A sampling distribution describes the distribution of:
A. a parameter.
B. a statistic.
C. either a parameter or a statistic.
D. neither a parameter nor a statistic.
The correct answer and explanation is :
The correct answer is:
B. a statistic.
Explanation:
A sampling distribution describes the distribution of a statistic that is computed from multiple samples drawn from a population. In statistical terms, a statistic is a value calculated from a sample, such as the sample mean, sample variance, or sample proportion. The sampling distribution is the probability distribution of that statistic across all possible random samples from the population.
To clarify further:
- Parameter vs. Statistic:
- A parameter is a numerical value that describes a characteristic of the entire population, such as the population mean (μ) or population variance (σ²). Parameters are generally fixed but unknown in practice because we often cannot measure the entire population.
- A statistic, on the other hand, is a numerical value calculated from a sample. Examples include the sample mean (x̄), sample standard deviation (s), and sample proportion (p̂).
- Sampling Distribution:
- The sampling distribution refers to the distribution of a statistic (such as the sample mean) computed from all possible random samples of a specific size drawn from the same population. It shows how that statistic varies across different samples from the population.
- For example, if you repeatedly draw samples of size 30 from a population and calculate the sample mean for each sample, the collection of these sample means forms a sampling distribution of the sample mean.
- The sampling distribution helps us understand how much the statistic (like the sample mean) can vary from sample to sample, even though the parameter (population mean) is fixed.
- Purpose and Use:
- The sampling distribution is critical in statistical inference because it allows us to estimate how much sampling variability exists and aids in constructing confidence intervals or performing hypothesis tests about the population parameter. For example, knowing the sampling distribution of the sample mean allows us to assess the probability of obtaining a sample mean as extreme as the one observed, given a certain population mean.
Thus, a sampling distribution describes the variability of a statistic, not a parameter, which is why the correct answer is B. a statistic.