True or False: The population proportion and sample proportion always have the same value.

Verified step by step guidance

1. Understand the definitions: The population proportion, denoted as \(p\), is the true proportion of a characteristic in the entire population, while the sample proportion, denoted as \(\hat{p}\), is the proportion observed in a sample drawn from that population.
2. Recognize that the sample proportion \(\hat{p}\) is a statistic calculated from sample data and is used to estimate the population proportion \(p\).
3. Note that due to sampling variability, the sample proportion \(\hat{p}\) can differ from the population proportion \(p\) because the sample may not perfectly represent the entire population.
4. Therefore, the sample proportion \(\hat{p}\) is generally an estimate and may not always equal the population proportion \(p\).
5. Conclude that the statement 'The population proportion and sample proportion always have the same value' is false because the sample proportion varies depending on the sample selected.

Key Concepts

Population Proportion

The population proportion is the true proportion of a characteristic or attribute in the entire population. It is a fixed value but usually unknown because measuring the entire population is often impractical.

Sample Proportion

The sample proportion is the proportion of a characteristic found within a subset (sample) drawn from the population. It is used as an estimate of the population proportion and can vary from sample to sample.

Sampling Variability

Sampling variability refers to the natural differences that occur between sample statistics and the true population parameters due to random sampling. Because of this, the sample proportion does not always equal the population proportion.