00:01
In this question, we are looking at a normal distribution, so i'm going to start by drawing that.
00:09
The total area under this curve is 1.
00:12
It's a probability curve.
00:14
And it is symmetric, so the area to the left of the mean is 0 .5, to the right of the mean, 0 .5.
00:21
The mean, mu, is 3 ,262.
00:26
Standard deviation sigma is 1 ,100.
00:30
In part a, we want the probability that this randomly selected college c owes at least 1 ,000.
00:37
So that is below the mean, and we want at least.
00:42
So we want the area to the right.
00:46
If we find this area, that will be the probability that we need.
00:51
So the first step is to find the corresponding z score.
00:54
Z is x minus mu over sigma.
01:00
So we have 1 ,000 minus 3 ,262, so that's minus 2 -262, divided by 1 ,100.
01:11
And it's not a very nice number, so i'm going to leave this in exact form, because then i can put it into my calculator without rounding it and maintain accuracy.
01:23
Which brings us to the next step.
01:25
We need to turn this z score into an area under the curve.
01:28
So you could use a z score table, but it will make you round your z score.
01:31
You could use your graphical calculator, you could use statistical software like excel or r.
01:37
Either way, there are two functions you can use.
01:40
They are the standard normal and the cumulative normal.
01:44
You can use either one.
01:46
Standard gives you the area between x and mu.
01:52
So i put this in, i get this area.
01:55
That's not quite what i want, but if i add 0 .5, i get my total area.
02:01
Cumulative is the area to the left.
02:06
So it would give me this bit here in red.
02:09
Again, not what i want, but if i take it away from 1, i've left with the area to the right.
02:17
I'll use the standard.
02:19
So whatever i get, i'm going to add 0 .5 to it.
02:24
Okay, so i put this z score into the standard normal function.
02:28
It's about minus 2 .06.
02:30
And i get 0 .4801, which gives me 0 .9801 as my answer.
02:44
Or part b, we now want the probability of more than 4 ,000.
02:51
So that is above the mean.
02:55
I'll get rid of this.
03:00
And we want more than so this area here.
03:06
Okay, first step again is to find the z score.
03:09
4 ,000 minus 3 ,262 is 738 out of 1 ,100.
03:20
So that is, again, not a nice number...