Question

Explain why there is sample-to-sample variability in \(\hat{p}\) but not in \(p\).

Short answer

\(\hat{p}\) displays sample-to-sample variability because each sample drawn might include different individuals, resulting in different sample proportions. On the other hand, \(p\) doesn't vary as it represents the whole population proportion, which remains constant despite the number or size of samples drawn from it.

Step-by-step solution

Understand what \(\hat{p}\) and \(p\) represent

\(\hat{p}\) represents the proportion in a sample, while \(p\) represents the proportion in a population.

Understand Sample Variability

Each time a sample is drawn from a population, there is a likelihood that different individuals will be selected. Therefore, the proportion \(\hat{p}\) computed from the sample will likely vary due to the different individuals included in the sample, this is what is referred to as sample-to-sample variability.

Understand constancy of Population Proportion

In contrast, the population parameter \(p\), the actual proportion in the population, does not vary. This is because it is a fixed value that represents the overall proportion in the entire population, regardless of how many samples are drawn or the size of these samples.