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Chapter 15: Problem 26

Testing Glass. How well materials conduct heat matters when designing houses. As a test of a new measurement process, 10 measurements are made on pieces of glass known to have conductivity 1 . The average of the 10 measurements is 1.07. For each of the boldface numbers, indicate whether it is a parameter or a statistic. Explain your answer.

Short Answer

Expert verified
1 is a parameter; 1.07 is a statistic.

Step by step solution

01

Understanding Parameters and Statistics

A **parameter** is a numerical summary of a population, which represents an entire group. A **statistic**, on the other hand, is a numerical summary of a sample, which is a subset of the population. In this problem, we need to identify whether the numbers given are describing the entire population or just a sample from it.
02

Analyzing Conductivity Value

The number '1' refers to the known conductivity of the glass. This value is obtained from the entire population of glass pieces and represents an inherent property of the material. Therefore, this number is a **parameter** since it characterizes the entire population of glass pieces.
03

Examining the Average Measurement

The average of the 10 measurements is 1.07. These measurements are taken from a subset of all possible pieces of glass, which means this average is based on a sample. Therefore, the number 1.07 is a **statistic** because it summarizes a specific sample of glass pieces rather than the entire population.

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Key Concepts

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

Parameter and Statistic
In statistics, understanding the difference between a parameter and a statistic is crucial. It helps to distinguish whether the data pertains to an entire population or just a sample. A **parameter** is characterized by its ability to describe an entire population, which means it encompasses all elements of that group. For instance, if all the glass pieces ever produced are considered for their conductivity, and their average value is calculated, this average would be termed as a parameter.

On the other hand, a **statistic** relates to a sample. It is a numerical representation of only a portion of the population. For example, if we only measure the conductivity of 10 specific glass pieces, the average of these measurements is a statistic. It represents only those 10 pieces out of potentially thousands or millions. Understanding this distinction helps in comprehending the broader context in statistical terms and ensures accurate data interpretation.
Population vs Sample
When dealing with statistics, it is essential to differentiate between a population and a sample. A **population** is the complete set of all possible subjects or items that possess certain characteristics. For example, all the glass pieces ever made constitute a population in the context of measuring conductivity.

Contrastingly, a **sample** is just a subset or a smaller group selected from this population. If we were to measure conductivity from only 10 pieces of glass out of the entire batch, then these 10 pieces form a sample.
  • Populations provide a comprehensive picture but are often too large to study entirely.
  • Samples are manageable and chosen because they can approximately represent the broader population.
Recognizing if your data represents a population or a sample is fundamental in conducting correct statistical analyses.
Statistical Analysis
The practice of statistical analysis involves using mathematical techniques to interpret, describe, and draw conclusions from data. It involves breaking down complex data sets into understandable segments. At the heart of statistical analysis is the need to differentiate whether our focus is on parameters or statistics, essentially reflecting whether we're dealing with populations or samples.

Let's delve into this through the previous example: the 10 measurements of glass conductivity are summarized by a statistical value, which is the average of these measures. This average is criticized and evaluated under statistical analysis to understand variability, reliability, and significance within the sample.
  • Analysts use statistical tools to estimate parameters by studying statistics from samples.
  • It helps in predicting trends and making decisions based on the data conclusions.
Hence, statistical analysis is fundamental, as it transforms raw data into meaningful information, enabling informed decisions.

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Most popular questions from this chapter

Statistics Anxiety. What can teachers do to alleviate statistics anxiety in their students? To explore this question, statistics anxiety for students in two classes was compared. In one class, the instructor lectured in a formal manner, including dressing formally. In the other, the instructor was less formal, dressed casually, was more personal, used humor, and called on students by their first names. Anxiety was measured using a questionnaire. Higher scores indicate a greater level of anxiety. The mean anxiety score for students in the formal lecture class was 25.40; in the informal class, the mean was 20.41. For each of the boldface numbers, indicate whether it is a parameter or a statistic. Explain your answer.

Guns in School. Researchers surveyed 14,765 American high school students (grades 9-12) and found that \(27.3 \%\) of those surveyed were in grade 9 . The percentage of all American high school students who are are in grade 9 is \(\mathbf{2 6 . 5} \%\). The percentage of those surveyed who were in grade 9 and had carried a gun to school was \(\mathbf{4 . 4 \%}\). Is each of the boldface numbers a parameter or a statistic?

Playing the Numbers. The numbers racket is a wellentrenched illegal gambling operation in most large cities. One version works as follows: you choose one of the 1000 three-digit numbers 000 to 999 and pay your local numbers runner a dollar to enter your bet. Each day, one three-digit number is chosen at random and pays off \(\$ 600\). The mean payoff for the population of thousands of bets is \(\mu=60\) cents. Joe makes one bet every day for many years. Explain what the law of large numbers says about Joe's results as he keeps on betting.

The Law of Large Numbers Made Visible. Roll two balanced dice and count the total spots on the up-faces. The probability model appears in Example 12.5 (page 277). You can see that this distribution is symmetric with 7 as its center, so it's no surprise that the mean is \(\mu=7\). This is the population mean for the idealized population that contains the results of rolling two dice forever. The law of large numbers says that the average \(x\) from a finite number of rolls tends to get closer and closer to 7 as we do more and more rolls. a. Click "More dice" once in the Law of Large Numbers applet to get two dice. Click "Show \(\mu_{x}\) " to see the mean 7 on the graph. Leaving the number of rolls at 1 , click "Roll dice" three times, recording each roll. How many spots did each roll produce? What is the average for the three rolls? You see that the graph displays at each point the average number of spots for all rolls up to the last one. This is exactly like Ejgure 15.1. b. Click "Reset" to start over. Set the number of rolls to 100 and click "Roll dice." The applet rolls the two dice 100 times. The graph shows how the average count of spots changes as we make more rolls. That is, the graph shows \(x\) as we continue to roll the dice. Sketch (or print out) the final graph. c. Repeat your work from part (b). Click "Reset" to start over, then roll two dice 100 times. Make a sketch of the final graph of the mean \(x\) against the number of rolls. Your two graphs will often look very different. What they have in common is that the average eventually gets close to the population mean \(\mu=7\). The law of large numbers says that this will always happen if you keep on rolling the dice.

Insurance. The idea of insurance is that we all face risks that are unlikely but carry high cost. Think of a fire or flood destroying your apartment. Insurance spreads the risk: we all pay a small amount, and the insurance policy pays a large amount to those few of us whose apartments are damaged. An insurance company looks at the records for millions of apartment owners and sees that the mean loss from apartment damage in a year is \(\mu=\$ 150\) per person. (Most of us have no loss, but a few lose most of their possessions. The \(\$ 150\) is the average loss.) The company plans to sell renters' insurance for \(\$ 150\) plus enough to cover its costs and profit. Explain clearly why it would be unwise to sell only 10 policies. Then explain why selling thousands of such policies is a safe business.
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