In SRSWOR for variables , in usual notations prove that Sample mean is an unbiased estimator of population mean.

Sure, here is the step-by-step proof that the sample mean is an unbiased estimator of the population mean in Simple Random Sampling Without Replacement (SRSWOR):

1. Let's denote the population mean as μ and the sample mean as x̄. The population mean is calculated as the sum of all the values in the population divided by the total number of values in the population (N). The sample mean is calculated as the sum of all the values in the sample divided by the total number of values in the sample (n).

2. The goal is to show that the expected value of the sample mean (E[x̄]) is equal to the population mean (μ). In other words, we want to prove that E[x̄] = μ.

3. The expected value of the sample mean can be written as the sum of the expected values of each individual sample value divided by the sample size. In mathematical terms, E[x̄] = E[Σxi/n] = ΣE[xi]/n.

4. In SRSWOR, each individual sample value xi is an unbiased estimator of the population mean. This means that the expected value of each individual sample value is equal to the population mean. In mathematical terms, E[xi] = μ.

5. Substituting E[xi] = μ into the equation from step 3 gives us E[x̄] = Σμ/n = μ.

6. Therefore, the sample mean is an unbiased estimator of the population mean in SRSWOR.

This proof assumes that the population is large enough that the difference between sampling with replacement and without replacement is negligible. If the population is not large enough, then the sample mean may be a biased estimator of the population mean.