Question

Consider the following statement: The proportion of all calls made to a county \(9-1-1\) emergency number during the year 2011 that were nonemergency calls was \(0.14 .\) a. Is the number that appears in boldface in this statement a sample proportion or a population proportion? b. Which of the following use of notation is correct, \(p=0.14\) or \(\hat{p}=0.14 ?\)

Short answer

a. The number 0.14 represents a population proportion. b. The correct notation for this scenario is \(p=0.14\).

Step-by-step solution

Understanding Population versus Sample

A population includes all members of a defined group. We will use the entire population if we count every subject of interest. For instance, all calls made to this number in 2011. A sample, on the other hand, is a subset of the population from which data is obtained.

Determine Proportion Type

In the problem statement, we have a given proportion based on all calls made to a county 9-1-1 number in 2011 showing that 14% were nonemergency calls. Since this includes all calls made in 2011, it refers to an entire group or population. So, the number 0.14 is a population proportion.

Notation for Proportions

The notation p is usually used to denote the probability of success for a population proportion, while the notation \(\hat{p}\) is used to denote the sample proportion or an estimated population proportion. Since we determined that 0.14 is a population proportion, the correct notation would be \(p = 0.14\).

Key concepts

Sample Proportion

Understanding the concept of a sample proportion is key to statistics and involves discerning a part of a larger population. In statistical studies, when we can't examine an entire population, we take a sample. A sample proportion, denoted by \(\hat{p}\), is the fraction of the sample that displays the characteristic of interest. For instance, if we randomly surveyed 100 individuals about their exercise habits and found that 50 regularly exercise, our sample proportion of individuals who exercise would be 0.50.

It's important to grasp that this is just an estimate of the true population proportion, as it only reflects a subset. The accuracy of the sample proportion in representing the whole population's behavior depends on many factors, such as sample size and sampling method. To reduce potential error and increase reliability, careful planning and random sampling methods are typically employed.

Statistical Notation

Statistical notation is the language through which statisticians effectively communicate and interpret statistical data and concepts. Being familiar with the symbols and formulas is essential for understanding and conveying information accurately. For example, in the context of proportions, the notation \(p\) represents the population proportion, which is the actual, true proportion found in the entire population.

On the other hand, the notation \(\hat{p}\) signifies the sample proportion, which is an estimation based on a sample. These notational differences are critical; they help distinguish between what is known from an entire population and what is estimated from a sample, potentially avoiding confusion in analysis and reporting.

Population vs Sample

Distinguishing between a population and a sample is fundamental in statistics, and it's pivotal for anyone working with data to understand the difference. A population is the entire group in which you're interested. For example, if you wanted to know the average height of all the trees in a forest, the population would be every single tree in that forest. On the other hand, a sample refers to a subset of the population selected for study, like measuring the height of 100 random trees to estimate the average.

While a population measurement (parameter) is usually not feasible due to resource constraints, a sample statistic can provide a practical, although approximate, result. This distinction also underlies the concepts of population proportion, which reflects the true value for the entire population, and sample proportion, which provides an estimate based on the observed sample.