Two populations are surveyed with simple random samples. A...

Two populations are surveyed with simple random samples. A sample of size $n_{1}$ is used for population I, which has a population standard deviation $\sigma_{1} ;$ a sample of size $n_{2}=2 n_{1}$ is used for population II, which has a population standard deviation $\sigma_{2}=2 \sigma_{1} .$ Ignoring finite population corrections, in which of the two samples would you expect the estimate of the population mean to be more accurate?

Key Concepts

Effect of Sample Size on Estimation Accuracy

The accuracy of an estimate, such as the sample mean, improves with increasing sample size because the standard error decreases proportionally to the square root of the sample size. This relationship is central to understanding how gathering more data can lead to more reliable inferences about the population mean, even when the underlying population variability remains constant.

Population Variability

Population variability is quantified by the population standard deviation, which reflects the spread or dispersion of individual observations around the mean. The inherent variability in the population directly affects the accuracy of estimates; a higher standard deviation implies greater dispersion and larger standard error for a fixed sample size, reducing the precision of the sample mean.

Simple Random Sampling

Simple random sampling is a method of selecting a sample in which every individual in the population has an equal chance of being chosen. This fundamental approach underlies many statistical procedures and ensures that the resulting sample is representative of the population, thereby reducing bias in estimates of population parameters.

Standard Error of the Mean

The standard error of the mean measures the precision of the sample mean as an estimate of the true population mean. It is calculated as the population standard deviation divided by the square root of the sample size (?/?n). A smaller standard error indicates a more accurate estimate, emphasizing the impacts of both the population variability and the sample size.

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