Question

SKILL BUILDER 1 In Exercises 3.41 to \(3.44,\) data from a sample is being used to estimate something about a population. In each case: (a) Give notation for the quantity that is being estimated. (b) Give notation for the quantity that gives the best estimate. A random sample of registered voters in the US is used to estimate the proportion of all US registered voters who voted in the last election.

Short answer

The quantity being estimated, the proportion of all registered voters who voted, is denoted by \(P\). The quantity that gives the best estimate, the sample proportion, is denoted by \(\hat{p}\).

Step-by-step solution

Notation for Quantity Being Estimated

In this problem, the parameter being estimated is the proportion of all US registered voters who voted in the last election. In statistics, this true population proportion is usually denoted by \(P\).

Notation for Best Estimate

The best estimate for this parameter will come from the sample proportion. The sample proportion is usually denoted using \(\hat{p}\) (here, p-hat), which is calculated by dividing the number of registered voters who voted by the total number of registered voters in the sample.

Key concepts

Statistical Notation

Understanding statistical notation is vital for interpreting and communicating findings in a clear and standardized way. In the context of estimating population proportions, statistical notation serves as a concise way to represent complex ideas. For instance, when referring to the true population proportion, statisticians typically use the symbol \(P\). This represents the parameter that researchers want to estimate, such as the proportion of all US registered voters who actually voted in the last election in the provided exercise.

Moreover, when we want to express the result calculated from a sample, we make use of the sample proportion notation \(\hat{p}\). This symbol, which resembles a lowercase 'p' with a hat on top, stands for the estimate of the population proportion derived from our sample data. To find \(\hat{p}\), one would divide the number of favorable outcomes in the sample (e.g., voters who participated) by the total sample size. It's the simplicity and consistency of this notation that allow anyone versed in statistics to understand which values represent the estimate versus the actual parameter.

Sample Proportion

The sample proportion is a cornerstone concept in statistics, especially when it comes to estimating characteristics of a larger population. Essentially, it tells us what fraction of our sample meets a certain criteria. For example, in an educational study, it could represent the proportion of students who pass a specific exam. When discussing sample proportion, it's critical to comprehend how it is calculated and what it represents.

To calculate the sample proportion \(\hat{p}\), you would divide the number of individuals in the sample with the characteristic of interest by the total number of individuals in the sample. Returning to our exercise example, if you have a sample of 100 voters and 60 voted in the last election, the sample proportion \(\hat{p}\) would be \(\frac{60}{100} = 0.60\), signifying that 60% of the sample voted.

The accuracy of \(\hat{p}\) as an estimate is contingent on sample size and randomness; hence, a larger and more random sample provides a better estimate of the population proportion. Ensuring these conditions in a study's design is key for the reliability of its conclusions.

Parameter Estimation

Parameter estimation is an analytical process used in statistics to ascertain the approximate values of population parameters based on sample data. There are several methods for parameter estimation, but they all share the goal of making the most accurate predictions possible about a population, given a finite set of observations. In our ongoing discussion, the population parameter of interest, designated by \(P\), is the proportion of all US registered voters who voted in the last election.

The sample proportion \(\hat{p}\) is an estimator of this population parameter. It becomes the best estimate we have for \(P\) when there's no feasible way to survey the entire population. Parameter estimation hinges on the quality of the sample; researchers must collect the sample in an unbiased, systematic way and ensure it's big enough to be representative of the population. Only then can they confidently infer population attributes from their sample statistics. The exercise improvement advice emphasizes the importance of understanding these concepts in order to correctly interpret the results of statistical studies and utilize them in practice.