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Chapter 7: Problem 5

Two symbols are used for the mean: \(\mu\) and \(\bar{x}\). a. Which represents a parameter, and which a statistic? b. In determining the mean age of all students at your school, you survey 30 students and find the mean of their ages. Is this mean \(\bar{x}\) or \(\mu\) ?

Short Answer

Expert verified
a. \(\mu\) represents a parameter and \(\bar{x}\) represents a statistic. b. The mean is \(\bar{x}\) as \(\bar{x}\) represents the mean of a sample and we have taken a sample of 30 students from the school.

Step by step solution

01

Understand The Terms

Firstly, it's key to understand these two statistical terms: a parameter is a characteristic of an entire population, whereas a statistic is a characteristic of a sample which is a subset of the population. \(\mu\) denotes the population mean, while \(\bar{x}\) signifies the sample mean.
02

Identify The Parameter and Statistic

Given the definitions, \(\mu\) represents a parameter because it describes the characteristic of an entire population. On the other hand, \(\bar{x}\) symbol represents a statistic as it describes the characteristic of a sample.
03

Determine The Mean

In the scenario of determining the mean age of all students at the school by surveying 30 students, this is sampling a subset of the population, not the entire population. Therefore, the mean of their ages would be \(\bar{x}\), which represents a statistic or the mean of a sample.

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Key Concepts

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

Population Parameter
When we talk about a population parameter, we're referring to a characteristic that applies to an entire population. If the goal is to measure a particular quality or feature broadly across every member of a group, we are dealing with a parameter. In statistics, we often use Greek letters to symbolize parameters. For example, the mean of a population is denoted by the Greek letter \( \mu \).

Some characteristics of population parameters include:
  • The parameter encompasses the full range of potential values as it considers the entire population.
  • It remains fixed unless there's a change in the entire population.
  • It's often impractical to measure because it's costly and time-consuming to include everyone.
In statistical terms, population parameters are the gold standard of measurement, providing the most accurate representation.
Sample Statistic
Unlike a population parameter, a sample statistic is derived from a smaller group, a subset of the larger population. This statistic approximates or infers what the actual parameter might be. The sample's mean is often referred to by the symbol \( \bar{x} \), as it relates specifically to the sample.

A few key points about sample statistics include:
  • Statistics can vary because they are based on a part of the population, not the whole.
  • They provide a practical and efficient way of making inferences about a population.
  • As samples may differ each time, results could slightly change with different samples.
Sample statistics are crucial in research and studies where it is impossible or impractical to measure everyone in a population.
Mean Calculation
Mean calculation is fundamental in understanding datasets in both statistics and everyday situations. The mean gives us an average number that represents a central point of the data. If we're discussing a sample, we calculate the sample mean, \( \bar{x} \), whereas for a whole population, the parameter mean is \( \mu \).

To calculate the mean of a sample:
  • Add all the values or data points together.
  • Divide the total by the number of data points.
Let's consider taking the ages of 30 students to find the average age. By summing their ages and dividing by 30, you attain \( \bar{x} \), the sample mean.

Mean calculations offer insights into data's tendency, helping to predict trends or draw comparisons in real-world situations.

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Most popular questions from this chapter

In 2003 and 2017 Gallup asked Democratic voters about their views on the FBI. In \(2003,44 \%\) thought the FBI did a good or excellent job. In \(2017,69 \%\) of Democratic voters felt this way. Assume these percentages are based on samples of 1200 Democratic voters. a. Can we conclude, on the basis of these two percentages alone, that the proportion of Democratic voters who think the FBI is doing a good or excellent job has increase from 2003 to \(2017 ?\) Why or why not? b. Check that the conditions for using a two-proportion confidence interval hold. You can assume that the sample is a random sample. c. Construct a \(95 \%\) confidence interval for the difference in the proportions of Democratic voters who believe the FBI is doing a good or excellent job, \(p_{1}-p_{2}\). Let \(p_{1}\) be the proportion of Democratic voters who felt this way in 2003 and \(p_{2}\) be the proportion of Democratic voters who felt this way in 2017 . d. Interpret the interval you constructed in part c. Has the proportion of Democratic voters who feel this way increased? Explain.

You want to find the mean weight of the students at your college. You calculate the mean weight of a sample of members of the football team. Is this method biased? If so, would the mean of the sample be larger or smaller than the true population mean for the whole school? Explain.

In carrying out a study of views on capital punishment, a student asked a question two ways: 1\. With persuasion: "My brother has been accused of murder and he is innocent. If he is found guilty, he might suffer capital punishment. Now do you support or oppose capital punishment?" 2\. Without persuasion: "Do you support or oppose capital punishment?" Here is a breakdown of her actual data. $$ \begin{aligned} &\text { Men }\\\ &\begin{array}{lcc} & \begin{array}{c} \text { With } \\ \text { persuasion } \end{array} & \begin{array}{c} \text { No } \\ \text { persuasion } \end{array} \\ \hline \text { For capital punishment } & 6 & 13 \\ \hline \text { Against capital punishment } & 9 & 2 \\ \text { Women } \end{array}\\\ &\begin{array}{lcc} & \begin{array}{c} \text { With } \\ \text { persuasion } \end{array} & \begin{array}{c} \text { No } \\ \text { persuasion } \end{array} \\ \hline \text { For capital punishment } & 2 & 5 \\ \hline \text { Against capital punishment } & 8 & 5 \end{array} \end{aligned} $$ a. What percentage of those persuaded against it support capital punishment? b. What percentage of those not persuaded against it support capital punishment? c. Compare the percentages in parts a and b. Is this what you expected? Explain.

The Centers for Disease Control and Prevention (CDC) conducts an annual Youth Risk Behavior Survey, surveying over 15,000 high school students. The 2015 survey reported that, while cigarette use among high school youth had declined to its lowest levels, \(24 \%\) of those surveyed reported using e-cigarettes. Identify the sample and population. Is the value \(24 \%\) a parameter or a statistic? What symbol would we use for this value?

The website scholarshipstats.com collected data on all 5341 NCAA basketball players for the 2017 season and found a mean height of 77 inches. Is the number 77 a parameter or a statistic? Also identify the population and explain your choice.
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