Final answer:
The correct answer is c) Inversely related to the sample size, meaning the larger the sample size, the smaller the sampling error, according to the Central Limit Theorem. As the sample size increases, the distribution of sample means will more closely approximate a normal distribution and the sampling error will decrease.
Step-by-step explanation:
Understanding the Central Limit Theorem and Sampling Error
The Central Limit Theorem (CLT) is a fundamental statistical principle that describes how the distribution of sample means approaches a normal distribution as the sample size increases. When considering the effect of sample size on the accuracy of statistical estimates, it is essential to understand concepts such as sampling error and standard error.
Sampling error refers to the discrepancy between a sample statistic and the corresponding population parameter. According to the CLT, as the sample size increases, the standard error, which is the standard deviation of the sampling distribution of the means, decreases. Specifically, the standard error is calculated as the population standard deviation divided by the square root of the sample size (n). This relationship is inversely proportional, meaning that a larger sample size will result in a smaller sampling error, assuming that the population standard deviation remains constant.
Contrary to options a and b of the question, neither the size of the sample being directly related to sampling error nor the population mean having a direct effect on the sampling error is correct. Additionally, option d, which suggests that sampling error is inversely related to the population standard deviation, is not accurate. The correct response is:
c) Inversely related to the sample size, i.e., the larger the sample size the smaller the sampling error.
The law of large numbers further reinforces this idea, indicating that a larger sample size will yield a sample mean that is closer to the population mean. Therefore, to reduce sampling error and enhance the reliability of sample statistics, increasing the sample size is advisable.