Question

Consider the following statement: ``At the 95% confidence level, we estimate that the true population proportion of United States teenagers that smoke cigarettes is at least 0.15 but no more than 0.24. If we were to construct a confidence interval at the 99% confidence level, the resulting interval would be narrower.’’ Are there any errors? Explain your answer.

Answer 1

The given statement is incorrect as a 99% confidence interval would be wider, not narrower, than a 95% confidence interval because higher confidence levels require wider intervals to capture the true population parameter with greater certainty.

Step-by-step explanation:

The statement contains an error regarding the comparison of confidence intervals at different confidence levels. If the confidence interval for the population proportion of United States teenagers that smoke cigarettes is estimated to be between 0.15 and 0.24 at a 95% confidence level, then increasing the confidence level to 99% would not result in a narrower interval, but a wider one. This is because higher confidence levels require wider intervals to ensure that they capture the true population parameter with greater certainty.

Examples of Confidence Interval Width Changes

• A 95% confidence interval is typically narrower than a 99% confidence interval.
• Reducing the confidence level from 99% to 90% results in a narrower confidence interval.
• Conversely, increasing the confidence level from 90% to 95% or to 99% makes the confidence interval wider.