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91Ó°ÊÓ

Suppose a 12 -year-old asked you to explain the difference between a statistic and a parameter. a. What information should your answer include? b. What reasons would you give him or her for why one would report the value of a statistic instead of the value of a parameter?

Short Answer

Expert verified
A statistic is an attribute of a sample from the data, while a parameter is an attribute of all the data. A statistic is often reported instead of a parameter because it is difficult, time-consuming, or impossible to collect all the data that a parameter would require.

Step by step solution

01

Define Statistic and Parameter

A statistic, in simple terms, can be described as a number that represents a part of the data, like an average or a sum. On the other hand, a parameter is a value that describes a characteristic of the entire set of data. For example, if we have the heights of all the trees in a forest, the average height would be a parameter.
02

Provide Examples

To make it easier to understand, let's think of it like this: imagine we want to know the average height of students in a school. It would be very time-consuming to measure every single student, right? So, instead, we choose a few students randomly and measure their heights. The average height we get from this smaller group is a statistic. The average height of all students in the entire school, which we did not measure, is called a parameter.
03

Explain Why One Might Report a Statistic Instead of a Parameter

Now, the reason we sometimes report the value of a statistic instead of a parameter is because it's often impossible or too much work to collect all the data needed for a parameter. For the school example, it would be hard to measure all the students. So we use a statistic, which comes from just a part of the data, as an estimate of the parameter, which comes from all of the data.

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Key Concepts

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

Sampling in Statistics
Sampling in statistics is quite similar to taking a small bite of something to understand the whole flavor. It involves selecting a small group, or a sample, from a larger group. This large group is known as the population. Sampling aims to gather information about the population without needing to measure every individual item or person. This makes it easier and more practical, especially when the population is very large.
Instead of surveying every single student in a school to find out their average height, we select a few students to represent the whole group. This selected group is our sample. The data from this sample helps us estimate characteristics of the entire population, like the average height of all students.
To make sure our sample accurately represents the population, the process of selection should be random. This reduces bias and leads to more reliable results. Here are some important points about sampling:
  • Random Sampling: Every individual has an equal chance of being chosen, which helps in reducing bias.
  • Sample Size: Larger samples often provide more accurate estimates than smaller ones.
  • Sampling Method: Different methods like simple random sampling, stratified sampling, and cluster sampling can be used depending on the situation.
Descriptive Statistics
Descriptive statistics is all about summarizing and organizing data in a way that makes it easier to understand. Think of it as using photographs instead of reading a long novel. It gives you a straightforward view of what the data looks like.
These statistics provide simple summaries and visualizations of data. Common measures in descriptive statistics include the mean (average), median (middle value), and mode (most frequent value). These help describe the central tendency of data, or where the middle of the data lies.
Descriptive statistics also include measures of variability like range, variance, and standard deviation, which tell us about how spread out the data is. Some key aspects of descriptive statistics include:
  • Central Tendency: This involves finding the average (mean), which gives us an idea about the central value of the data.
  • Spread: It includes measuring how much the data varies through measures like range and standard deviation.
  • Data Visualization: Using graphs and charts, such as histograms and pie charts, to make data easier to see and understand.
Population Parameters
Population parameters are values that describe certain characteristics of an entire population. These are like the 'facts' about the whole group we're interested in. Because they represent the whole set, finding these values can be difficult if the population is large.
Parameters could be statistics like the population mean, which signifies the average value of a data set, or the population variance, telling how data spread around that mean. For example, measuring the height of every student in a school to find the average height is finding a population parameter.
Since collecting data on all individuals is usually impractical, especially with large populations, we often use statistics from samples to estimate these parameters. Here’s a closer look:
  • True Representation: Parameters provide a precise depiction of the population’s characteristics, but they're hard to obtain due to population size.
  • Estimation: Through sampling, we estimate these parameters using statistics from samples.
  • Fixed Values: Unlike sample-derived statistics, parameters are fixed and do not change unless the population itself changes.

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Most popular questions from this chapter

Harris Interactive conducted an online poll of U.S. adults during August 2008 in anticipation of September being Library Sign-up Month. Libary Card Use These are some of the results of a Harris Interactive poll of 2,710 U.S. adults conducted online between August 11 and 17,2008 . \(68 \%\) of Americans currently own a library card. Certain groups are more likely to have a library card than others-Echo Boomers (those \(18-31\) ) are more likely to have one over other age categories \((70 \%\) versus \(68-65 \%) ;\) women over men \((73 \% \text { versus } 62 \%) ;\) Hispanics over African Americans and Whites (72\% versus 67\% and \(66 \%\) ); Midwesterners \((72 \%)\) over Easterners \((65 \%)\) and Southerners (63\%). Politically there is also a difference as Democrats are more likely to have a library card over Republicans and Independents \((71 \% \text { versus } 67 \% \text { and } 61 \%)\) Over one-third \((35 \%)\) of people with a library card have used the library 1 to 5 times in the past year and \(15 \%\) have used it more than 25 times in the past year. a. What is the population? b. Name at least 3 variables that must have been used. c. Classify all the variables of the study as either attribute or numerical.

Identify each of the following as examples of (1) attribute (qualitative) or ( 2 ) numerical (quantitative) variables: a. The breaking strength of a given type of string b. The hair color of children auditioning for the musical Annie c. The number of stop signs in towns of fewer than 500 people d. Whether or not a faucet is defective e. The number of questions answered correctly on a standardized test f. The length of time required to answer a telephone call at a certain real estate office

Statistics is defined on page 1 as "the science of collecting, describing, and interpreting data." Using your own words, write a sentence describing each of the three statistical activities. Retain your work for Exercise 1.79

Of the adult U.S. population, \(36 \%\) has an allergy. A sample of 1200 randomly selected adults resulted in \(33.2 \%\) having an allergy. a. Describe the population. b. What is the sample? c. Describe the variable. d. Identify the statistic and give its value. e. Identify the parameter and give its value.

We want to describe the so-called typical student at your college. Describe a variable that measures some characteristic of a student and results in a. Attribute data b. Numerical data

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