00:01
So for the first part here, we want to compute what is the probability that the x bar for the man group, or the mayo graduates, will be within the mean from the population, which is given by this, within $1.
00:18
So we are saying that the difference could be at maximum 1 or minus 1 here.
00:25
So this means that it is within.
00:27
So now to compute this, we need to know what is the distribution of x bar.
00:33
We need at least to know what is the standard deviation from the distribution of x bar.
00:38
So the x bar here will be normally distributed because the sample size is 50, so it is considerably a large sample.
00:47
The mean here will be the same mean as in the population for the male.
00:52
This here is the male.
00:54
So we can even put one here.
00:57
And the standard deviation for the male is given by the standard deviation, the population, which is, in this case, $460 divided by the sample size, which is the square root of the sample size, which is 50.
01:14
So this here is our standard deviation for the male distribution.
01:19
The male, in this case, the average, in this case, for the male distribution.
01:24
So, okay, now the trick here to some.
01:27
Solve this question is that to compute this, we need to transform this probability to be in a standard normal distribution.
01:34
So to do this, what we need to have? we need to have like the x bar, which is our variable, minus the mean from the x bar.
01:44
And when you divide this by the standard deviation from the x ball distribution, you're going to say that this here now, this variable here is a standard norm distribution.
02:00
But, okay, now we are in the standard normal distribution.
02:03
But as you saw, i had to divide by the simple standard deviation.
02:08
So i need to do this in all the sides of this probability.
02:14
So what i'm saying is that this probability here is the same as if you compute minus 1 divided by 460 divided by the square root of 50, now in this standard normal distribution.
02:29
And again, here we'll have the same number, but it's the positive one.
02:34
So if you compute this number here and this number here, you're going to find the probability that we want is the same as finding this probability in a standard normal distribution.
02:47
And now we can use a way to compute this.
02:52
So this here is the same as this is the standard normal distribution.
02:56
We have this number, 1 .504, minus 1 .54.
03:02
So we want this area.
03:04
But to compute this using a z table, the z table gives us the area left to a number.
03:10
So we need to transform this probability to be probabilities left to a number.
03:14
So what we need to do, we need to compute this entire probability, which we include this part.
03:19
And then we need to compute the probability of this port alone to some.
03:26
Subtract what we don't need.
03:28
So what i'm saying is that this probability that we want is the same as this entire probability here, so until the positive 154, minus the part that we don't want, which is the probability of z being less than the negative.
03:45
So if you compute this using a z table, you're going to find that the first one is 9379.
03:51
The second one is 0 .0621.
03:54
So the probability that the average in the sample would be within $1 from the mean of the population is 87 .56%...