Answer :
In AP Statistics, one of the checks we perform when analyzing a set of data or when conducting inference is the 'independence check.' This check is crucial to ensure that the observations or data values are independent of each other, which is often a key assumption in statistical analyses such as regression, hypothesis tests, and confidence intervals.
For many practical purposes, particularly in simple random samples, the rule of thumb is that the sample size should be less than or equal to 10% of the population size. This is often referred to as the '10% condition.' The logic behind this is that when you sample less than 10% of the population, the sampling distributions of statistics can be assumed to mimic the true population distributions without introducing significant bias due to the sampling method.
Here's how you can check for independence:
Identify the Size of the Population (N): Determine the total number of observations or subjects in the entire group you're considering.
Identify the Sample Size (n): Determine how many observations you actually have in your sample.
Apply the 10% Condition: Ensure that your sample size (n) is less than or equal to 10% of the population size (N). In other words, check if [tex]n \leq 0.1N[/tex].
By applying this rule, you can say with reasonable confidence that your sample is independent enough to proceed with certain types of statistical analyses.
Remember that this is a guideline rather than a strict rule, and there are some contexts, like when observations are clearly not independent due to study design or data collection methods, where the independence of observations cannot be assumed no matter the sample size.
