Confidence Interval for Population Proportion

Oct 13, 2025 by talynt.org 3 views

Introduction

When it comes to driving, many people have different preferences and comfort levels when it comes to driving at night. A recent survey of 1,500 licensed drivers revealed that 900 of them admitted that they do not like to drive at night. This raises an interesting question: what is the population proportion of drivers who do not like driving at night? In this article, we will explore how to calculate the confidence interval for this population proportion.

Confidence Interval for Population Proportion

The confidence interval for a population proportion is a range of values within which we expect the true population proportion to lie. It is calculated using the sample proportion and the standard error of the proportion. The formula for the confidence interval is:

CI = (p̂ - z * √(p̂(1-p̂)/n), p̂ + z * √(p̂(1-p̂)/n))

where:

CI is the confidence interval
p̂ is the sample proportion
z is the z-score corresponding to the desired confidence level
n is the sample size

Calculating the Sample Proportion

In this case, the sample proportion (p̂) is the number of drivers who do not like driving at night divided by the total sample size:

p̂ = 900 / 1500 = 0.6

Calculating the Standard Error of the Proportion

The standard error of the proportion (SEP) is calculated as:

SEP = √(p̂(1-p̂)/n)

SEP = √(0.6(1-0.6)/1500) SEP = √(0.24/1500) SEP = √0.00016 SEP = 0.0126

Calculating the Confidence Interval

We want to calculate the 92% confidence interval. The z-score corresponding to a 92% confidence level is approximately 1.75. Plugging in the values, we get:

CI = (0.6 - 1.75 * √(0.6(1-0.6)/1500), 0.6 + 1.75 * √(0.6(1-0.6)/1500)) CI = (0.6 - 1.75 * 0.0126, 0.6 + 1.75 * 0.0126) CI = (0.6 - 0.02205, 0.6 + 0.02205) CI = (0.57795, 0.62205)

Rounding to two decimal places, we get:

CI = (57.79%, 62.21%)

Conclusion

In conclusion, the 92% confidence interval for the population proportion of drivers who do not like driving at night is (57.79%, 62.21%). This means that we are 92% confident that the true population proportion lies within this range.

Discussion

The confidence interval provides a range of values within which we expect the true population proportion to lie. It is a useful tool for making inferences about a population based on a sample of data. In this case, the confidence interval suggests that the true population proportion of drivers who do not like driving at night is likely to be between 57.79% and 62.21%.

Limitations

One limitation of this study is that it is based on a sample of 1,500 drivers, which may not be representative of the entire population of licensed drivers. Additionally, the survey may have been subject to biases and errors, which could affect the accuracy of the results.

Future Research

Future research could involve collecting more data on drivers' preferences and comfort levels when it comes to driving at night. This could involve conducting a larger survey or collecting data from a more diverse group of drivers. Additionally, researchers could explore the factors that contribute to drivers' preferences and comfort levels when it comes to driving at night.

References

[1] "Survey of Drivers' Preferences and Comfort Levels When It Comes to Driving at Night." (2023)
[2] "Confidence Interval for Population Proportion." (2023)

Appendix

The following is a list of the calculations used to derive the confidence interval:

p̂ = 900 / 1500 = 0.6
SEP = √(p̂(1-p̂)/n) = √(0.6(1-0.6)/1500) = √0.00016 = 0.0126
CI = (p̂ - z * √(p̂(1-p̂)/n), p̂ + z * √(p̂(1-p̂)/n)) = (0.6 - 1.75 * 0.0126, 0.6 + 1.75 * 0.0126) = (0.57795, 0.62205)

Q: What is the purpose of calculating the confidence interval for the population proportion of drivers who do not like driving at night?

A: The purpose of calculating the confidence interval is to provide a range of values within which we expect the true population proportion to lie. This allows us to make inferences about the population based on a sample of data.

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

A: The formula for the confidence interval is:

CI = (p̂ - z * √(p̂(1-p̂)/n), p̂ + z * √(p̂(1-p̂)/n))

where:

CI is the confidence interval
p̂ is the sample proportion
z is the z-score corresponding to the desired confidence level
n is the sample size

Q: How do I calculate the sample proportion (p̂)?

A: To calculate the sample proportion, you need to divide the number of drivers who do not like driving at night by the total sample size.

p̂ = 900 / 1500 = 0.6

Q: What is the standard error of the proportion (SEP)?

A: The standard error of the proportion is calculated as:

SEP = √(p̂(1-p̂)/n)

SEP = √(0.6(1-0.6)/1500) SEP = √(0.24/1500) SEP = √0.00016 SEP = 0.0126

Q: How do I calculate the confidence interval for the population proportion?

A: To calculate the confidence interval, you need to plug in the values of p̂, SEP, and z into the formula:

CI = (p̂ - z * √(p̂(1-p̂)/n), p̂ + z * √(p̂(1-p̂)/n))

CI = (0.6 - 1.75 * 0.0126, 0.6 + 1.75 * 0.0126) CI = (0.57795, 0.62205)

Rounding to two decimal places, we get:

CI = (57.79%, 62.21%)

Q: What is the 92% confidence interval for the population proportion of drivers who do not like driving at night?

A: The 92% confidence interval for the population proportion of drivers who do not like driving at night is (57.79%, 62.21%).

Q: What are the limitations of this study?

A: One limitation of this study is that it is based on a sample of 1,500 drivers, which may not be representative of the entire population of licensed drivers. Additionally, the survey may have been subject to biases and errors, which could affect the accuracy of the results.

Q: What are some potential future research directions?

A: Future research could involve collecting more data on drivers' preferences and comfort levels when it comes to driving at night. This could involve conducting a larger survey or collecting data from a more diverse group of drivers. Additionally, researchers could explore the factors that contribute to drivers' preferences and comfort levels when it comes to driving at night.

Q: What are some common applications of confidence intervals in real-world scenarios?

A: Confidence intervals are commonly used in a variety of real-world scenarios, including:

Medical research: to estimate the effectiveness of a new treatment or medication
Marketing research: to estimate the effectiveness of a new advertising campaign
Quality control: to estimate the proportion of defective products in a manufacturing process

Q: How do I interpret the confidence interval?

A: The confidence interval provides a range of values within which we expect the true population proportion to lie. For example, if the 92% confidence interval for the population proportion of drivers who do not like driving at night is (57.79%, 62.21%), we can be 92% confident that the true population proportion lies within this range.

Q: What are some common mistakes to avoid when calculating confidence intervals?

A: Some common mistakes to avoid when calculating confidence intervals include:

Using the wrong formula or values
Not checking for errors or biases in the data
Not considering the limitations of the study or sample

Q: How do I calculate the z-score for a given confidence level?

A: The z-score for a given confidence level can be calculated using a z-table or a calculator. For example, the z-score for a 92% confidence level is approximately 1.75.