00:02
Once again, welcome to a new problem.
00:03
This time we're dealing with standard normal distributions.
00:10
We're looking at standard normal distributions.
00:13
And when you think about these distributions, we have bell -shaped distribution, and the middle is the mean and the standard deviation.
00:26
One standard deviation from the mean is the same as 68 % of the data, and then we have two standard deviations from the mean.
00:47
This represents 95 % of the data.
00:56
And then, of course, we have three standard deviations from the mean.
01:08
Which is 97 point or rather 99 .7 % of the data.
01:21
We're given a normal cholesterol model.
01:31
And in this particular model, the mean happens to be 185 milligrams per dl.
01:42
And then of course the standard deviations is the same as 28 milligrams per dl.
01:48
That's the standard deviation.
01:50
In part a, draw the normal model.
02:01
So in terms of the normal model and the percentages were looking at, we have the mean in the middle, which happens to be 185.
02:16
And then of course one standard deviation from the mean one standard deviation from the mean that's going to be the same as 213 and 157 and one standard deviation will give a gap of 68 % of the data and then we're looking at the second distributions or two standard deviations from the mean on both sides.
02:54
That's going to be 241 and we're also going to have 129 and then the gap between those two that represents 95 % of the data.
03:11
And then of course we have three standard deviations from the mean on both sides and that's the same as 269 and then the other value is 101 that represents 99 .7 % of the data so this is 99 .7 % of the data and then in part b was saying determine the percent of adults with cholesterol, with cholesterol above 190 milligrams for dl.
04:08
So in terms of distributions, we have the mean and the standard deviation.
04:14
The mean is 185.
04:16
The standard deviation is 28.
04:18
We have x being 190.
04:21
So we want to get this percentage.
04:24
So the first thing we have to do is to get the z score, which is x minus.
04:27
Mu over sigma x is the same as 190 minus mu is 185 over sigma is 28 so 190 minus 185 that's 5 over 28 and that gives me the same value as 5 and 288 is the same thing as 0 0 .1786 and so in terms of probability z is greater than 0 .1786.
05:06
We have to go back to the z score table and look at those positive probabilities and so we're going to have 1 minus 0 .17 0 .1786 in terms of z scores 0 .17 or 1a we get 5714, so 0 .5714.
05:38
So we have 1 minus .5714.
05:45
So we get 0 .4286.
05:54
That's the percentage of probability, which is approximately 42 .86%.
06:01
The next part of the problem is part b or rather part c.
06:13
And in part c we're saying determine the percent of adults between 150 and 160 milligrams per dl cholesterol.
06:38
So we say probability that x lies between 150 and 160.
06:46
We can transform this to z scores.
06:52
So we have a z score 150 minus the mean, which is 185, all of the standard deviation.
07:08
And that's giving us a z score of the same as these values, 185 of a 288.
07:20
Remember, in terms of distribution, the mean is 185, and the standard deviation happens to be 28.
07:30
Of course, we're looking for numbers below 185, both of them.
07:35
X1 is 150, and x2, is 160.
07:44
So this is the portion that you're mostly interested in.
07:49
And the probability we're going to get from this is the same as p 150 minus 185 divided by 28.
08:04
That gives us a z score of negative 1 .25.
08:08
And then the other one is 160 minus 185.
08:13
Divided by 28 and that gives us a z score of negative 0 .8928 and so in terms of probabilities you have to go back to the z table and we're going to say that these values are the same as these values in the z table are going to be the same as we have to go back and look at the z tables...