Complete The Statements.The Estimated Population Proportion, $\widehat{p}$, Is _____%.When The Margin Of Error Is Calculated Using The Formula $E=z^{\star} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$, To The Nearest Tenth Of A

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Introduction

In statistics, the estimated population proportion, denoted by $\widehat{p}$, is a crucial concept used to describe the proportion of a population that possesses a particular characteristic. This value is often used as a basis for making inferences about a larger population. In this article, we will delve into the concept of the estimated population proportion and explore how it is used in conjunction with the margin of error to provide a range of values for a population proportion.

The Estimated Population Proportion

The estimated population proportion, $\widehat{p}$, is a statistic that estimates the true population proportion, denoted by $p$. It is calculated by dividing the number of successes (i.e., the number of individuals in the sample that possess the characteristic of interest) by the total sample size, $n$. Mathematically, this can be represented as:

p^=xn\widehat{p} = \frac{x}{n}

where $x$ is the number of successes in the sample.

Example

Suppose we are interested in estimating the proportion of people in a city who own a smartphone. We conduct a survey of 1000 residents and find that 750 of them own a smartphone. In this case, the estimated population proportion would be:

p^=7501000=0.75\widehat{p} = \frac{750}{1000} = 0.75

This means that we estimate that 75% of the city's residents own a smartphone.

The Margin of Error

The margin of error, denoted by $E$, is a measure of the maximum amount by which the estimated population proportion may differ from the true population proportion. It is calculated using the formula:

E=z⋆⋅p^(1−p^)nE=z^{\star} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

where $z^{\star}$ is the critical value from the standard normal distribution, $\hat{p}$ is the estimated population proportion, and $n$ is the sample size.

Interpretation of the Margin of Error

The margin of error provides a range of values within which the true population proportion is likely to lie. For example, if we estimate that 75% of the city's residents own a smartphone with a margin of error of 3%, we can be 95% confident that the true population proportion lies between 72% and 78%.

Complete the Statement

The estimated population proportion, $\widehat{p}$, is 75%.

Conclusion

In conclusion, the estimated population proportion and margin of error are two important concepts in statistics that are used to make inferences about a larger population. By understanding these concepts, researchers and analysts can gain valuable insights into the characteristics of a population and make informed decisions.

References

  • [1] Moore, D. S., & McCabe, G. P. (2012). Introduction to the practice of statistics. W.H. Freeman and Company.
  • [2] Larson, R. (2013). Elementary statistics: A step-by-step approach. McGraw-Hill Education.

Additional Resources

  • [1] Khan Academy. (n.d.). Statistics and probability. Retrieved from https://www.khanacademy.org/math/statistics-probability
  • [2] Stat Trek. (n.d.). Statistics tutorials and calculators. Retrieved from https://stattrek.com/
    Frequently Asked Questions: Estimated Population Proportion and Margin of Error ====================================================================================

Q: What is the estimated population proportion, and how is it calculated?

A: The estimated population proportion, denoted by $\widehat{p}$, is a statistic that estimates the true population proportion, denoted by $p$. It is calculated by dividing the number of successes (i.e., the number of individuals in the sample that possess the characteristic of interest) by the total sample size, $n$. Mathematically, this can be represented as:

p^=xn\widehat{p} = \frac{x}{n}

where $x$ is the number of successes in the sample.

Q: What is the margin of error, and how is it calculated?

A: The margin of error, denoted by $E$, is a measure of the maximum amount by which the estimated population proportion may differ from the true population proportion. It is calculated using the formula:

E=z⋆⋅p^(1−p^)nE=z^{\star} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

where $z^{\star}$ is the critical value from the standard normal distribution, $\hat{p}$ is the estimated population proportion, and $n$ is the sample size.

Q: What is the purpose of the margin of error?

A: The margin of error provides a range of values within which the true population proportion is likely to lie. For example, if we estimate that 75% of the city's residents own a smartphone with a margin of error of 3%, we can be 95% confident that the true population proportion lies between 72% and 78%.

Q: How do I determine the sample size, $n$, for a given margin of error?

A: The sample size, $n$, can be determined using the formula:

n=(z⋆E)2⋅p^(1−p^)n = \left(\frac{z^{\star}}{E}\right)^2 \cdot \hat{p}(1-\hat{p})

where $z^{\star}$ is the critical value from the standard normal distribution, $E$ is the desired margin of error, and $\hat{p}$ is the estimated population proportion.

Q: What is the relationship between the confidence level and the margin of error?

A: The confidence level and the margin of error are inversely related. As the confidence level increases, the margin of error decreases. For example, if we want to be 99% confident that the true population proportion lies within a certain range, we will need a smaller margin of error than if we were only 95% confident.

Q: Can I use the estimated population proportion and margin of error to make inferences about a larger population?

A: Yes, the estimated population proportion and margin of error can be used to make inferences about a larger population. By understanding these concepts, researchers and analysts can gain valuable insights into the characteristics of a population and make informed decisions.

Q: What are some common applications of the estimated population proportion and margin of error?

A: The estimated population proportion and margin of error have a wide range of applications in fields such as:

  • Marketing research: to estimate the proportion of customers who will respond to a particular marketing campaign
  • Public health: to estimate the proportion of people who have a certain disease or condition
  • Economics: to estimate the proportion of people who will purchase a particular product or service

Conclusion

In conclusion, the estimated population proportion and margin of error are two important concepts in statistics that are used to make inferences about a larger population. By understanding these concepts, researchers and analysts can gain valuable insights into the characteristics of a population and make informed decisions.

References

  • [1] Moore, D. S., & McCabe, G. P. (2012). Introduction to the practice of statistics. W.H. Freeman and Company.
  • [2] Larson, R. (2013). Elementary statistics: A step-by-step approach. McGraw-Hill Education.

Additional Resources