In Stratified random sampling show that sample mean is NOT an unbiased estimator of (10) population mean. Hence define an unbiased estimator of population mean and obtain its variance.

Stratified random sampling is a method of sampling that involves the division of a population into smaller groups known as strata. In statistical terms, these groups are formed based on members' shared attributes or characteristics.

Let's consider a population divided into 'k' strata. The size of the 'i'th stratum is Ni and the total population size is N. The mean of the 'i'th stratum is μi.

The population mean (μ) is given by:

μ = Σ (Ni/N) * μi

In stratified random sampling, we select 'ni' samples from the 'i'th stratum. The sample mean (x̄) is given by:

x̄ = Σ (ni/n) * x̄i

where 'n' is the total sample size and x̄i is the sample mean of the 'i'th stratum.

Now, let's consider the case where the sample size from each stratum is proportional to the size of the stratum i.e., ni/Ni = n/N. In this case, the sample mean becomes:

x̄ = Σ (Ni/N) * x̄i

Here, we can see that the sample mean is a biased estimator of the population mean because the weights in the sample mean are the sample means of each stratum (x̄i), not the population means (μi).

To make the sample mean an unbiased estimator of the population mean, we need to adjust the weights. The unbiased estimator (x̄') of the population mean is given by:

x̄' = Σ (Ni/N) * x̄i

Now, the weights are the population proportions (Ni/N), and x̄' is an unbiased estimator of μ.

The variance of this unbiased estimator is given by:

Var(x̄') = Σ (Ni/N)² * σi²/ni

where σi² is the population variance of the 'i'th stratum. This formula shows that the variance of the unbiased estimator depends on the population variances and the sample sizes of each stratum.

Stratified random sampling is a method of sampling that involves the division of a population into smaller groups known as strata. In statistical terms, these groups are formed based on members' shared attributes or characteristics.

Let's consider a population divided into 'k' strata. The size of the 'i'th stratum is Ni and the total population size is N. The mean of the 'i'th stratum is μi.

The population mean (μ) is given by:

μ = Σ (Ni/N) * μi

In stratified random sampling, we select 'ni' samples from the 'i'th stratum. The sample mean (x̄) is given by:

x̄ = Σ (ni/n) * x̄i

where 'n' is the total sample size and x̄i is the sample mean of the 'i'th stratum.

Now, let's consider the case where the sample size from each stratum is proportional to the size of the stratum i.e., ni/Ni = n/N. In this case, the sample mean becomes:

x̄ = Σ (Ni/N) * x̄i

Here, we can see that the sample mean is a biased estimator of the population mean because the weights in the sample mean are the sample means of each stratum (x̄i), not the population means (μi).

To make the sample mean an unbiased estimator of the population mean, we need to adjust the weights. The unbiased estimator (x̄') of the population mean is given by:

x̄' = Σ (Ni/N) * x̄i

Now, the weights are the population proportions (Ni/N), and x̄' is an unbiased estimator of μ.

The variance of this unbiased estimator is given by:

Var(x̄') = Σ (Ni/N)² * σi²/ni

where σi² is the population variance of the 'i'th stratum. This formula shows that the variance of the unbiased estimator depends on the population variances and the sample sizes of each stratum.