00:01
So in this question we consider we have a population of 1 million observations, and in that population, the mean is 10, and this 10 and deviation is 3.
00:11
We are going to compute here what it is the sample mean distribution, or the sampling distribution of the means.
00:20
So they are considering that they were going to get 1 ,000 random samples of size 100 from this population, and they are going to compute the sample mean for each one of those samples.
00:30
So in the end, they're going to have like in this case 1 ,000 sample means.
00:37
And in item a, we want to answer what is the percentage of the sample means that will be less than 10 .25.
00:45
This is the same as finding this probability.
00:48
So the idea is that by considering that the sample size that we have in each one of those random samples that we are collecting is 100, this is considered to be large enough in terms of like a central limit theorem, in a way that we can find that the distribution of the sample mean will be approximately normal with the same mean as the population mean.
01:16
Okay, 10.
01:17
And the variance here for the sample mean distribution is given by the variance in the population over the sample size that we have that we are collecting.
01:27
So that is the explanation for how we can compute this.
01:32
And for the other questions as well, we are assuming that we have a large enough sample size.
01:38
In that case, we can assume that the simple mean has a normal distribution for these specific parameters.
01:45
Now, because this is a normal distribution, if you want to complete probability based on the normal distribution, we usually consider the z score approach, which will write this probability based on this standard normal distribution.
01:58
So we can use the z table.
02:00
So in the standard normal distribution, we need to convert this value to be between 0 and 1, not between 0 ,0 ,0 ,1, but to be part of the synonyonal distribution, which most of the values will be between negative 4 and plus 4.
02:15
So to transform this value to be one possible value in the standalone distribution, we computed the difference of the value that we have here with respect to the mean of the population of the simple mean.
02:28
And we divide this by the standard deviation of the sample mean.
02:32
That will give us z being less than 0 .83, and now we use the z table to find that the proportion of observations in the sample mean distribution that will be less than 1025 is 0 .7967, which means, in this case, that because they are asking us the percentage, if you multiply this by 100, you get the answer that you want.
02:56
And the explanation is this part.
03:02
Now in item b, assuming again, n is equals to 100, and then we have that we can say that the distribution here of the simple mean is normal.
03:16
With this variance and this mean, we can then define here...