If The Confidence Level And The Point Estimate Remain The Same, But The Sample Size Is Increased To 100 People, Which Statement Is True?$E = Z^* \sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}$A. The New Survey's Margin Of Error Will Be 50% The

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If the Confidence Level and Point Estimate Remain the Same, But the Sample Size is Increased to 100 People

Understanding the Relationship Between Sample Size and Margin of Error

When conducting surveys or experiments, researchers often rely on statistical methods to estimate population parameters. One crucial aspect of statistical analysis is the margin of error, which represents the maximum amount by which the sample estimate may differ from the true population parameter. In this article, we will explore the relationship between sample size and margin of error, specifically focusing on the impact of increasing the sample size from 50 to 100 people.

The Formula for Margin of Error

The margin of error is typically calculated using the following formula:

E=zβˆ—p^(1βˆ’p^)nE = z^* \sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}

where:

  • EE is the margin of error
  • zβˆ—z^* is the critical value from the standard normal distribution corresponding to the desired confidence level
  • p^\widehat{p} is the sample proportion
  • nn is the sample size

The Effect of Increasing Sample Size

If the confidence level and point estimate remain the same, but the sample size is increased to 100 people, we need to determine how this change affects the margin of error. To do this, we can substitute the new sample size into the formula and examine the resulting margin of error.

Comparing the Margin of Error with Different Sample Sizes

Let's assume that the sample proportion p^\widehat{p} remains the same, and the confidence level is still 95%. We can calculate the margin of error for both sample sizes using the formula above.

For the original sample size of 50 people:

E50=zβˆ—p^(1βˆ’p^)50E_{50} = z^* \sqrt{\frac{\widehat{p}(1-\widehat{p})}{50}}

For the new sample size of 100 people:

E100=zβˆ—p^(1βˆ’p^)100E_{100} = z^* \sqrt{\frac{\widehat{p}(1-\widehat{p})}{100}}

Analyzing the Results

By comparing the two formulas, we can see that the margin of error for the new sample size of 100 people is:

E100=12E50E_{100} = \frac{1}{\sqrt{2}} E_{50}

This means that the new survey's margin of error will be approximately 71% of the original margin of error.

Conclusion

In conclusion, if the confidence level and point estimate remain the same, but the sample size is increased to 100 people, the new survey's margin of error will be approximately 71% of the original margin of error. This demonstrates the positive impact of increasing the sample size on the accuracy of the survey results.

Frequently Asked Questions

  • Q: What happens to the margin of error when the sample size is increased? A: The margin of error decreases as the sample size increases.
  • Q: How much does the margin of error decrease when the sample size is increased from 50 to 100 people? A: The margin of error decreases by approximately 29%.
  • Q: What is the relationship between the sample size and the margin of error? A: The margin of error is inversely proportional to the square root of the sample size.

References

  • [1] Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
  • [2] Larson, R. E., & Farber, B. A. (2018). Elementary statistics: Picturing the world. Cengage Learning.
  • [3] Agresti, A., & Franklin, C. (2018). Statistics: The art and science of learning from data. Pearson Education.
    Margin of Error: Frequently Asked Questions

Understanding the Margin of Error

The margin of error is a crucial concept in statistics that helps researchers and analysts understand the accuracy of their estimates. In this article, we will address some of the most frequently asked questions about the margin of error.

Q: What is the margin of error?

A: The margin of error is the maximum amount by which the sample estimate may differ from the true population parameter.

Q: How is the margin of error calculated?

A: The margin of error is typically calculated using the following formula:

E=zβˆ—p^(1βˆ’p^)nE = z^* \sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}

where:

  • EE is the margin of error
  • zβˆ—z^* is the critical value from the standard normal distribution corresponding to the desired confidence level
  • p^\widehat{p} is the sample proportion
  • nn is the sample size

Q: What factors affect the margin of error?

A: The margin of error is affected by the following factors:

  • Sample size: Increasing the sample size decreases the margin of error.
  • Confidence level: Increasing the confidence level increases the margin of error.
  • Sample proportion: The margin of error is largest when the sample proportion is 0.5.

Q: How does the margin of error relate to the sample size?

A: The margin of error is inversely proportional to the square root of the sample size. This means that as the sample size increases, the margin of error decreases.

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

A: The margin of error is directly proportional to the critical value from the standard normal distribution corresponding to the desired confidence level. This means that as the confidence level increases, the margin of error also increases.

Q: Can the margin of error be reduced?

A: Yes, the margin of error can be reduced by:

  • Increasing the sample size
  • Decreasing the confidence level
  • Improving the quality of the sample

Q: What is the difference between the margin of error and the standard error?

A: The standard error is the standard deviation of the sampling distribution of the sample estimate. The margin of error is the maximum amount by which the sample estimate may differ from the true population parameter.

Q: How is the margin of error used in practice?

A: The margin of error is used in a variety of applications, including:

  • Polling and survey research
  • Quality control and process improvement
  • Medical research and clinical trials
  • Business and marketing research

Q: What are some common mistakes to avoid when calculating the margin of error?

A: Some common mistakes to avoid when calculating the margin of error include:

  • Failing to account for the sample size
  • Using an incorrect confidence level
  • Ignoring the sample proportion
  • Failing to consider the standard error

Conclusion

The margin of error is a critical concept in statistics that helps researchers and analysts understand the accuracy of their estimates. By understanding the factors that affect the margin of error and how to calculate it, researchers can make more informed decisions and improve the accuracy of their results.

Frequently Asked Questions: Margin of Error

  • Q: What is the margin of error? A: The margin of error is the maximum amount by which the sample estimate may differ from the true population parameter.
  • Q: How is the margin of error calculated? A: The margin of error is typically calculated using the following formula:

E=zβˆ—p^(1βˆ’p^)nE = z^* \sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}

  • Q: What factors affect the margin of error? A: The margin of error is affected by the sample size, confidence level, and sample proportion.

References

  • [1] Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
  • [2] Larson, R. E., & Farber, B. A. (2018). Elementary statistics: Picturing the world. Cengage Learning.
  • [3] Agresti, A., & Franklin, C. (2018). Statistics: The art and science of learning from data. Pearson Education.