Katie Selects A Simple Random Sample Of 25 Students At Her Large School And Finds That 5 Of Them Are Planning To Try Out For The Soccer Team Next Year. She Wants To Construct A Confidence Interval For $p$, The Proportion Of All Students At Her

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Introduction

In statistics, a confidence interval is a range of values within which a population parameter is likely to lie. When dealing with proportions, it's essential to construct a confidence interval to estimate the population proportion. In this article, we'll explore how to construct a confidence interval for a population proportion using a simple random sample.

What is a Simple Random Sample?

A simple random sample is a subset of individuals selected from a larger population, where every individual has an equal chance of being selected. In Katie's case, she selects 25 students from her large school, and this is an example of a simple random sample.

What is a Confidence Interval?

A confidence interval is a range of values within which a population parameter is likely to lie. It's a way to express the uncertainty associated with a sample statistic. In this case, we want to estimate the population proportion, denoted by pp, which represents the proportion of all students at Katie's school who plan to try out for the soccer team next year.

The Formula for a Confidence Interval for a Population Proportion

The formula for a confidence interval for a population proportion is:

p^±zα/2p^(1−p^)n\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

where:

  • p^\hat{p} is the sample proportion
  • zα/2z_{\alpha/2} is the critical value from the standard normal distribution
  • α\alpha is the desired level of confidence
  • nn is the sample size

Calculating the Sample Proportion

To calculate the sample proportion, we need to divide the number of students planning to try out for the soccer team by the total sample size.

p^=525=0.2\hat{p} = \frac{5}{25} = 0.2

Choosing the Desired Level of Confidence

The desired level of confidence is typically expressed as a percentage. For example, a 95% confidence interval means that we're 95% confident that the true population proportion lies within the interval.

Choosing the Critical Value

The critical value zα/2z_{\alpha/2} is determined by the desired level of confidence. For a 95% confidence interval, the critical value is approximately 1.96.

Calculating the Margin of Error

The margin of error is the maximum amount by which the sample proportion may differ from the true population proportion. It's calculated as:

zα/2p^(1−p^)nz_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

Plugging in the values, we get:

1.960.2(1−0.2)25=0.1341.96 \sqrt{\frac{0.2(1-0.2)}{25}} = 0.134

Constructing the Confidence Interval

Now that we have the sample proportion, the critical value, and the margin of error, we can construct the confidence interval.

0.2±0.1340.2 \pm 0.134

This gives us a confidence interval of (0.066, 0.334).

Interpretation of the Confidence Interval

The confidence interval tells us that we're 95% confident that the true population proportion lies between 0.066 and 0.334. This means that we can be 95% confident that between 6.6% and 33.4% of all students at Katie's school plan to try out for the soccer team next year.

Conclusion

In this article, we've explored how to construct a confidence interval for a population proportion using a simple random sample. We've calculated the sample proportion, chosen the desired level of confidence, and calculated the margin of error. Finally, we've constructed the confidence interval and interpreted its meaning. By following these steps, you can construct a confidence interval for a population proportion and gain a better understanding of the uncertainty associated with a sample statistic.

References

  • Moore, D. S., & McCabe, G. P. (2012). Introduction to the practice of statistics. W.H. Freeman and Company.
  • Larson, R. J., & Farber, B. A. (2018). Elementary statistics: Picturing the world. Cengage Learning.

Additional Resources

  • Khan Academy: Confidence intervals for proportions
  • Stat Trek: Confidence intervals for proportions
  • Math Is Fun: Confidence intervals for proportions
    Frequently Asked Questions: Confidence Intervals for Population Proportions ====================================================================

Q: What is the purpose of a confidence interval for a population proportion?

A: The purpose of a confidence interval for a population proportion is to estimate the population proportion with a certain level of confidence. It provides a range of values within which the true population proportion is likely to lie.

Q: What is the difference between a confidence interval and a margin of error?

A: A confidence interval is a range of values within which the true population proportion is likely to lie, while the margin of error is the maximum amount by which the sample proportion may differ from the true population proportion.

Q: How do I choose the desired level of confidence?

A: The desired level of confidence is typically expressed as a percentage. For example, a 95% confidence interval means that we're 95% confident that the true population proportion lies within the interval.

Q: What is the critical value, and how do I determine it?

A: The critical value is determined by the desired level of confidence. For a 95% confidence interval, the critical value is approximately 1.96.

Q: How do I calculate the sample proportion?

A: To calculate the sample proportion, you need to divide the number of individuals in the sample who have the characteristic of interest by the total sample size.

Q: What is the formula for a confidence interval for a population proportion?

A: The formula for a confidence interval for a population proportion is:

p^±zα/2p^(1−p^)n\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

where:

  • p^\hat{p} is the sample proportion
  • zα/2z_{\alpha/2} is the critical value from the standard normal distribution
  • α\alpha is the desired level of confidence
  • nn is the sample size

Q: How do I interpret the confidence interval?

A: The confidence interval tells us that we're a certain percentage confident that the true population proportion lies within the interval. For example, a 95% confidence interval means that we're 95% confident that the true population proportion lies within the interval.

Q: What are some common mistakes to avoid when constructing a confidence interval for a population proportion?

A: Some common mistakes to avoid when constructing a confidence interval for a population proportion include:

  • Not choosing a desired level of confidence
  • Not calculating the sample proportion correctly
  • Not determining the critical value correctly
  • Not interpreting the confidence interval correctly

Q: What are some real-world applications of confidence intervals for population proportions?

A: Confidence intervals for population proportions have many real-world applications, including:

  • Estimating the proportion of a population with a certain characteristic
  • Comparing the proportions of two or more populations
  • Determining the effectiveness of a treatment or intervention
  • Making informed decisions based on data

Q: How do I calculate the margin of error?

A: The margin of error is calculated as:

zα/2p^(1−p^)nz_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

Q: What is the relationship between the sample size and the margin of error?

A: The sample size and the margin of error are inversely related. As the sample size increases, the margin of error decreases.

Q: How do I determine the sample size required for a confidence interval for a population proportion?

A: The sample size required for a confidence interval for a population proportion can be determined using the following formula:

n=(zα/2)2p^(1−p^)E2n = \frac{(z_{\alpha/2})^2 \hat{p}(1-\hat{p})}{E^2}

where:

  • nn is the sample size
  • zα/2z_{\alpha/2} is the critical value from the standard normal distribution
  • p^\hat{p} is the sample proportion
  • EE is the desired margin of error

Q: What are some common software packages used to calculate confidence intervals for population proportions?

A: Some common software packages used to calculate confidence intervals for population proportions include:

  • R
  • Python
  • Excel
  • SPSS
  • SAS

Q: How do I report the results of a confidence interval for a population proportion?

A: The results of a confidence interval for a population proportion should be reported in the following format:

"Based on a sample of [sample size], we estimate that [proportion] of the population has the characteristic of interest. The 95% confidence interval is [lower bound, upper bound]."

Q: What are some common pitfalls to avoid when reporting the results of a confidence interval for a population proportion?

A: Some common pitfalls to avoid when reporting the results of a confidence interval for a population proportion include:

  • Not reporting the sample size
  • Not reporting the desired level of confidence
  • Not reporting the margin of error
  • Not interpreting the results correctly

Q: How do I use a confidence interval for a population proportion to make informed decisions?

A: A confidence interval for a population proportion can be used to make informed decisions by:

  • Estimating the proportion of a population with a certain characteristic
  • Comparing the proportions of two or more populations
  • Determining the effectiveness of a treatment or intervention
  • Making informed decisions based on data