00:01
So for this ap statistics question, we're looking at a population of values that has a normal distribution within a given range.
00:11
We're trying to solve this probability problem, and we're going to do so with using the properties of a normal distribution and a z -score formula.
00:22
So for starters, the z -score formula is...
00:31
Oh, didn't mean to...
00:33
Z equals x minus u divided by standard deviation.
00:44
So that's the mean and the standard deviation, and x is the value we are interested in.
00:52
So for starters, the probability that a single randomly selected value is between 188 and 202 .3 is as follows.
01:09
Probability...
01:34
Well, for starters, let's calculate the z -score of 202 .3, which is z of 188 point...
02:02
Another one? no.
02:06
Z of 202 .3.
02:16
Sorry.
02:19
My apologies.
02:23
202 .3 minus 188 .1.
02:30
There we go.
02:32
Divided by 54 .7, and that is 0 .2590.
02:43
Okay.
02:47
Then we need z of 188 .1, which is 188 .1 minus 188 .1 divided by 54 .7, which is 0.
03:07
So using a standard normal table, we can find that p of z is less than 0 .2590 is 0 .6039.
03:31
And then we've got p of z is less than 0 is 0 .50.
03:40
So therefore, p, the probability of 188 .1 less than x, less than 202 .3, is 0 .6039 minus 0 .500, which is 0 .1039 to four decimal places.
04:31
Okay.
04:32
Moving forward.
04:34
So let's find the probability that a sample of size n equals 100 is randomly selected with a mean of 188 .1 and 202 .3.
04:45
So we have to know that the mean of the sample distribution is the same as the population mean, and the standard deviation of the sample distribution is given by the formula.
05:00
Standard deviation over root n.
05:09
And once again, that is our standard error, which would be 54 .7 divided by square root of 100, which is 5 .47.
05:37
Okay.
05:40
So now let's calculate the z -score for 188 .1 and 202 .3 using the sample mean and standard error.
05:49
All right.
05:50
So let's do this using the parameters.
05:52
Z -scores.
05:57
We're looking at z -scores.
05:59
202 .3 equals 202 .3 minus 188 .1 divided by 5 .47 equals 2 .5902.
06:35
And the z -score of 188 .1 equals 188 .1 minus 188 .1 over 5 .47, which is 0.
07:00
And so let's find these probabilities using a standard normal table.
07:06
That the p, or probability, that z is less than 2 .5902 is very high at 0 .9959...