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Chapter 8: Problem 69

What is the point estimator of the population proportion, \(p\) ?

Short Answer

Expert verified
The point estimator for the population proportion, \(p\), is the sample proportion, denoted by \(\hat{p}\), calculated using the formula \(\hat{p} = \dfrac{x}{n}\).

Step by step solution

01

Definition of Sample Proportion

The sample proportion, often denoted by \(\hat{p}\), is used as a point estimator for the population proportion, \(p\). The sample proportion is calculated by the number of successful events over the sample size.
02

Formula for Sample Proportion

The sample proportion, \(\hat{p}\), is calculated using the formula: \(\hat{p} = \dfrac{x}{n}\), where \(x\) is the number of successful outcomes in the sample and \(n\) is the size of the sample.
03

Point Estimator of Population Proportion

The point estimator for the population proportion, \(p\), is the sample proportion, \(\hat{p}\), meaning we estimate the proportion of the population by calculating the proportion of the sample.

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Key Concepts

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

Understanding Population Proportion
The population proportion is a critical concept in statistics. It is denoted by the symbol \( p \) and represents the ratio of a particular characteristic within an entire population. For example, if we are looking at the proportion of left-handed people in a country, \( p \) would be the proportion of left-handed individuals compared to the total population of the country. This measure gives us a wider view of how common a characteristic is in the whole population.
  • Population proportion \( p \) refers to the characteristic ratio in the entire population.
  • It is usually unknown, so we need tools like point estimation to estimate it.
  • Understanding the proportion gives insights into the prevalence of a trait or feature among the population.
Thus, knowing the population proportion helps in decision-making and predicting future trends based on historical data.
The Role of Sample Proportion
The sample proportion, symbolized as \( \hat{p} \), acts as a mirror for population proportion in statistics because we often do not have access to complete population data. Instead, we collect samples. The sample proportion is calculated by taking the number of successful events or occurrences of a feature in the sample, \( x \), and dividing it by the total sample size, \( n \). This calculation gives us \( \hat{p} = \dfrac{x}{n} \), which serves as an estimate for the unknown population proportion.
  • Sample proportion \( \hat{p} \) is used when the entire population cannot be studied.
  • Helps in bridging the gap between sample data and population characteristics.
  • Provides flexibility by utilizing a smaller group to make population inferences.
By using sample proportion, we try to representatively infer the characteristics of the whole population.
What is Point Estimation?
Point estimation plays a vital role in the field of statistics, helping to estimate an unknown population parameter using sample data. The sample proportion, \( \hat{p} \), is the point estimator for the population proportion, \( p \). It tries to provide a single value as a close approximation of the actual unknown population value. This concept is essential because it allows statisticians to make educated guesses based on available data.
  • A single best guess or estimate of an unknown parameter is made through point estimation.
  • The sample proportion \( \hat{p} \) functions as the point estimator for the population proportion \( p \).
  • Provides a more straightforward approach compared to interval estimation.
Thus, point estimation simplifies the process of interpreting statistical data and drawing conclusions from it.
Using Statistical Formulas
Statistical formulas are the backbone of data analysis and are used to perform calculations essential for interpreting sample data. These formulas guide us in estimating unknown parameters such as the population proportion. For calculating the sample proportion \( \hat{p} \), we use the simple yet powerful formula \( \hat{p} = \dfrac{x}{n} \). Understanding and applying this formula helps in translating sample observations into meaningful estimates of population parameters.
  • Formulas like \( \hat{p} = \dfrac{x}{n} \) help estimate unknown population parameters.
  • They provide a structured way to analyze and interpret data from samples.
  • A critical tool for ensuring the accuracy and reliability of statistical conclusions.
Ultimately, mastering statistical formulas is fundamental for anyone looking to delve into data analysis and statistical computations.

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Most popular questions from this chapter

a. A sample of 300 observations taken from a population produced a sample proportion of .63. Make a \(95 \%\) confidence interval for \(p\). b. Another sample of 300 observations taken from the same population produced a sample proportion of .59. Make a \(95 \%\) confidence interval for \(p\). c. A third sample of 300 observations taken from the same population produced a sample proportion of .67. Make a \(95 \%\) confidence interval for \(p\). d. The true population proportion for this population is .65. Which of the confidence intervals constructed in parts a through c cover this population proportion and which do not?

In June 2008 , SBRI Public Affairs conducted a telephone poll of 1004 adult Americans aged 18 and older. One of the questions asked was, "In the past year, was there ever a time when you ...?" Respondents could choose more than one of the answers mentioned. Of the respondents, \(64 \%\) said "cut back on vacations or entertainment because of their cost," \(37 \%\) said "failed to pay a bill on time," and \(25 \%\) said "have not gone to a doctor because of the cost." (Source: http://www.srbi.com/AmericansConcernEconomic.html.) Using these results, find a \(95 \%\) confidence interval for the corresponding population percentage for each answer. Write a one-page report to present these results to a group of college students who have not taken statistics. Your report should answer questions such as: (1) What is a confidence interval? (2) Why is a range of values more informative than a single percentage? (3) What does \(95 \%\) confidence mean in this context? (4) What assumptions, if any, are you making when you construct each confidence interval?

Tony's Pizza guarantees all pizza deliveries within 30 minutes of the placement of orders. An agency wants to estimate the proportion of all pizzas delivered within 30 minutes by Tony's. What is the most conservative estimate of the sample size that would limit the margin of error to within \(.02\) of the population proportion for a \(99 \%\) confidence interval?

For each of the following, find the area in the appropriate tail of the \(t\) distribution. a. \(t=2.467\) and \(d f=28\) b. \(t=-1.672\) and \(d f=58\) c. \(t=-2.670\) and \(n=55\) d. \(t=2.383\) and \(n=23\)

A department store manager wants to estimate at a \(90 \%\) confidence level the mean amount spent by all customers at this store. The manager knows that the standard deviation of amounts spent by all customers at this store is \(\$ 31\). What sample size should he choose so that the estimate is within \(\$ 3\) of the population mean?
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