30
So i could take those percentages and find the z -score.
00:33
That's what i'm going to do.
00:36
So we'll start there.
00:38
So we want the z score for 2 ,500, we want the z score for 1 ,500.
00:43
Because there's a formula, z is equal to x minus mu over sigma.
00:51
So i know x is 2 ,500, i can get z from the probability, which would leave me with two unknowns, the mean and the standard deviation.
01:01
If i have two of those equations, two equations, two unknowns, i can solve for that.
01:07
So to get from area under this curve to the z score, i need to use the inverse normal function.
01:15
This could be found on a graphical calculator.
01:17
You can also find it on software like excel.
01:20
This is a cumulative function, so you put in the area to the left, and it gives you the z score.
01:27
So for 2 ,500, i put in 8%, 0 .088, and it gives me z is 1 .4051 to four decimal places.
01:39
For 1 ,500, i put in 1 .4%, 0 .014, get minus 2 .1973 to four decimal places.
01:51
Now i'm going to construct my simultaneous equations.
01:57
So we have that back there.
02:03
2 ,500 is equal to the mean minus, 1 .4051 sigma, 1 ,500, so it's just rearranging this, is the mean minus 2 .1973, sigma.
02:22
And i'm just going to take the second equation away from the first, and get 1 ,000 is equal to.
02:30
So we have a minus 1 .4051 minus 2 .1973, 0 .7 .0 .7 -922 sigma.
02:46
So from this we get the standard deviation.
02:49
So i've got 1 ,000 divided by.
02:52
That value is 1 -262 .31.
02:57
And the mean, i can now get from this.
03:01
So i'll just take this and multiply it by 1 .4051.
03:07
So this bit here is 1773 and so on...