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Chapter 8: Problem 1

Explain the difference between a population characteristic and a statistic.

Short Answer

Expert verified
A 'population characteristic' is any attribute or feature that describes the entire population, like the average age of citizens in an entire country. A 'statistic', however, is numerical data derived from specific observations of a sample from the population, like the average age of a group of randomly selected citizens from the same country. Thus, the main difference is that a population characteristic pertains to the entire population, while a statistic pertains to a specific sample from that population.

Step by step solution

01

Understanding 'Population Characteristic'

A 'population characteristic' is essentially a specific attribute or a feature that describes the entire population. This could refer to any substantial aspect such as age, gender, income level, etc. For instance, the average income level of all households in the US is a 'population characteristic'.
02

Understanding 'Statistic'

'Statistic' refers to numerical data that is derived from a series of observations for a specific sample from a particular population. It provides summary measures of sample data. For instance, if we take a random group of 100 households from the US for a survey, their average income will be considered a 'statistic'.
03

Difference between a population characteristic and a statistic

The difference between 'population characteristic' and a 'statistic' is that the former pertains to the entire population, whereas the latter refers to a specific sample taken from that population. A population characteristic is generally a fixed value and defines the entirety of the group, while a statistic is variable, being dependent on the selected sample group.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Characteristic
A population characteristic refers to an inherent trait or measurement that provides information about an entire population. This population could be anything from all the residents of a city to every student in a school. Characteristics such as age, height, income, or education levels count towards this.
  • Population characteristics are often labeled as parameters because they summarize an attribute for everyone in the population.
  • These are constants and remain unchanged unless the entire population data is reevaluated.
Imagine knowing the average age of every citizen in a country 鈥� this is a population characteristic. It helps policymakers design better public health strategies, businesses to tailor products for specific age groups, and educators to create suitable learning programs. Understanding these parameters is crucial as they form the basis for making informed and effective decisions.
Statistic
A statistic is a concise numerical summary that derives from a sample, which is a subset of a broader population. Unlike population characteristics, statistics offer estimates because they reflect the features of just a portion of the entire group.
  • Statistics are vital for making predictions about the population since they depict what could be true for the entire group.
  • They are variable and can change with different samples even when drawn from the same population.
Whether assessing the average income of people from a neighborhood or the favorite sport of students in a class, statistics provide insights. They are representations that help researchers and analysts better understand, but they need careful interpretation as they carry margins of error due to representing only a fraction of the population.
Sample
A sample is a smaller, manageable subset that is selected from an overall population. It acts as a representative for studying and analyzing population characteristics without needing to survey the entire group.
  • Samples allow researchers to gather data efficiently, saving on time and resources.
  • Choosing a representative sample is crucial as it influences the reliability of the statistic that is derived from it.
The process of selecting a sample involves careful planning to ensure it reflects the true diversity and characteristics of the entire population. For example, choosing students from different courses, ages, and backgrounds when researching campus trends helps ensure a well-rounded and accurate result that can be generalized more reliably to the entire student body. Sampling methodologies like random sampling, stratified sampling, and systematic sampling each have their purpose and role in data collection.

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Most popular questions from this chapter

Explain the difference between \(\sigma\) and \(\sigma_{\bar{x}}\) and between \(\mu\) and \(\mu_{\bar{x}}^{-}\)

Consider the following population: \(\\{1,2,3,4\\} .\) Note that the population mean is \(\mu=\frac{1+2+3+4}{4}=2.5\) a. Suppose that a random sample of size 2 is to be selected without replacement from this population. There are 12 possible samples (provided that the order in which observations are selected is taken into account): \(\begin{array}{rrrrrr}1,2 & 1,3 & 1,4 & 2,1 & 2,3 & 2,4 \\ 3,1 & 3,2 & 3,4 & 4,1 & 4,2 & 4,3\end{array}\) Compute the sample mean for each of the 12 possible samples. Use this information to construct the sampling distribution of \(\bar{x}\). (Display the sampling distribution as a density histogram.) b. Suppose that a random sample of size 2 is to be selected, but this time sampling will be done with replacement. Using a method similar to that of Part (a), construct the sampling distribution of \(\bar{x}\). (Hint: There are 16 different possible samples in this case.) c. In what ways are the two sampling distributions of Parts (a) and (b) similar? In what ways are they different?

In the library on a university campus, there is a sign in the elevator that indicates a limit of 16 persons. In addition, there is a weight limit of 2500 pounds. Assume that the average weight of students, faculty, and staff on campus is 150 pounds, that the standard deviation is 27 pounds, and that the distribution of weights of individuals on campus is approximately normal. If a random sample of 16 persons from the campus is to be taken: a. What is the expected value of the distribution of the sample mean weight? b. What is the standard deviation of the sampling distribution of the sample mean weight? c. What average weights for a sample of 16 people will result in the total weight exceeding the weight limit of 2500 pounds? \(\bar{x}>156.25\) d. What is the chance that a random sample of 16 people will exceed the weight limit?

Suppose that a sample of size 100 is to be drawn from a population with standard deviation 10 . a. What is the probability that the sample mean will be within 1 of the value of \(\mu\) ? b. For this example \((n=100, \sigma=10),\) complete each of the following statements by computing the appropriate value: i. Approximately \(95 \%\) of the time, \(\bar{x}\) will be within \(\quad\) of \(\mu\). ii. Approximately \(0.3 \%\) of the time, \(\bar{x}\) will be farther than \(\quad\) from \(\mu\).

The article "Should Pregnant Women Move? Linking Risks for Birth Defects with Proximity to Toxic Waste Sites" (Chance [I992]: \(40-45\) ) reported that in a large study carried out in the state of New York, approximately \(30 \%\) of the study subjects lived within 1 mile of a hazardous waste site. Let \(p\) denote the proportion of all New York residents who live within 1 mile of such a site, and suppose that \(p=.3\). a. Would \(\hat{p}\) based on a random sample of only 10 residents have approximately a normal distribution? Explain why or why not. \(\quad n p=10(0.3)=3<10 .\) b. What are the mean value and standard deviation of \(\hat{p}\) based on a random sample of size \(400 ?\) c. When \(n=400\), what is \(P(.25 \leq \hat{p} \leq .35)\) ? d. Is the probability calculated in Part (c) larger or smaller than would be the case if \(n=500\) ? Answer without actually calculating this probability.
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