2.(a) Define systematic sampling. Discuss sample selection procedures in this sampling. Suggest an unbiased estimator of the population mean. Show that the systematic sample mean \(\bar{y}_{sys}\) is more efficient than the simple random sample mean \(\bar{y}\) if the variance within the systematic samples is more than the total variation in the population, i.e., \(S_w^2 > S^2\).
Answer (1)
Systematic sampling is a statistical method involving the selection of elements from an ordered sampling frame. The most straightforward form of systematic sampling is to list the population units, decide on the sample size (n), calculate the sampling interval (k) by dividing the population size (N) by the sample size (n), and then randomly select a number between 1 and k to start with (this number is called the random start). Subsequent sample elements are selected by adding the sampling interval to the initially selected number.
For example:
Suppose we have a population of 1000 units and we want to select a sample of size 100.
The sampling interval (k) is calculated as 1000 (N) / 100 (n) = 10.
We randomly select a starting point between 1 and 10, say 7.
Then the sample will include the items at positions 7, 17, 27, ..., up to 997.
Systematic sampling is particularly useful when the population elements are arranged in a logical order and is easy to execute and understand.
An unbiased estimator of the population mean using systematic sampling is the sample mean, denoted as y ˉ sys , calculated as the average of the systematically selected sample data values:
y ˉ sys = n 1 i = 1 ∑ n y i
To determine the efficiency of the systematic sample mean y ˉ sys compared to the simple random sample mean y ˉ , we assess the relationship between the variance of the systematic sample mean and the simple random sample mean. The systematic sample mean is said to be more efficient (i.e., it has a lower variance) if the variance within systematic samples, denoted S w 2 , is greater than the total variance of the population, S 2 .
S^2"> Va r ( y ˉ sys ) < Va r ( y ˉ ) if S w 2 > S 2
This condition implies that when there is more variability within the systematic samples as compared to the total population, the systematic sampling approach provides a more precise estimate of the population mean than simple random sampling.
