A population parameter is a numerical summary or characteristic of an entire population. It is a fixed, unknown value that describes a population and is the true, underlying value that a researcher is interested in estimating or making inferences about.
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Population parameters are fixed, unknown values that describe the entire population, while sample statistics are calculated from a subset of the population.
Estimating population parameters is a key goal of statistical inference, as parameters cannot be directly observed or measured for the entire population.
The normal distribution is a widely used probability distribution that is characterized by its population parameters: the mean (μ) and standard deviation (σ).
Confidence intervals are used to estimate an unknown population parameter, such as the mean or proportion, based on a sample drawn from the population.
The margin of error in a confidence interval is directly related to the population standard deviation and the sample size used to estimate the parameter.
Review Questions
Explain how a population parameter differs from a sample statistic in the context of a sampling experiment.
In a sampling experiment, a population parameter is a fixed, unknown numerical characteristic of the entire population that the researcher is interested in estimating or making inferences about. In contrast, a sample statistic is a numerical summary calculated from a subset (sample) of the population, which is used to estimate the corresponding population parameter. While a sample statistic can vary from sample to sample, the population parameter is a constant, true value that the researcher is trying to learn about through the sampling process.
Describe the role of population parameters in the context of the normal distribution and its applications, such as the analysis of pinkie length.
The normal distribution is a commonly used probability distribution that is characterized by two population parameters: the mean (μ) and the standard deviation (σ). These parameters describe the central tendency and variability of the entire population, respectively. In the context of analyzing pinkie length, the population mean (μ) would represent the average pinkie length for the entire population, while the population standard deviation (σ) would describe the spread or variability of pinkie lengths around the mean. Knowing the values of these population parameters is crucial for making inferences about the distribution of pinkie lengths in the population.
Explain how population parameters are used to construct confidence intervals for estimating unknown population characteristics, such as average home costs.
Confidence intervals are statistical tools used to estimate an unknown population parameter, such as the mean or proportion, based on a sample drawn from the population. The width of the confidence interval is directly related to the population standard deviation (σ) and the sample size used to estimate the parameter. A smaller population standard deviation and a larger sample size will result in a narrower confidence interval, providing a more precise estimate of the unknown population parameter. In the context of estimating average home costs, the population parameter of interest would be the true mean home cost for the entire population, and a confidence interval would be used to provide a range of plausible values for this unknown parameter based on a sample of home cost data.