00:01
So in this question, we have a population size of the population is 4, 5, and 9.
00:27
And then we have a population size of 3 because it does say that the n equals to 2.
00:43
Yeah, so definitely the population size is 3 because there are three figures.
00:47
And then it does say that n equal to 2 so yeah and the first question was supposed to find the value of the population on variance and to find the population variance we need to find the population mean first because you need the mean to get the variance so we're gonna start with that so population mean and this is equal to summation x over n and then we know that the summation x is 4 plus 5 plus 9 and we know that n is equal to 3 because you have 3 figures so 4 plus 5 plus 9 is 18 over 3 and that's 6 that's the population main now the population variance because that's what we need to find the population variance this is equal to the formula is submission open bracket x1 um x1 minus the mean divided by n and squared that so we're supposed to find the population variance of this equation is going to be open bracket 4 because the first x is 4 we have 4 5 9 so it's 4 minus 6 that's the mean squared plus 5 plus 5 minus 6 squared plus 9 minus 6 squared then you divide all of this by 3 and this gives you um 4 minus 6 is minus 2 squared minus 2 squared is um 4 so you're going to have 4 plus um 1 plus 9 divided by 3 and equals to 14 over 3 and to find what channel i put in a one decimal place so it's easier to work with and that will be 4 .7 that's going to be the population variance the second question wants us to construct wants us to describe the sampling distribution of the sample mean and also construct the table representing the sampling distribution so um to start with that remember that two values are picked a random.
04:36
That's what the question said.
04:38
Two values are selected a random.
04:41
And with replacement from the population, 459.
04:47
So if n equals to 2, then the samples, the possible samples from picking two options from the population 3 ,5 ,000, no sorry four five nine is going to be four four you're going to have four four four five four nine um we're also going to have five four five five and five nine and then for nine you're going to have nine four nine five and then nine and then from there we can tell that the number of the number of probability, the total number of probability is nine because there are nine possible of samples.
06:29
So therefore, to construct the table, you need a sample variance because you're trying to construct a table representing the sample in distribution of the sample variance.
06:42
So definitely you need the sample variance for each sample.
06:45
So the formula for sample mean is to find sample variance, we need to find sample mean.
06:53
So definitely the formula for sample mean, this is, i'm sorry about that.
07:16
This is x, and this is equal to summation x1 over n.
07:31
And then the sample variance, the formula for finding that, the formula for finding that is summation x minus the mean squared over n minus 1.
08:05
So we're going to put this in finalizing each sample to figure out the total variance.
08:13
And then for four for the samples, four, we're going to be starting with that.
08:19
The sample mean, the sample mean is equal to four plus four.
08:30
Divided by 2 and this equals to 4 and the sample variance is equal to 4 minus 4 4 4 minus 4 squared plus 4 minus 4 squared divided by 2 minus 1 because n minus 1 so that's 2 minus 1 we will use your calculator this is going to give you 0 um you can you come down to the sample 4 -5, the mean, the sample mean is going to be equal to 4 plus 5 divided by 2, and then this is 4 .5.
09:30
And then the sample variance is going to be 4 minus 4 .5, that's the mean squared plus 5 minus 4 minus 4 .5.
09:54
Squared and i'm going to divide this by 2 minus 1 and where you saw for this is going to give you 0 .50 and we come to the samples 4 and 9 the mean is equal to 4 plus 9 divided by 2 which is equal to 6 .5 and this sample variance is 4 minus 4 minus 6 .5 squared plus 9 minus 6 .5 squared divided by 2 minus 1 and this is going to give you 12 .5.
11:14
We're going to move to.
11:15
So we can already tell something common is that the next sample which is 5 .5, the sample mean is going to be 0.
11:25
So the sample mean is going to be 5 and the sample variance is going to be 0.
11:30
So we just move.
11:32
We move to 5 .9.
11:36
And the reason why we skip 5 -4, because the next one is 5 -4, is because you already have 4, 5.
11:42
So there's no point in doing 5 -4 anymore.
11:45
So we move to 5 -9.
11:49
So for 5 -9, the mean is equal to the sample mean is equal to 5 plus 9.
12:02
Divided by 2 and this is 14 divided by 2 that's 7 and the sample variance is equal to 5 minus 7 squared plus 9 minus 7 square divided by 2 minus 1 and this is going to give you 8 so you put that in mind and then the next one is going to be 9 .4 but we're going to skip that because we're we already sold for 4 .9 as we can see here so we're going to skip that the next one is 9 .9 .5 we already sold for 5 .9 so we skip that and then go straight to 9 .9.
13:01
9 .9 we can already tell the sample variance is going to be equal to 0 so we skip that and then now we're going to have to draw the table with the samples so this side we're going to write samples and our samples of 4 .4, 4 .5, 4 .9, 5 .4, 5 .5, 5 .9, 9 .4, 9 .5, 9 .9.
13:59
And then, sorry about that.
14:11
We're going to move to the sample mean.
14:25
The formula so the sample mean for 4 .4 we saw it was um four so we're just going to write some answers they sample mean was 4 the sample mean for 4 .5 was 4 .5 the sample mean for this was 6 .5 and this was 4 .5 this was 5 this was 7 this was 6 .5 this was 7 this was 6 .5 this was 7 this was 7 this was 7 and then this was 9 and then we'll go back to the table and then we'll have our variance, the sample variance.
15:19
Our sample variance for this was 0 .5...