Final answer:
Statistical analysis often relies on random sampling from a population. The sample size and method of sampling determine the reliability of inferences. Larger sample sizes and simple random samples are preferred for unbiased results and accurate population representation.
Step-by-step explanation:
Understanding Random Sampling in Statistics
Random sampling is a crucial concept in statistics used to represent a larger population without bias. When a random sample is taken from a normal distribution, we want to ensure that the sample size is sufficient enough to make valid inferences about the population. For example, in the provided data, a sample of 10 observations might be good for preliminary analysis, but larger sample sizes, like in the case of 1,000 students from the eastern United States, are usually better for reliable generalizations.
To address the question of whether the sample size of 10 observations is sufficient, it is essential to assess the variability in the data and the purpose of the study. Statistical methods, such as calculating the standard deviation and confidence intervals, can help in determining the adequacy of the sample size.
Sampling methods vary, and each method has its pros and cons. A simple random sample ensures that every member of the population has an equal chance of being selected, which is ideal for unbiased results. However, in reality, other sampling methods such as convenience sampling or stratified sampling might be used, some of which may introduce bias into the sample, affecting the results interpretation.