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.


OxyContin The drug OxyContin (oxycodone) is used to treat pain, but it is dangerous because it is addictive and can be lethal. In clinical trials, 227 subjects were treated with OxyContin and 52 of them developed nausea (based on data from Purdue Pharma L.P.).


b. Compare the result from part (a) to this 95% confidence interval for 5 subjects who developed nausea among the 45 subjects given a placebo instead of OxyContin: . What do you conclude?

Verified step by step guidance

1. Step 1: Identify the problem type. This problem involves constructing a confidence interval for the proportion (p) of subjects who developed nausea after being treated with OxyContin. Additionally, it requires comparing this confidence interval to another provided interval for subjects given a placebo.
2. Step 2: Calculate the sample proportion (p̂) for the OxyContin group. Use the formula p̂ = x / n, where x is the number of subjects who developed nausea (52) and n is the total number of subjects treated with OxyContin (227).
3. Step 3: Determine the standard error (SE) for the proportion. Use the formula SE = sqrt((p̂ * (1 - p̂)) / n), where p̂ is the sample proportion and n is the sample size.
4. Step 4: Construct the confidence interval for the proportion using the formula: CI = p̂ ± Z * SE, where Z is the critical value corresponding to the 95% confidence level (typically Z = 1.96 for 95%). Substitute the values of p̂ and SE calculated in the previous steps.
5. Step 5: Compare the confidence interval for the OxyContin group to the provided confidence interval for the placebo group (.). Analyze whether the intervals overlap or are distinct, and interpret what this means in terms of the likelihood of nausea being associated with OxyContin versus the placebo.

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. For example, a 95% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 95% of those intervals would contain the true parameter. This concept is crucial for understanding the reliability of estimates in statistics.

Proportion

Proportion refers to the fraction of the total that exhibits a certain characteristic, often expressed as a percentage. In the context of the question, it relates to the number of subjects who developed nausea after taking OxyContin compared to the total number of subjects treated. Understanding proportions is essential for comparing outcomes between different groups, such as those receiving OxyContin versus a placebo.

Statistical Comparison

Statistical comparison involves evaluating two or more groups to determine if there are significant differences between them. In this case, it requires comparing the proportion of subjects who developed nausea from OxyContin to those who received a placebo. This concept is vital for drawing conclusions about the effectiveness or safety of treatments based on empirical data.