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Chapter 1: Problem 4

For a set population, does a parameter ever change? If there are three different samples of the same size from a set population, is it possible to get three different values for the same statistic?

Short Answer

Expert verified
The parameter never changes unless the population changes, but statistical values from samples can vary.

Step by step solution

01

Understanding Parameters and Statistics

A parameter is a fixed measure that describes a characteristic of a population, such as the mean, median, or standard deviation. It does not change unless the population itself changes. In contrast, a statistic is a measure that describes a characteristic of a sample drawn from the population and can vary from one sample to another.
02

Analyzing the Parameter

Since a parameter is calculated using the entire population data, it does not change unless the population changes. This stability is because it reflects a fixed attribute of the whole population.
03

Statistics and Sample Variation

A statistic varies because it is calculated from a sample instead of the full population. Different samples may include different individuals, leading to variability in the statistic even when the sample size remains the same.
04

Conclusion on Multiple Sample Statistics

Even if you have three different samples of the same size from the same population, you can obtain three different values for the same statistic due to natural sample variation. This is why statistics like the sample mean can vary between different samples, even if the samples are from the same population.

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Key Concepts

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

Population Parameter
A population parameter is a crucial concept in statistics. It represents a fixed numerical value that describes an entire population. Examples of population parameters include:
  • Mean (average)
  • Median (middle value)
  • Standard deviation (measure of data spread)
Population parameters are like a snapshot of the entire population's characteristics. They remain constant unless the population itself undergoes a change. This is because they encapsulate traits of every member with no sampling involved. For instance, if we knew the exact average height of all adults in a country, that would be the population parameter for height. In most real-world scenarios, measuring the entire population is impractical, necessitating the use of samples.
Sample Variability
Sample variability is a fundamental concept that helps explain why two samples from the same population can yield different results. When a sample is drawn from a population, it may capture different aspects or subsets. Here are important points concerning sample variability:
  • Each sample may capture different observations and thus provide varied results.
  • Randomness in choosing samples leads to natural variability.
  • The more diverse or spread out the population, the greater the expected variability in samples.
This natural variability is why we often observe different statistics when comparing multiple samples. For statistical analysis, understanding and accounting for sample variability is essential to make accurate predictions and conclusions.
Sample Mean
The sample mean is one of the most commonly used statistics to estimate the population mean. When calculating the sample mean, we take the sum of all data points in the sample and divide by the number of points. It is denoted as \( \bar{x} \). The sample mean is a valuable tool for:
  • Making inferences about the population mean.
  • Showing central tendency within the sample.
  • Providing an estimate of where most data points lie.
However, the sample mean is susceptible to sample variability. This means if you take several samples from the same population, the sample mean might differ for each. Despite this, with a large enough sample size, the sample mean tends to give a good approximation of the population mean, thanks to the law of large numbers.

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Most popular questions from this chapter

Surveys: Manipulation The New York Times did a special report on polling that was carried in papers across the nation. The article pointed out how readily the results of a survey can be manipulated. Some features that can influence the results of a poll include the following: the number of possible responses, the phrasing of the questions, the sampling techniques used (voluntary response or sample designed to be representative), the fact that words may mean different things to different people, the questions that precede the question of interest, and finally, the fact that respondents can offer opinions on issues they know nothing about. (a) Consider the expression "over the last few years." Do you think that this expression means the same time span to everyone? What would be a more precise phrase? (b) Consider this question: "Do you think fines for running stop signs should be doubled?" Do you think the response would be different if the question "Have you ever run a stop sign?" preceded the question about fines? (c) Consider this question: "Do you watch too much television?" What do you think the responses would be if the only responses possible were yes or no? What do you think the responses would be if the possible responses were "rarely," "sometimes," or "frequently"?

Government agencies carefully monitor water quality and its effect on wetlands (Reference: Environmental Protection Agency Wetland Report EPA \(832-\mathrm{R}-93-005\) ). Of particular concern is the concentration of nitrogen in water draining from fertilized lands. Too much nitrogen can kill fish and wildlife. Twenty-eight samples of water were taken at random from a lake. The nitrogen concentration (milligrams of nitrogen per liter of water) was determined for each sample. (a) Identify the variable. (b) Is the variable quantitative or qualitative? (c) What is the implied population?

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