Final answer:
Option A is correct option. The key to any problem in statistical inference is a representative sample, which is necessary to ensure that sample data can be used to make valid conclusions about the population. Options like Standard Error of the Mean, Confidence Interval, and Probability Distribution are important, but secondary to the representativeness of the sample.
Step-by-step explanation:
The key to any problem in statistical inference is ensuring that a sample is representative of the population from which it was drawn. This is the concept represented by option C) Representative Sample. A representative sample accurately reflects the characteristics of the population, allowing for valid inferences about the population based on the sample data. In the context of sampling and statistical inference, other options such as A) Standard Error of the Mean (SEM), B) Confidence Interval, and D) Probability Distribution play a role, however, without a representative sample, the results may be biased and not applicable to the broader population.
We apply various statistics, such as the sample mean (x) and the sample standard deviation (s), as point estimates for the population mean (μ) and population standard deviation (σ). Also, the confidence interval provides a range within which the population parameter is expected to lie with a certain level of confidence. For instance, if repeated samples are taken, a 90% confidence interval would contain the true population mean in about 90% of the cases. This depends on having a representative sample, otherwise the statistics and confidence intervals calculated could be misleading.
To assess variability in data from different samples, we use the standard error of the mean, which tells us how much the sample mean is expected to vary from sample to sample. Ultimately, the most critical aspect remains the representativeness of the sample to make legitimate inferences about the population.