00:01
All right, so we have some questions about gpa.
00:03
All right, so the average grade point average of undergrad students in new york is normally distributed with the population mean of 2 and 1 half and the population of student deviation of 0 .5.
00:16
So we're going to compute the following.
00:18
We're going to compute the percentage of students with gpas between 1 .3 and 1 .8.
00:24
So here's a normal distribution.
00:29
Here's our mean of 2 .5.
00:32
We want 1 .3 to 1 .8 we want to know this region right here.
00:41
Let me do that over here with a little more space.
00:45
So we want to know this.
00:47
Here's our mean.
00:49
Here's the 1 .3 over here.
00:53
And this is not to scale but just to give you an idea here what we're looking at.
00:59
The main part is that's below the mean.
01:01
It's kind of an important piece here.
01:03
We know that region.
01:04
So it's a we're going to do a little z.
01:07
Score calculation so we want to find the z and the z score calculator is given is x minus the mean over the standard deviation so we find the z score for each of these values of 1 .8 and 1 .3 and we get them to be negative 2 .4 and negative 1 .4 respectively and the mean has z score of 0 and then you do a little table lookup to find the probabilities.
01:39
And i use the spreadsheet.
01:41
And so what you do is you want to find the probability, the probability that x is between 1 .3 and 1 .8.
01:55
We just compute it to a z score of negative 2 .4 less than z less than negative 1 .4.
02:03
So how do we compute this? the actual find the probabilities, given these p values we found.
02:08
So what we do is we take the probability that z is less than negative 1 .4 and we subtract from that the probability that z is less than negative 2 .4 and the reason we do that in a colorful example this value this point 08 is all this stuff from negative 1 point or 1 .8 to the left but we don't want the whole that whole thing we want to actually subtract off the part from 1 .3 into the left and and so that's why we do 0 .08 minus 0 .008.
02:50
And we end up with .072.
02:57
Or if we round it to, so when i did this, just so you know, i calculated this with the precision that my spreadsheet gave me, so i found it found it to be this value.
03:06
So i'll just do that, but depending on the precision you want to go to and what you're asked to do, beware of that.
03:12
So the answer for one, the percentage of students with gpas between 1 .3 and 1 .8 is about 7 .2, 7 .3%.
03:29
We're going to find the percentage of students with gpas above 3 .0.
03:32
Let me erase this.
03:36
We'll stick with our graph though.
03:38
It's kind of nice to see this for our normal distribution.
03:46
So we still have our mean right here in the center.
03:57
We want to know gpa's above three, so here's 3 .0 that is, and we want to know all this stuff to the right.
04:05
So we're going to standardize it.
04:06
And it ends up with the z score of 1, because 3 minus 2 .5 over 1 1 is 1.
04:17
And that's the z value.
04:22
And we still have to do a little bit of algebra, because my p -value, my z table doesn't give me the error to the right.
04:33
The way we do this, is because what we're going to find is the probability or the percentage of, the x is greater than 3 .0, but we convert it to a z score, so it became z greater than 1.
04:51
So the way to calculate this is we do 1 minus the probability that z is less than 1.
05:02
And similar to before, that's area to the left is easy to find from the table, because most tables give area to the right.
05:11
That's what mine does anyway.
05:13
So it's 1 minus 0 .84, which gives us 0 .16.
05:21
Like i said, i rounded to the higher precision.
05:24
So the value we get us here.
05:28
So for our purposes, we'll round it to 16 % while we're doing hundreds.
05:36
So it will be consistent.
05:38
We'll do 1 .5 .9%.
05:43
15 .9%.
05:46
But what gpa will the top 3 % of students be? so for that, we want to know some z score, some x value, so right here's our mean, some x value such that the area to the right is .03.
06:16
So the way we do this is we do a reverse table look at.
06:24
We look for the z score that corresponds with an area of .03.
06:31
And be consistent with my colors from the floor.
06:34
Here's our mean at the center.
06:37
Here's the unknown x, this area.
06:42
.3 is what we're trying to find.
06:43
And we find that z score is 1 .881.
06:50
So we undo this.
06:57
This, oh, and just so, you know, the way i got that is i looked for the probability, i did the probably, you know, doing the exact opposite, the probability that z is less than some value, such that it equals 0 .97, because that's the complement of 0 .03.
07:18
So just to be clear on that.
07:20
And then we do, what we do is we know the z score.
07:24
We use this formula.
07:25
We know the z score.
07:26
We don't know the x.
07:27
So we just fill in what we have.
07:28
1 .881 equals some unknown x minus the mean of 2 and a half all over the standard deviation of 0 .5.
07:36
And we get this x value, we just do algebra.
07:41
So we do 1 .881 times .5, and we add 2 1ā2.
07:49
And that's our x value, which is 3 .445.
07:55
So that means the gpa would be 3 .45.
08:04
Well, you might be saying, i mean, it depends how precise you want to be like, this place will actually be a little bit less than 3%.
08:18
If i want to be a little more precise, it would be 405, 4405.
08:24
But most gpas are to the 100s point.
08:29
These are all measured in the 10th place.
08:32
So just 3 .4 wouldn't quite be that there would be a little less, a little more than 3, but we could even say 3 .5 to make sure we've captured it with the top 3...