Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.


Measured Results vs. Reported Results The same study cited in the preceding exercise produced these results after six months for the 198 patients given sustained care: 25.8% were no longer smoking, and these results were biochemically confirmed, but 40.9% of these patients reported that they were no longer smoking. Construct the two 95% confidence intervals. Compare the results. What do you conclude?

Verified step by step guidance

1. Step 1: Identify the problem's key components. We are tasked with constructing two 95% confidence intervals for the population proportion (p). The first confidence interval is based on the biochemically confirmed proportion (25.8%), and the second is based on the self-reported proportion (40.9%). The sample size is 198 patients.
2. Step 2: Recall the formula for a confidence interval for a population proportion. The formula is: CI = p̂ ± Z * sqrt((p̂(1 - p̂)) / n), where p̂ is the sample proportion, Z is the critical value for the desired confidence level (for 95%, Z ≈ 1.96), and n is the sample size.
3. Step 3: Calculate the first confidence interval for the biochemically confirmed proportion. Here, p̂ = 0.258 (25.8% as a decimal) and n = 198. Substitute these values into the formula: CI = 0.258 ± 1.96 * sqrt((0.258(1 - 0.258)) / 198). Simplify the expression to find the margin of error and the confidence interval bounds.
4. Step 4: Calculate the second confidence interval for the self-reported proportion. Here, p̂ = 0.409 (40.9% as a decimal) and n = 198. Substitute these values into the same formula: CI = 0.409 ± 1.96 * sqrt((0.409(1 - 0.409)) / 198). Again, simplify the expression to find the margin of error and the confidence interval bounds.
5. Step 5: Compare the two confidence intervals. Analyze whether the intervals overlap and what this implies about the difference between the biochemically confirmed and self-reported proportions. Conclude whether there is a significant discrepancy between the two proportions and discuss potential reasons for the difference, such as overreporting in self-reported data.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence, typically expressed as a percentage (e.g., 95%). It provides an estimate of uncertainty around a sample statistic, allowing researchers to infer about the population from which the sample was drawn.

Proportion

Proportion refers to the fraction of a whole, often expressed as a percentage, that represents a specific characteristic within a population. In this context, it relates to the percentage of patients who reported or were confirmed to have stopped smoking, which is crucial for understanding the effectiveness of the intervention and comparing self-reported versus biochemically confirmed results.

Biochemical Confirmation

Biochemical confirmation involves using biological tests to verify self-reported behaviors or conditions, such as smoking cessation. This method provides a more objective measure compared to self-reports, which can be biased. Understanding the difference between reported and confirmed results is essential for evaluating the reliability of the data and drawing valid conclusions.