00:01
All right, so we're told that there were some 36 ,000 high school students who took the positive two portion of the park exam in 2018 -19 school year.
00:11
And scores are given one of five levels.
00:14
And the levels that came through were missing one, because there should be five levels, but only four given, and the questions are asking us to do between levels three and five.
00:27
So it's missing.
00:28
So what i did is i went on the website and found the 2019 data.
00:33
And the website i found had these scores to determine the levels.
00:39
So i use these in our data.
00:42
But one of the beautiful things about deeply understanding this work is regardless of what numbers you get, you can plug in the numbers you want and you'll find the probabilities.
00:51
So we're told that the mean score was 750 .29, and the standardvation is 23 .83.
00:58
So here's a normal distribution, because we're told that it's normally distributed.
01:02
And so the mean is right in the center.
01:04
We're going to find the probability that a randomly selected high school student in new jersey who took the algebra two part of the park exam had a score that falls on level four.
01:15
So level four is between 750 and 809.
01:18
So what we're going to do is take our z score formula, which is x minus mu over sigma.
01:27
And we're given the mean and the standard deviation.
01:32
So the mean is 750 .29 and the standard evasion is 23 .83.
01:42
And the way we do this is we want to convert 750 and 809 into a z score.
01:52
So we put, and the reason for that is because then here's the mean of 750.
01:56
We'll put that right here, 750.
01:58
0 .29.
02:03
750 not to scale will be like right here just to the left of the mean and then 809 would be roughly here and what we're looking for is this area right here we need a picture pictures help us so much when we're doing with this so there is there's the what we're looking for right there so we do 750 minus 750 .29 divided by 22 .3 we get this is our z score same thing with 809 plug in 809 in for x here we get z score of 2 .46.
02:35
And then we can do a table lookup or i use my spreadsheet function, norm s dist, and you put in the z score and then out pops the area to the left, i should say.
02:51
Because the way we find the error in the middle is we take the error to the left of the lower value and subtract it from the area to the left of the upper value.
03:00
And then what's left is going to be what we want, which is that stuff right in the middle.
03:07
That's how that works.
03:10
So we basically want this in probability terms.
03:12
So we take the difference of these values, and we get 0 .49 -79 -7 -9 -28 -76.
03:21
And then part two or question two, we want to find the probability that i randomly selected high school student falls on level three.
03:28
So now we do the same thing, but we look at the level three levels, 725 and 749.
03:32
So we put 725 and for x in our z score calculator, we get negative 1 .06.
03:38
Same thing with 749.
03:39
We get negative 0 .05.
03:43
And we take the difference of what those, after we run the norm s -dist function, we get the areas to the left of each.
03:51
In other words, the probability being less than 725 and the probability being less than 7 .49.
03:57
And we take the difference of those, we get 0 .33.
03:59
There's a picture.
04:00
Again, pictures are nice.
04:04
So this goes up to 749.
04:06
So basically we want just to the left of it here.
04:10
And down to 725, which is roughly here, we'll say.
04:21
We don't want this.
04:23
We just want this area, and we want that.
04:30
Again, not to scale, but this is just to give you a picture of what we're looking at.
04:34
And it's 0 .334.
04:36
And then the third question asks us in the range of 3 to 5.
04:40
So that means between 725 and 850.
04:45
850 is the max and that's the next of our level 5.
04:49
So we do the same thing as we do in the other examples, except we go from 725 to 850.
04:54
So again, we put the, we already have the 725 already.
04:58
So we'd put the 850 in our x value.
05:00
We get this z score, 4 .184, do the norm as dysfunction on the z scores, get the error to the left of each.
05:07
If i take the difference, we get 0 .85...