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Chapter 6: Problem 3

What is a population parameter? Give three examples.

Short Answer

Expert verified
A population parameter is a measure of a population's characteristics, like \(\mu\), \(P\), and \(\sigma^2\).

Step by step solution

01

Understanding Population Parameter

A population parameter is a characteristic or measure of an entire population. It is usually a constant value that provides information about a specific trait of the population, such as the average or proportion.
02

Example 1: Population Mean

The population mean, denoted as \(\mu\), is the average of all values in a population. It represents the central tendency of the population data.
03

Example 2: Population Proportion

The population proportion, denoted as \(P\), is the ratio of members in a population that have a particular attribute, expressed as a fraction or percentage.
04

Example 3: Population Variance

The population variance, denoted as \(\sigma^2\), is a measure of the dispersion or variability of the values in a population from the population mean.

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Key Concepts

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

Population Mean
The population mean is an important measure in statistics, representing the average of all data points within a population. It is denoted by the symbol \(\mu\). Calculating the population mean involves summing all the values and then dividing by the number of values present in the population. This metric helps us understand the central tendency of data, giving us insight into where most values are clustered.
In a formula, the population mean is expressed as:
  • \( \mu = \frac{\sum X}{N} \)
Where \(\sum X\) is the sum of all values in the population, and \(N\) is the total number of values.
  • Important because it helps identify real-world patterns.
  • Used in everyday decision-making, from product development to healthcare analysis.
Population Proportion
Population proportion is another key concept that deals with understanding how prevalent a certain characteristic is within a population. The population proportion is denoted by \(P\) and is calculated as the number of individuals in the population with a particular attribute divided by the total number of individuals in the population.
Consider it as measuring a part of a whole and expressing this part as a fraction or percentage of the entire population.
Mathematically, population proportion can be expressed as:
  • \( P = \frac{X}{N} \)
Where \(X\) represents the number of individuals with the attribute, and \(N\) is the overall population size.
  • Helps to determine percentages of traits, like survey results or voting preferences.
  • Crucial for conducting accurate market research and behavioral studies.
Population Variance
Population variance is a measure indicating how data points in a population differ from the population mean. It is symbolized by \(\sigma^2\) and reflects the degree of spread or variability within the dataset.
A low population variance means that most data points are close to the mean, revealing a consistent population. Conversely, a high variance signifies that data points are more scattered, indicating greater diversity within the population.
The population variance is calculated with this formula:
  • \( \sigma^2 = \frac{\sum (X - \mu)^2}{N} \)
Where \(X\) is an individual data point, \(\mu\) is the population mean, and \(N\) is the total population size.
  • Used to identify consistency across a data set.
  • Instrumental in fields like finance and genetics, where variability impacts decision-making.

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Most popular questions from this chapter

Sketch the areas under the standard normal curve over the indicated intervals and find the specified areas. To the left of \(z=0.72\)

Find the \(z\) value described and sketch the area described.Find \(z\) such that \(97.5 \%\) of the standard normal curve lies to the left of \(z\).

Let \(x\) represent the dollar amount spent on supermarket impulse buying in a 10 -minute (unplanned) shopping interval. Based on a Denver Post article, the mean of the \(x\) distribution is about \(\$ 20\) and the estimated standard deviation is about \(\$ 7\) (a) Consider a random sample of \(n=100\) customers, each of whom has 10 minutes of unplanned shopping time in a supermarket. From the central limit theorem, what can you say about the probability distribution of \(\bar{x}\) the average amount spent by these customers due to impulse buying? What are the mean and standard deviation of the \(\bar{x}\) distribution? Is it necessary to make any assumption about the \(x\) distribution? Explain. (b) What is the probability that \(\bar{x}\) is between \(\$ 18\) and \(\$ 22 ?\) (c) Let us assume that \(x\) has a distribution that is approximately normal. What is the probability that \(x\) is between \(\$ 18\) and \(\$ 22 ?\) (d) Interpretation: In part (b), we used \(\bar{x},\) the average amount spent, computed for 100 customers. In part (c), we used \(x,\) the amount spent by only one customer. The answers to parts (b) and (c) are very different. Why would this happen? In this example, \(\bar{x}\) is a much more predictable or reliable statistic than \(x\). Consider that almost all marketing strategies and sales pitches are designed for the average customer and not the individual customer. How does the central limit theorem tell us that the average customer is much more predictable than the individual customer?

Insurance: Claims Do you try to pad an insurance claim to cover your deductible? About \(40 \%\) of all U.S. adults will try to pad their insurance claims! (Source: Are You Normal?, by Bernice Kanner, St. Martin's Press.) Suppose that you are the director of an insurance adjustment office. Your office has just received 128 insurance claims to be processed in the next few days. What is the probability that (a) half or more of the claims have been padded? (b) fewer than 45 of the claims have been padded? (c) from 40 to 64 of the claims have been padded? (d) more than 80 of the claims have not been padded?

Is \(\hat{p}\) an unbiased estimator for \(p\) when \(n p>5\) and \(n q>5 ?\) Recall that a statistic is an unbiased estimator of the corresponding parameter if the mean of the sampling distribution equals the parameter in question.
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