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Chapter 7: Problem 21

A newspaper poll reported that 73\(\%\) of respondents liked business tycoon Donald Trump. The number 73\(\%\) is (a) a population. (b) a parameter. (c) a sample. (d) a statistic. (e) an unbiased estimator.

Short Answer

Expert verified
The number 73% is a statistic, describing respondents' likes in the sample.

Step by step solution

01

Understanding the terms

Before solving the problem, it's essential to understand the terms provided in the options: (a) **Population**: All possible subjects in a group. (b) **Parameter**: A numerical value describing a characteristic of a population. (c) **Sample**: A subset of the population. (d) **Statistic**: A numerical value describing a characteristic of a sample. (e) **Unbiased Estimator**: A statistic used to estimate a parameter without systematic errors.
02

Examining the problem statement

The survey states that 73% of respondents liked Donald Trump. This means the percentage is derived from the people who responded to the poll, a subset of a larger group of possible respondents.
03

Applying definitions to options

Since the 73% is derived from the respondents (a sample), it is a numerical summary of this sample. A **statistic** is a number that describes a sample. It helps us make inferences about the population.
04

Identifying the correct answer

Based on the definition and the context given, the number 73% is a statistic, because it represents a summary of the survey results from the sample (respondents).

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Key Concepts

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

Population
In statistics, the term *population* refers to the entire group of individuals or items that we want to study and make conclusions about. This could be as broad as all humans living on Earth, or as specific as the students in a particular classroom. The key idea is that a population includes every possible member that fits the criteria of your research interest.

Some important points about populations in statistics include:
  • The population is all-embracing and often too large to examine in its entirety.
  • Populations can be both finite (e.g., all the students in a school) or infinite (e.g., all possible rolls of a die).
  • Studying an entire population can be impractical, which leads researchers to use samples.
The goal of statistical analysis is often to learn about the population by examining a smaller group known as a sample.
Sample
A *sample* is a smaller group chosen from a larger population and is used to make inferences about the population. Imagine a giant jar filled with marbles, and you take out a handful to count; this handful represents a sample.

When choosing a sample, it's important to ensure that it accurately reflects the population. This is typically done through various sampling methods such as random sampling, stratified sampling, or cluster sampling.
  • A good sample should be representative of the population to minimize bias.
  • Sample size matters; larger samples can provide more reliable insights.
  • Analysis of sample data results in what is known as a *statistic*.
Statistics derived from samples, like the 73% mentioned in the newspaper poll example, help us estimate and make conclusions about the entire population's behavior or characteristics.
Parameter
The term *parameter* refers to a numerical value that provides information about a particular characteristic of a population. It is an unknown fixed value for which statistics are used to make estimates.
  • Parameters are the true values in the population that we're often trying to uncover through sample statistics.
  • Examples of parameters include the population mean (average), population standard deviation, or population proportion.
  • Unlike statistics, parameters remain consistent unless the entire population changes.
In our context, if the 73% had been based on the opinion of every individual in the population concerning Donald Trump, this value would represent a *parameter*. However, because it was derived from a sample (respondents of the poll), 73% serves as a statistic estimating the population parameter.

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Most popular questions from this chapter

Bottling cola A bottling company uses a flling machine to fill plastic bottles with cola. The bottles are supposed to contain 300 milliters \((\mathrm{ml}) .\) In fact, the contents vary according to a Normal distribution with mean \(\mu=298 \mathrm{ml}\) and standard deviation \(\sigma=3 \mathrm{ml}\) (a) What is the probability that an individual bottle contains less than 295 \(\mathrm{ml}\) ? Show your work. (b) What is the probability that the mean contents of six randomly selected bottles is less than 295 \(\mathrm{ml}\) ? Show your work.

Do you go to church? The Gallup Poll asked a random sample of 1785 adults whether they attended church or synagogue during the past week. Of the respondents, 44\(\%\) said they did attend. Suppose that 40\(\%\) of the adult population actually went to church or synagogue last week. Let \(\hat{p}\) be the proportion of people in the sample who attended church or synagogue. (a) What is the mean of the sampling distribution of \(\hat{p}\) ? Why? (b) Find the standard deviation of the sampling distribution of \(\hat{p} .\) Check to see if the 10\(\%\) condition is met. (c) Is the sampling distribution of \(\hat{p}\) approximately Normal? Check to see if the Normal condition is met. (d) Find the probability of obtaining a sample of 1785 adults in which 44\(\%\) or more say they attended church or synagogue last week. Do you have any doubts about the result of this poll?

Larger sample Suppose that the blood cholesterol level of all men aged 20 to 34 follows the Normal distribution with mean \(\mu=188\) milligrams per deciliter \((\mathrm{mg} / \mathrm{dl})\) and standard deviation \(\sigma=41 \mathrm{mg} / \mathrm{dl}\) (a) Choose an SRS of 100 men from this population What is the sampling distribution of \(\overline{x} ?\) (b) Find the probability that \(\overline{x}\) estimates \(\mu\) within \(\pm 3 \mathrm{mgldl}\) . This is the probability that \(\overline{x}\) takes a value between 185 and 191 \(\mathrm{mg} / \mathrm{dl} .\) ) Show your work. (c) Choose an SRS of 1000 men from this population. Now what is the probability that \(\overline{x}\) falls within \(\pm 3 \mathrm{mg} / \mathrm{dl}\) of \(\mu ?\) Show your work. In what sense is the larger sample "better"?

Sharing music online \((5.2)\) A sample survey reports that 29\(\%\) of Internet users download music fles online, 21\(\%\) share music fles from their computers, and 12\(\%\) both download and share music. 6 Make a Venn diagram that displays this information. What percent of Internet users neither download nor share music files?

If we take a simple random sample of size \(n=500\) from a population of size \(5,000,000,\) the variability of our estimate will be (a) much less than the variability for a sample of size \(n=500\) from a population of size \(50,000,000\) . (b) slightly less than the variability for a sample of size \(n=500\) from a population of size \(50,000,000\) . (c) about the same as the variability for a sample of size \(n=500\) from a population of size \(50,000,000\) . (d) slightly greater than the variability for a sample of size \(n=500\) from a population of size \(50,000,000\) . (e) much greater than the variability for a sample of size \(n=500\) from a population of size \(50,000,000\) .
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