The center of the interval

Oct 13, 2025 by wbrum.org 101 views

Understanding the $$95 %$$ Confidence Interval

A $$95 %$$ confidence interval for the proportion of teens who pass the driving test on the first attempt is 0.607 to 0.769. This interval suggests that we are $$95 %$$ confident that the true proportion of teens who pass the driving test on the first attempt lies between 60.7% and 76.9%. The confidence interval provides a range of values within which the true population proportion is likely to lie.

What Does the Confidence Interval Tell Us?

The confidence interval tells us that the true proportion of teens who pass the driving test on the first attempt is likely to be between 60.7% and 76.9%. This means that we can be $$95 %$$ confident that the true proportion is not less than 60.7% or greater than 76.9%. The interval also suggests that the true proportion is likely to be closer to the sample proportion, which is 0.67 (or 67%).

Interpreting the Confidence Interval

To interpret the confidence interval, we need to consider the following:

The width of the interval: The width of the interval is 0.162 (or 16.2%). This means that we are $$95 %$$ confident that the true proportion lies within 16.2% of the sample proportion.
The center of the interval: The center of the interval is 0.688 (or 68.8%). This means that we are $$95 %$$ confident that the true proportion lies close to 68.8%.
The lower and upper bounds: The lower bound of the interval is 0.607 (or 60.7%), and the upper bound is 0.769 (or 76.9%). This means that we are $$95 %$$ confident that the true proportion is not less than 60.7% or greater than 76.9%.

What Does the Confidence Interval Not Tell Us?

The confidence interval does not tell us the following:

The probability that a teen will pass the driving test on the first attempt: The confidence interval only provides a range of values within which the true proportion is likely to lie. It does not provide the probability that a teen will pass the driving test on the first attempt.
The probability that a teen will fail the driving test on the first attempt: The confidence interval only provides a range of values within which the true proportion is likely to lie. It does not provide the probability that a teen will fail the driving test on the first attempt.
The probability that a teen will pass the driving test on a subsequent attempt: The confidence interval only provides a range of values within which the true proportion is likely to lie. It does not provide the probability that a teen will pass the driving test on a subsequent attempt.

Conclusion

In conclusion, the $$95 %$$ confidence interval for the proportion of teens who pass the driving test on the first attempt is 0.607 to 0.769. This interval suggests that we are $$95 %$$ confident that the true proportion of teens who pass the driving test on the first attempt lies between 60.7% and 76.9%. The confidence interval provides a range of values within which the true population proportion is likely to lie.

Discussion

The confidence interval provides a range of values within which the true population proportion is likely to lie. This means that we can be $$95 %$$ confident that the true proportion is not less than 60.7% or greater than 76.9%. The interval also suggests that the true proportion is likely to be closer to the sample proportion, which is 0.67 (or 67%).

Limitations of the Confidence Interval

The confidence interval has the following limitations:

It does not provide the probability that a teen will pass the driving test on the first attempt: The confidence interval only provides a range of values within which the true proportion is likely to lie. It does not provide the probability that a teen will pass the driving test on the first attempt.
It does not provide the probability that a teen will fail the driving test on the first attempt: The confidence interval only provides a range of values within which the true proportion is likely to lie. It does not provide the probability that a teen will fail the driving test on the first attempt.
It does not provide the probability that a teen will pass the driving test on a subsequent attempt: The confidence interval only provides a range of values within which the true proportion is likely to lie. It does not provide the probability that a teen will pass the driving test on a subsequent attempt.

Implications of the Confidence Interval

The confidence interval has the following implications:

It suggests that the true proportion of teens who pass the driving test on the first attempt is likely to be between 60.7% and 76.9%: This means that we can be $$95 %$$ confident that the true proportion is not less than 60.7% or greater than 76.9%.
It suggests that the true proportion of teens who pass the driving test on the first attempt is likely to be closer to the sample proportion, which is 0.67 (or 67%): This means that we can be $$95 %$$ confident that the true proportion is likely to be closer to 67%.

Future Research Directions

Future research directions include:

Investigating the factors that affect the probability of passing the driving test on the first attempt: This could include investigating the effects of age, sex, and prior experience on the probability of passing the driving test on the first attempt.
Investigating the effects of different teaching methods on the probability of passing the driving test on the first attempt: This could include investigating the effects of different teaching methods, such as online tutorials or in-person instruction, on the probability of passing the driving test on the first attempt.

Conclusion

Understanding the $$95 %$$ Confidence Interval

Q&A

Q: What is a confidence interval?

A: A confidence interval is a range of values within which a population parameter is likely to lie. It is calculated from a sample of data and provides a measure of the uncertainty associated with the estimate.

Q: What does the $$95 %$$ confidence interval tell us?

A: The $$95 %$$ confidence interval tells us that we are $$95 %$$ confident that the true proportion of teens who pass the driving test on the first attempt lies between 60.7% and 76.9%.

Q: What is the width of the confidence interval?

A: The width of the confidence interval is 0.162 (or 16.2%). This means that we are $$95 %$$ confident that the true proportion lies within 16.2% of the sample proportion.

Q: What is the center of the confidence interval?

A: The center of the confidence interval is 0.688 (or 68.8%). This means that we are $$95 %$$ confident that the true proportion lies close to 68.8%.

Q: What does the confidence interval not tell us?

A: The confidence interval does not tell us the following:

The probability that a teen will pass the driving test on the first attempt: The confidence interval only provides a range of values within which the true proportion is likely to lie. It does not provide the probability that a teen will pass the driving test on the first attempt.
The probability that a teen will fail the driving test on the first attempt: The confidence interval only provides a range of values within which the true proportion is likely to lie. It does not provide the probability that a teen will fail the driving test on the first attempt.
The probability that a teen will pass the driving test on a subsequent attempt: The confidence interval only provides a range of values within which the true proportion is likely to lie. It does not provide the probability that a teen will pass the driving test on a subsequent attempt.

Q: What are the implications of the confidence interval?

A: The confidence interval has the following implications:

It suggests that the true proportion of teens who pass the driving test on the first attempt is likely to be between 60.7% and 76.9%: This means that we can be $$95 %$$ confident that the true proportion is not less than 60.7% or greater than 76.9%.
It suggests that the true proportion of teens who pass the driving test on the first attempt is likely to be closer to the sample proportion, which is 0.67 (or 67%): This means that we can be $$95 %$$ confident that the true proportion is likely to be closer to 67%.

Q: What are the limitations of the confidence interval?

A: The confidence interval has the following limitations:

It does not provide the probability that a teen will pass the driving test on the first attempt: The confidence interval only provides a range of values within which the true proportion is likely to lie. It does not provide the probability that a teen will pass the driving test on the first attempt.
It does not provide the probability that a teen will fail the driving test on the first attempt: The confidence interval only provides a range of values within which the true proportion is likely to lie. It does not provide the probability that a teen will fail the driving test on the first attempt.
It does not provide the probability that a teen will pass the driving test on a subsequent attempt: The confidence interval only provides a range of values within which the true proportion is likely to lie. It does not provide the probability that a teen will pass the driving test on a subsequent attempt.

Q: What are the future research directions?

A: Future research directions include:

Investigating the factors that affect the probability of passing the driving test on the first attempt: This could include investigating the effects of age, sex, and prior experience on the probability of passing the driving test on the first attempt.
Investigating the effects of different teaching methods on the probability of passing the driving test on the first attempt: This could include investigating the effects of different teaching methods, such as online tutorials or in-person instruction, on the probability of passing the driving test on the first attempt.