# Stratified sampling ratio with srswor

The variable X under study have rectangular distribution with interval (a, a+ d ) the interval is divided into k equal subintervals which form k equal strata of equals sizes . From each stratum simple random sample of n/ k units is drawn . Let V1 and V2 be the variance in estimator of population mean based on stratified and simple random sampling of size n .

prove that V1/V2 = 1/K²

My attempt

V2 = Var($$\ y_R$$ ) = $$\ ( \frac{1}{n} - \frac{1}{\ k^2} ) \ S^2$$

As it is given in the question that total k stratum are of k subsamples in each of total population should be $$\ k^2$$

V1 = Var($$\ y_st$$ ) = $$\ ( \frac{1}{n} - \frac{1}{\ k^2} ) \sum_{i= 0 }^{k} \ p_i \ S_i^2$$

So v1/v2 = $$\frac{\sum_{i=0}^{\ k} \ S_i^2 }{k \ S^2}$$

Is this correct approach ??

• The sum in the definition of $V_1$ should be from 1 to $k$, not $0$ to $k^2$ Jul 4, 2021 at 3:32
The ratio of the variances depends on the ratio of the within-stratum variances of $$X$$ to the overall variance of $$X$$. In this case, the stratum-specific variance for an interval of length $$k$$ is proportional to $$1/k^2$$ (specifically, it's $$(b-a)/12k^2$$)