91Ó°ÊÓ

Population mean estimation is essential when using stratified sampling in surveys. In this context, the process involves breaking down the entire population into separate strata, collecting samples from those strata, and estimating the overall mean based on these. For the given problem, we know there are two strata: H and L. Each stratum contributes to the overall population mean weighted by its size.

To calculate the population mean, the formula is:

• \[ \bar{X} = \frac{N_H}{N} \bar{X}_H + \frac{N_L}{N} \bar{X}_L \]

This means multiplying the mean of each stratum by the proportion its population size contributes to the total, and summing these values. Here, \( N_H \) and \( N_L \) are the sizes of strata H and L, respectively, while \( \bar{X}_H \) and \( \bar{X}_L \) are their respective mean estimates. Using this approach ensures that the population mean is accurately estimated, reflecting the distribution and contribution of each stratum.

Understanding this method is vital for anyone conducting surveys, as it ensures that each subgroup is properly represented in statistical models and outcomes.

Calculating the standard error of the population mean in stratified sampling is crucial for understanding the estimate's precision. The standard error provides a measure of how much sampling variability is expected in the estimation procedure.

The formula used is:

• \[ SE = \sqrt{\left(\frac{N_H^2}{N^2}\right)\frac{S_H^2}{n_H} + \left(\frac{N_L^2}{N^2}\right)\frac{S_L^2}{n_L}} \]

Here, \( S_H \) and \( S_L \) represent the standard deviations for strata H and L, and \( n_H \) and \( n_L \) are the sample sizes allocated to these strata. This expression combines the variances of each stratum, weighted by their respective sizes, scaled by the inverse of their sample sizes.

When you apply this formula, you get the standard error, which helps you understand the reliability of the population mean estimate. A smaller standard error indicates a more precise estimate, crucial for making informed decisions based on sample data.

Proportional allocation is a technique that assigns samples in a survey in proportion to the strata size, ensuring that each subgroup is equally represented. This method helps to minimize bias and potentially reduce sampling error.

In our scenario, the proportional allocation would involve allocating samples based on the size of strata H and L out of the total population. This is done using:

• \( n_H = \frac{100000}{600000} \times 300 = 50 \)
• \( n_L = \frac{500000}{600000} \times 300 = 250 \)

By assigning 50 samples to stratum H and 250 to stratum L, each stratum's relative size is acknowledged, and thus properly represented in the sample. This method is often preferred as it ensures that the sample reflects the diversity within the population, which can improve the accuracy and reliability of the population mean estimate.

Implementing proportional allocation leads to more representative survey data and plays a crucial role in the analysis of stratified samples.

Sample allocation in surveys is the process of deciding how to distribute the total number of samples across different strata in a population. This is a critical decision that impacts the survey's accuracy and standard error.

In the context of this exercise, two potential allocations were tested:

• Allocating 100 samples to stratum H and 200 to stratum L.
• Allocating 200 samples to stratum H and 100 to stratum L.

Each allocation impacts the standard error calculation differently because reallocating samples changes the weights and variances used in the formula.

The goal of sample allocation is to reduce the standard error, providing a precise estimate while acknowledging practical constraints like resources. The decision about sample allocation often balances proportions, variances, and logistical considerations, aiming to provide the most reliable estimate given all available information.

In practice, evaluating different allocations helps to find a configuration that provides the best trade-off between cost and accuracy, ensuring the survey delivers meaningful results.