As we are discussing that we utilize the ratio method of estimation when we have a study variable with us which is correlated with it and we can utilize it for its support. Students consider auxiliary variable in ratio method of estimation as it is already available. We don't have to make any effort for it.
so we don't consume much resource on it we utilize it for estimation so we have seen this is as the ratio estimator I have told you in last module so our cap is this that is maybe the division of the totals y is total and x is also x and y is y notice that these are both sample estimates means this is the total from the sample and this is also the total from the sample of x similarly if you divide the means then also the estimate of the ratio will come or if you divide the totals then also the estimate of the ratio will come so in this way we can estimate the ratio that in one variable according to the other variable how the change is coming further students you can estimate the population mean and population total from the ratio estimation okay If we multiply the ratio with capital X bar, then we will estimate the mean. As I said, we have the information for the auxiliary variable for all about the population. Capital X bar is known. The information of the population is known of the auxiliary variable. We can utilize it.
We can multiply it with capital X bar. So small y bar is divided by small x bar. These are the sample information.
So we got the estimate of the ratio. and we are multiplying it with the population mean to getting the estimate of the population mean about the study variable, variable of interest this is exactly the same, if we know that we have registered births suppose one and when we find the ratio with the population we get that registered births against means something like this, if we divide registered births with our total population and guess what So the answer of this ratio is 2 It means that the population is double The number of births that are registered The population is double It is 2 times So when you multiply the total registered births When you multiply the ratio from the total registered Then you will get the total population That would be the 2 times So this is how we are taking the benefit from the ratio And we are multiplying with the mean Or the total of the so when you multiply it, you get the mean of population variable of interest and the total of population so it means that we can utilize it not only for ratio but also for estimating mean and population total students, this ratio estimator is biased estimator but as the sample size increases its bias is almost removed so we can say that for the large sample size this will be approximately unbiased that is expected value of R cap would approximately equal to capital R when the sample size is large so we will process it and see and prove it here so students we know that r cap is equal to small y bar over small x bar if you subtract capital r from this then you subtract capital x bar from small y bar over small x bar take capital r as LCM it is a simple step and if you consider that n is large you can replace small x bar with capital x bar so here we are replacing it small x bar with capital X bar so small we got this small y bar minus our Small x bar over capital X bar so these as we said that by increasing or by taking the large sample Small x bar will approximately equal to capital X bar so our cap minus R will approximately equal to this expression So students I got up this okay, don't know the expression case equation to donut of up expected value you will get like this expected value on the left side expected value of small y bar minus our expected value of small x bar So small x bar is an unbiased estimate. You can replace small expected value of small x bar with capital X bar because we know that by simple random sampling this is an unbiased estimate. So we will replace it with capital X bar and we will get like this.
Capital Y bar, expected value small y bar will become capital Y bar. Just like expected value of small x bar will become the capital X bar. Capital R will remain the same or capital X bar.
Right. If you notice here, R means this part R capital X bar you can write capital Y bar over capital X bar into capital X bar these two terms will cancel out you will remain with capital Y bar means this part is actually capital Y bar so capital Y bar minus capital Y bar will become 0 so ultimately you can say expected value of R cap minus R is zero. If you apply this expected value you can easily get expected value of R cap approximately equal to R.
Similarly this will also be an approximation sign because we are continuing the same sign so this is how we can say that expected value of R cap would approximately equal to R when the sample size is large. So here we we can conclude that when the sample size will be large R cap the ratio estimator will be almost unbiased