Final answer:
The size of a sample to represent a wider population is determined by the population variability and the precision required. Larger samples reduce sampling variability and increase confidence in generalization. Practical considerations, however, often limit sample size.
Step-by-step explanation:
The size of your sample if you want to generalize from the sample to a wider population is governed primarily by two things: the variability of the population and the level of confidence or precision you need in your generalizations.
The first consideration is the underlying variability in the population, which refers to how much the traits or behaviors you're studying differ within the population. If the variability is high, you'll need a larger sample size to ensure that your sample accurately reflects the population. This is known as the Law of Large Numbers, which states that as the sample size increases, the sample mean is more likely to approximate the population mean.
The second consideration is the degree of precision or confidence you require. If you need results with a high degree of certainty, you would typically require a larger sample size. In statistical terms, this is often discussed in relation to confidence intervals, which provide a range within which you can expect the true population parameter to fall.
It is also important to note that for a sample to be considered truly representative, it must be a simple random sample that is adequately large and properly conducted. Furthermore, when dealing with proportion problems, the sample size should be large enough so that the number of successes and failures is at least five.
Moreover, there's a growing consensus in the literature that the population should be at least 10 or 20 times the size of the sample, to avoid over-sampling and incorrect results. While larger sample sizes are more desirable for precision, practical considerations such as time and money often necessitate the use of smaller samples.